Hoops Package Creation

R.H.Beresford 21/02/2007.

Quit[] (* prevents shadowing *)

Off[General::spell, General::spell1, Syntax::com]
tf = TraditionalForm;

1. Version, Copyright notice.

Version 1c. 21/02/07.
© Copyright 2006,2007 Roger Herbert Beresford, 30, Brook Rd., Woodhouse Eaves, Loughborough, Leics, LE12 8RS, England. 
Considerable effort has gone into checking this work, but some errors probably persist. No responsibility will be accepted for any problems arising from its use. Permission is granted to distribute this file for any purpose except for inclusion in commercial software or program collections. This copyright notice must remain intact.
Please send any comments to me at rhberesford@btinternet.com

2. Revisions.

Version 1 20/04/06.
Version 1a. 07/05/06. Minor edits. Introduced BoldFace by "//bf".
Version 1b. 23/09/06. Minor edits. Exra hoops added, from latest GroupLoopHoop.m.
Version 1c. 21/02/07. Minor edits.  Extended "as[A_,B_]"; hoopNames global.

3. Keywords.

Algebras, Clifford algebras, conservative loops & algebras, Frobenius-conservation, generalized signs, Hoops, Hoop shapes & Hoop sizes, Moufang Loops, plex-conjugate, partial-division-by-zero, remainders,  roots of unity, symmetries, zero-vectors.

4. Warnings.

This package is incompatible with GroupLoopHoop.m in MathSource 4894.

5. Limitations.

Many hoops lack polar forms.

6. Note on Implementation.

Data on each hoop is stored as Hoop["H"]; Use[H] puts this data into the set of variables used by the procedures hoopTimes, hoopInverse, hoopPower that implement the Hoop algebra for the "active" hoop. hoopNames lists all the names.
	For each hoop H, the first two elements of Hoop["H"] are mm & nn, i.e. the length of the hoop mm and its position nn in the GAP Atlas, as extended in the GroupLoopHoop.m package to include hoops and some other small loops. (This is adapted from ([GAP99] The Gap Group. GAP --- Groups, Algorithms,  and Programming, Version 4.2; Aachen, St Andrews, 1999. http://www-gap. dcs.st-and. ac.uk/~gap).
	Next is the index table "protoloop", i.e. the Cayley table isomorph used to define the hoop. This is a list of lists giving the product index, i.j=k in the i'th row and j'th column. Tables have {1,2...,mm} in the first row and column, and every index occurs (possibly with a sign) once in every row and column. Hoop tables have the Moufang division property x(yz) = (xy)z (for all elements x,y,z in the table, allowing for signs). By definition, they have the Frobenius conservation property Det[A] Det[B] = Det[AB], up to a sign, for the multiplication of vectors A & B. (Five tables without the conservation property are appended to provide counter-examples. Their names end in n)
	Next comes the factorized determinant for a Cartesian form {a,b,..m}, followed, where known, by the closely related Polar form which includes angles that take up the degrees of freedom released by quadratic sizes. (A null entry means that the polar form has not been developed. More may be available in later versions of MathSource/4894).
	If it is known, the Polar-Cartesian reversion will come next. This gives the Cartesian form for a polar form {a,b,..m}.
	Two lists, gi, gp, follow. These summarize information about the hoop, needed by the hoopInverse, topol and tovec routines. Finally, a Plex involution may be supplied. This is the Complex Conjugate for hoops C4c (complex), Qr (quaternion) and Oct (octonion) and is an equivalent of the Complex Conjugate for some other algebras.

7. Set up the Hoops.m package, usage.

7.1 Begin, Hoops Usage.

BeginPackage["Hoops`"];

Hoops::usage = "Hoops.m is a package that creates a database of over 80 hoops (symmetry-conserving partial-fraction division algebras) or counter-examples as index-Tables (Cayley tables with signed indices as elements). It provides Hoop procedures for multiplication & division, powers & roots, & some polar-dual formulations, of generalized vectors.";

7.2 Alphabetic list of Usage for functions and procedures.

a::usage = "Symbol in HoopTbl etc."; b::usage = ""; 
    c::usage = ""; d::usage = ""; e::usage = ""; 
    f::usage = ""; g::usage = ""; h::usage = ""; 
    i::usage = ""; j::usage = ""; k::usage = ""; 
    l::usage = ""; m::usage = ""; o::usage = ""; 
    p::usage = ""; q::usage = ""; r::usage = ""; 
    s::usage = ""; t::usage = ""; u::usage = ""; 
    v::usage = ""; w::usage = ""; x::usage = ""; 
    y::usage = "";

alph::usage="A list of 24 symbols, {a,b,...,x} used as symbolic elements of Cayley Tables.";

arcTanh::usage = "Quadrant sensitive Hyperbolc Tangent";

as::usage="as[w_] sets up a substitution table to replace alph symbols by elements of the vector w, to convert symbolic functions to specific cases.
 Extended so that /.as[v_,w_] replaces elements of v by elements of w.";

bf::usage = "BoldFormat. xx//bf will print in bold";

hoopInverse::usage="hoopInverse[A_] finds the multiplicative left inverse Ainv of A. hoopTimes[Ainv,B] divides B by A and creates remainders Rl & Rr to conserve any sizes that are lost from A or B";

\(hoopPower::usage = \*""hoopPower[A_, r_:1/2] calculates (A^r) if the active loop has a polar form."";\)

hoopTimes::usage  =  "hoopTimes[A,B] uses the active loop \
to multiply the vectors 'A' & 'B' to give a product 'P'. Remainders Rl & Rr \
are created, so that A=P/B+Rl, B=P/A+Rr, maintaining size conservation.";

gi::usage = "Indices used by hoopTimes.";

gp::usage = "Powers used by hoopTimes.";

hoopNames::usage = "A list of strings naming each loop.";

hoopTbl::usage="Set by use as the indexTable of the active hoop.";

genCos::usage = "Multiphase Sinusoid";

genSin::usage = "Multiphase Sinusoid";

hmin::usage = "hmin is arbitrarily set at 4$MachineEpsilon as the \
criterion for 'hoopInverse' to ignore (renormalize) partial fractions with \
near-zero denominators.";

J::usage = "J^3=1. ";

JJ::usage = "J J with J^3=1.";

js::usage = "re-write rule to make J & JJ act as signs with J^3=1. \
";

mm::usage = "The length of the active hoop";

nn::usage = "The position of the active hoop in loop[[mm]] of \
groupLoopHoop.nb";

plex::usage = "The Hoop-plex conjugate; duplicates the complex \
conjugates where they exist.";

r3::usage = "Sqrt[3.]";

r6::usage = "2 Sqrt[3.]";

Rl::usage = "Left remainder";

Rr::usage = "Right remainder";

sh::usage = "Symbolic shape (list of conserved sizes) for the \
active hoop";

topol::usage = "Symbolic polar form for the active hoop";

tovec::usage = "Vector form for the Symbolic polar form of the \
active hoop";

Use::usage = "After Use[hoopname_] other instructions employ this \
hoop as the active hoop.";

8. Start Private definitions.  Create Database.

8.1 Begin Private.

Begin["`Private`"];

Unprotect[NumberQ, Positive, Abs, Sign, Re, Im];

8.2 Data File, alph, hoopNames, Hoop[name].

alph =  (* 
          Dummy variables used in all symbolic tables and functions*) {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, 
          x};

Data is supplied for the following Hoops:-

\(\(hoopNames = {"C2", "C4c", "C3", "C3C2c", "C4C3c", "C9j", "C3i", "C12c", "K", "C4", "Qr", "Dav", "P4", "P16c", "O2", "CL2", "C4C2r", "C8c", "C5", "D3", "C3C2", "Q12c", "Q12r", "D6r", "KC3c", "C3Kr", "C3C4c", "D3C2c", "Clm3r", "Cl3r", "g2401c", "C8", "C4C2", "D4", "Q8", "KC2", "C4C4c", "KiC4c", "C4iC4c", "QC2r", "Q8C2r", "O4", "P8", "C2D4r", "CL3", "Octr", "KiC4r", "CL21", "CL03", "Octi", "C3C3", "Q12", "C3C4", "A4", "D3C2", "C3K", "C3Qr", "C3P4", "C3KC2r", "C6Kr", "Q12C2c", "C4C4", "KiC4", "C4iC4", "D8", "Q16", "C4K", "D4C2", "Q8C2", "P16", "KK", "Oct", "r31r", "CL4", "CL31", "CL22", "CL13", "CL04", "O8", "Q8M2", "C3C4C2", "Q4n", "C6n", "Q8n", "Alt8n", "D3Mn"};\)\)

Many more hoops are listed in [3], but not included here because they do not have known polar forms or are too closely related to those that are included to be of interest.

Nomenclature.
Cn is a cyclic group; CnCm (etc.) are direct compositions; CniCm are indirect compositions; Dn is the dihedral group with 2n elements; Qn is the generalazed quaternion group with n elements; Dav, O4 & O8 are tables with 2, 4 & 8 complex planes; Pn are tables created from Pauli-σ matrices; CLm are Clifford algebras. Names ending in r, j, c are signed tables obtained from hoops by 2, 3, or 4-fold collapse. D3Mn and Q8M2 are nonassociative Moufang loops; the first is not conservative; the second is Chein-doubled Q8 and folds to split-octonion algebra Octi.

Hoop["C2"] = {2, 
          1, {{1, 2}, {2, 1}}, {a - b, a + b}, {a - b, 
            a + b}, {(a + b)/2, (-a + b)/2}, {1, 2, 1, 1}, {1, 
            1}, {a, -b}};

Hoop["C4c"] = {2, 
          2, {{1, 2}, {2, -1}}, {-(a^2) - (b^2)}, {(a^2) + (b^2), 
            ArcTan[a, b]}, {Sqrt[a] Cos[b], Sqrt[a] Sin[b]}, {1, -2, 
            1}, {1, 0. }, {a, -b}};

Hoop["C3"]={3,1,{{1,2,3},{2,3,1},{3,1,2}},{a+b+c,(1/2)*((a-b)^2+(b-c)^2+(-a+c)^2)},{a+b+c,(1/2)*((a-b)^2+(b-c)^2+(-a+c)^2),ArcTan[2*a-b-c,(-Sqrt[3])*(b-c)]},{(1/3)*(a+2*Sqrt[b]*Cos[c]),(1/3)*(a+2*Sqrt[b]*Cos[c+(2*Pi)/3]),(1/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3])},{1,3,2,1,1},{1,1,0.},{a,c,b}};

Hoop["C3C2c"] = {3, 
          2, {{1, 2, 3}, {2, 3, -1}, {3, -1, -2}}, {a - b + 
              c, (1/2)*((a + b)^2 + (b + c)^2 + (a - 
                        c)^2)}, {a - b + 
              c, (1/2)*((a + b)^2 + (b + c)^2 + (a - 
                        c)^2), 
            ArcTan[2*a + b - 
                c, (-Sqrt[3])*(b + c)]}, {(1/3)*(a + 
                  2*Sqrt[b]*Cos[c]), (1/3)*(-a - 
                  2*Sqrt[b]*Cos[c - 2*Pi/3]), (1/3)*(a + 
                  2*Sqrt[b]*Cos[c - 2*Pi/3])}, {1, -3, -2, 1, 
            1}, {1, 1, 0. }, {a, -c, -b}};

Hoop["C4C3c"]={3,4,{{1,2,3},{2,3*I,-1},{3,-1,2*I}},{a+I*b+I*c,((a-I*b)^2-(b-c)^2+(a-I*c)^2)/2},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(a+c)^2)/2,ArcTan[2*a+I*b+c,Sqrt[3]*((-I)*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3]),(-a-2*Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,-3,-2,1,1},{1,1,0.},Null};

Hoop["C3C3j"]={3,5,{{1,2,3},{2,3*J^2,J},{3,J,2*^2}},{(a+b J^2+c J^2),((a-b J^2)^2+(b J^2-c J^2)^2+(c J^2-a)^2)/2},,,{1,3J,2J,1},,};

Hoop["C9j"]={3,6,{{1,2,3},{2,3*J^2,1},{3,1,2*J}},{a^3-3*a*b*c+c^3*J+b^3*J^2},,,{1,3,2,1},,};

Hoop["C3i"]={3,9,{{1,2,3},{2,3,I},{3,I,2*I}},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(a+c)^2)/2},{a-I*b-c,((a+I*b)^2+(I*b-c)^2+(a+c)^2)/2,ArcTan[2*a+I*b+c,Sqrt[3]*((-I)*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3]),(-a-2*Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,-3*I,-2*I,1,1},{1,1,0.},,};

Hoop["C12c"]={3,8,{{1,2,3},{2,3,-I},{3,-I,-2*I}},{a+I*b-c,((a-I*b)^2+(a+c)^2+(I*b+c)^2)/2},{a+I*b-c,((a-I*b)^2+(a+c)^2+(I*b+c)^2)/2,ArcTan[2*a-I*b+c,Sqrt[3]*(I*b+c)]},{(a+2*Sqrt[b]*Cos[c])/3,(-I/3)*(a+2*Sqrt[b]*Cos[c-(2*Pi)/3]),(-a-2*Sqrt[b]*Cos[c+(2*Pi)/3])/3},{1,3*I,2*I,1,1},{1,1,0.},};

Hoop["K"]={4,1,{{1,2,3,4},{2,1,4,3},{3,4,1,2},{4,3,2,1}},{a+b-c-d,a-b+c-d,a-b-c+d,a+b+c+d},{a+b-c-d,a-b+c-d,a-b-c+d,a+b+c+d},{(a+b+c+d)/4,(a-b-c+d)/4,(-a+b-c+d)/4,(-a-b+c+d)/4},{1,2,3,4,1,1,1,1},{1,1,1,1},{a,-b,-c,d}};

Hoop["C4"]={4,2,{{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}},{a+b+c+d,a-b+c-d,(a-c)^2+(b-d)^2},{a+b+c+d,a-b+c-d,(a-c)^2+(b-d)^2,ArcTan[a-c,b-d]},{(a+b+2*Sqrt[c]*Cos[d])/4,(a-b+2*Sqrt[c]*Sin[d])/4,(a+b-2*Sqrt[c]*Cos[d])/4,(a-b-2*Sqrt[c]*Sin[d])/4},{1,4,3,2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};

Hoop["Qr"]={4,3,{{1,2,3,4},{2,-1,4,-3},{3,-4,-1,2},{4,3,-2,-1}},{a^2+b^2+c^2+d^2},
{a^2+b^2+c^2+d^2,ArcTan[a,b],c^2+d^2,ArcTan[c,d]},{Sqrt[a-c]*Cos[b],Sqrt[a-c]*Sin[b],Sqrt[c]*Cos[d],Sqrt[c]*Sin[d]},{1,-2,-3,-4,2},{1,0.,1,0.},{a,-b,-c,-d}};

Hoop["Dav"]={4,6,{{1,2,3,4},{2,-1,4,-3},{3,4,-1,-2},{4,-3,-2,1}},{(b+c)^2+(a-d)^2,(b-c)^2+(a+d)^2},{(b+c)^2+(a-d)^2,ArcTan[a-d,b+c],(b-c)^2+(a+d)^2,ArcTan[a+d,b-c]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d])/2,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d])/2,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d])/2,(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d])/2},{1,-2,-3,4,1,1},{1,0.,1,0.},{a,-b,-c,d}};

Hoop["P4"]={4,8,{{1,2,3,4},{2,1,-4*I,3*I},{3,4*I,1,-2*I},{4,-3*I,2*I,1}},{a^2-b^2-c^2-d^2},{Sqrt[(b-c)^2+(c-d)^2+(-b+d)^2]/Sqrt[2],ArcTan[(Sqrt[2]*a)/Sqrt[(b-c)^2+(c-d)^2+(-b+d)^2]],(b+c+d)/3,If[c==d,0,ArcTan[2*b-c-d,(-Sqrt[3])*(c-d)]]},{a*Tan[b],c+(2*a*Cos[d])/3,c+(2*a*Cos[d+(2*Pi)/3])/3,c+(2*a*Cos[d+(4*Pi)/3])/3},{1,2,3,4,2},{1,0.,1,0.},{a,-b,-c,-d}};

Hoop["P16c"]={4,13,{{1,2,3,4},{2,-1,-4*I,-3*I},{3,4*I,-1,2*I},{4,3*I,-2*I,1}},{a^2+b^2+c^2-d^2},{a^2+b^2,If[b!=0,ArcTan[a,b],0],c^2-d^2,If[d!=0,arcTanh[c,d],0]},{Sqrt[a]*Cos[b],Sqrt[a]*Sin[b],Sqrt[c]*Cosh[d],Sqrt[c]*Sinh[d]},{1,-2,-3,4,2},{1,0.,1,0.},{a,-b,-c,-d}};

Hoop["O2"]={4,5,{{1,2,3,4},{2,-1,4,-3},{3,4,1,2},{4,-3,2,-1}},{(a-c)^2+(b-d)^2,(a+c)^2+(b+d)^2},{(a-c)^2+(b-d)^2,ArcTan[a-c,b-d],(a+c)^2+(b+d)^2,ArcTan[a+c,b+d]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d])/2,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d])/2,(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d])/2,(-(Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d])/2},{1,-2,3,-4,1,1},{1,0.,1,0.},{a,-b,-c,-d}};

Hoop["CL2"]={4,7,{{1,2,3,4},{2,1,4,3},{3,-4,-1,2},{4,-3,-2,1}},{a^2-b^2+c^2-d^2},{a^2-b^2+c^2-d^2,ArcTan[a,c],b^2+d^2,ArcTan[b,d]},{Sqrt[a+c]*Cos[b],Sqrt[c]*Cos[d],Sqrt[a+c]*Sin[b],Sqrt[c]*Sin[d]},{1,2,-3,4,2},{1,0.,1,0.},{a,-b,-c,-d}};

Hoop["C4C2r"]={4,9,{{1,2,3,4},{2,3,-4,-1},{3,-4,1,-2},{4,-1,-2,3}},{a+b+c-d,a-b+c+d,(a-c)^2+(b+d)^2},{a+b+c-d,a-b+c+d,(a-c)^2+(b+d)^2,ArcTan[a-c,b+d]},{(a+b+2*Sqrt[c]*Cos[d])/4,(a-b+2*Sqrt[c]*Sin[d])/4,(a+b-2*Sqrt[c]*Cos[d])/4,(-a+b+2*Sqrt[c]*Sin[d])/4},{1,-4,3,-2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};

Hoop["C3C4j"]={4,10,
{{1,2,3,4},{2,-3,4,-1},{3,4,1,2},{4,-1,2,-3}},{a+b-c-d,a-b-c+d,(a+c)^2+(b+d)^2},{a+b-c-d,a-b-c+d,(a+c)^2+(b+d)^2,ArcTan[a+c,b+d]},{(a+b+2*Sqrt[c]*Cos[d])/4,(a-b+2*Sqrt[c]*Sin[d])/4,(-a-b+2*Sqrt[c]*Cos[d])/4,(-a+b+2*Sqrt[c]*Sin[d])/4},{1,-4,3,-2,1,1,1},{1,1,1,0.},{a,-b,c,-d}};

Hoop["C8c"]={4,11,{{1,2,3,4},{2,3,4,-1},{3,4,-1,-2},{4,-1,-2,-3}},{a^2+b^2+c^2+d^2+Sqrt[2]*(a*b+b*c-a*d+c*d),a^2+b^2+c^2+d^2-Sqrt[2]*(a*b+b*c-a*d+c*d)},,,{1,-4,-3,-2,1,1},{11,1,0.},{a,-b,c,-d}};

Hoop["C5"]={5,1,{{1,2,3,4,5},{2,3,4,5,1},{3,4,5,1,2},{4,5,1,2,3},{5,1,2,3,4}},{a+b+c+d+e,(-(5/4))*(a*b-a*c+b*c-a*d-b*d+c*d+a*e-b*e-c*e+d*e)^2+(1/16)*((a-b)^2+(a-c)^2+(b-c)^2+(b-d)^2+(c-d)^2+(-a+d)^2+(c-e)^2+(d-e)^2+(-a+e)^2+(-b+e)^2)^2},,{a+b*Sin[c],a+b*Sin[c+2π/5],a+b*Sin[c+4π/5],a+b*
Sin[c+6π/5],a+b*Sin[c+8π/5]}/5 ,{1,5,4,3,2,1,1},,};

Hoop["D3"]={6,1,{{1,2,3,4,5,6},{2,1,6,5,4,3},{3,4,5,6,1,2},{4,3,2,1,6,5},{5,6,1,2,3,4},{6,5,4,3,2,1}},{a+b+c+d+e+f,a-b+c-d+e-f,((a-c)^2-(b-d)^2+(c-e)^2+(-a+e)^2-(d-f)^2-(-b+f)^2)/2},{a+b+c+d+e+f,a-b+c-d+e-f,((a-c)^2+(c-e)^2+(-a+e)^2-(b-d)^2-(d-f)^2-(-b+f)^2)/2,ArcTan[2*a-c-e,Sqrt[3]*(-c+e)],((a-c)^2+(c-e)^2+(-a+e)^2)/2,ArcTan[2*b-d-f,Sqrt[3]*(-d+f)]},{(a+b+4*Sqrt[e]*Cos[d])/6,(a-b+4*Sqrt[-c+e]*Cos[f])/6,(a+b+4*Sqrt[e]*Cos[d+(2*Pi)/3])/6,(a-b+4*Sqrt[-c+e]*Cos[f+(2*Pi)/3])/6,(a+b+4*Sqrt[e]*Cos[d-(2*Pi)/3])/6,(a-b+4*Sqrt[-c+e]*Cos[f-(2*Pi)/3])/6},{1,2,5,4,3,6,1,1,2},{1,1,1,0.,1,0.},,{(a+c+e)^2-(b+d+f)^2,arcTanh[a+c+e,b+d+e],((a-c)^2+(c-e)^2+(-a+e)^2)/2,ArcTan[2*a-c-e,Sqrt[3]*(e-c)],((b-d)^2+(d-f)^2+(-b+f)^2)/2,ArcTan[2*a-c-e,Sqrt[3]*(c-e)]+ArcTan[2*b-d-f,Sqrt[3]*(f-d)]},{2*Sqrt[c]*Cos[d]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f]+Sqrt[a]*Sinh[b],2*Sqrt[c]*Cos[d+2*Pi/3]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f+2*Pi/3]+Sqrt[a]*Sinh[b],2*Sqrt[c]*Cos[d-2*Pi/3]+Sqrt[a]*Cosh[b],2*Sqrt[e]*Cos[d+f-2*Pi/3]+Sqrt[a]*Sinh[b]}/3};

Hoop["C3C2"]={6,2,{{1,2,3,4,5,6},{2,3,1,5,6,4},{3,1,2,6,4,5},{4,5,6,1,2,3},{5,6,4,2,3,1},{6,4,5,3,1,2}},{a+b+c+d+e+f,a+b+c-d-e-f,((a-b+d-e)^2+(a-c+d-f)^2+(b-c+e-f)^2)/2,((a-b-d+e)^2+(a-c-d+f)^2+(b-c-e+f)^2)/2},{a+b+c+d+e+f,a+b+c-d-e-f,((a-b+d-e)^2+(b-c+e-f)^2+(-a+c-d+f)^2)/2,ArcTan[2*a-b-c+2*d-e-f,Sqrt[3]*(-b+c-e+f)],((a-b-d+e)^2+(-a+c+d-f)^2+(b-c-e+f)^2)/2,ArcTan[2*a-b-c-2*d+e+f,Sqrt[3]*(-b+c+e-f)]},{(a+b+2*Sqrt[c]*Cos[d]+2*Sqrt[e]*Cos[f])/6,(a+b+2*Sqrt[c]*Cos[d+(2*Pi)/3]+2*Sqrt[e]*Cos[f+(2*Pi)/3])/6,(a+b+2*Sqrt[c]*Cos[d-(2*Pi)/3]+2*Sqrt[e]*Cos[f-(2*Pi)/3])/6,(a-b+2*Sqrt[c]*Cos[d]-2*Sqrt[e]*Cos[f])/6,(a-b+2*Sqrt[c]*Cos[d+(2*Pi)/3]-2*Sqrt[e]*Cos[f+(2*Pi)/3])/6,(a-b+2*Sqrt[c]*Cos[d-(2*Pi)/3]-2*Sqrt[e]*Cos[f-(2*Pi)/3])/6},{1,3,2,4,6,5,1,1,1,1},{1,1,1,0.,1,0.},{a,c,b,d,f,e}};

Hoop["Q12c"]={6,3,{{1,2,3,4,5,6},{2,3,1,6,4,5},{3,1,2,5,6,4},{4,5,6,-1,-2,-3},
{5,6,4,-3,-1,-2},{6,4,5,-2,-3,-1}},{(a+b+c)^2+(d+e+f)^2,
((a-b)^2+(a-c)^2+(b-c)^2+(d-e)^2+(d-f)^2+(e-f)^2)/2},
{(a+b+c)^2+(d+e+f)^2,ArcTan[a+b+c,d+e+f],
((a-b)^2+(b-c)^2+(-a+c)^2)/2+((d-e)^2+(e-f)^2+(-d+f)^2)/2,
If[c==b,0,ArcTan[2*a-b-c,Sqrt[3]*(c-b)]],((d-e)^2+(e-f)^2+(-d+f)^2)/2,
If[e==f,0,ArcTan[2*d-e-f,Sqrt[3]*(f-e)]]},
{(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d])/3,(Sqrt[a]*Cos[b])/3+
(2*Sqrt[c-e]*Cos[d+(2*Pi)/3])/3,(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d-(2*Pi)/3])/3,
(2*Sqrt[e]*Cos[f])/3+(Sqrt[a]*Sin[b])/3,(2*Sqrt[e]*Cos[f+(2*Pi)/3])/3+(Sqrt[a]*Sin[b])/3,
(2*Sqrt[e]*Cos[f-(2*Pi)/3])/3+(Sqrt[a]*Sin[b])/3},{1,3,2,-4,-5,-6,1,2},
{1,2,1,0.,1,0.},{a,c,b,-d,-e,-f}};

Hoop["Q12r"]={6,4,{{1,2,3,4,5,6},{2,-1,4,-3,6,-5},{3,-6,5,2,-1,4},{4,5,6,-1,-2,-3},{5,-4,-1,-6,-3,2},{6,3,-2,5,-4,-1}},{(a-c+e)^2+(b-d+f)^2,((a+c)^2+(b+d)^2+(a-e)^2+(c+e)^2+(b-f)^2+(d+f)^2)/2},{(a-c+e)^2+(b-d+f)^2,If[b+f==d,0,ArcTan[a-c+e,b-d+f]],((a+c)^2+(a-e)^2+(c+e)^2)/2+((b+d)^2+(b-f)^2+(d+f)^2)/2,If[c==-e,0,ArcTan[2*a+c-e,Sqrt[3]*(c+e)]],((b+d)^2+(b-f)^2+(d+f)^2)/2,If[d==-f,0,ArcTan[2*b+d-f,Sqrt[3]*(d+f)]]},{(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d])/3,(2*Sqrt[e]*Cos[f])/3+(Sqrt[a]*Sin[b])/3,-(Sqrt[a]*Cos[b])/3-(2*Sqrt[c-e]*Cos[d+(2*Pi)/3])/3,(-2*Sqrt[e]*Cos[f+(2*Pi)/3])/3-(Sqrt[a]*Sin[b])/3,(Sqrt[a]*Cos[b])/3+(2*Sqrt[c-e]*Cos[d-(2*Pi)/3])/3,(2*Sqrt[e]*Cos[f-(2*Pi)/3])/3+(Sqrt[a]*Sin[b])/3},{1,-2,-5,-4,-3,-6,1,2},{1,2,1,0.,1,0.},};

Hoop["D6r"]={6,5,{{1,2,3,4,5,6},{2,1,4,3,6,5},{3,-6,5,2, -1,4},{4,-5,6,1,-2,3},{5,-4,-1,-6,-3,2},{6,-3,-2,-5,-4,1}},{a-b-c+d+e-f,a+b-c-d+e+f,(1/2)*((a+c)^2-(b+d)^2+(a-e)^2+(c+e)^2-(b-f)^2-(d+f)^2)},,,{1,2,-5,4,-3,6,1,1,2},{1, 1,1,0.,1,0.},};

Hoop["KC3c"]={6,6,{{1,2,3,4,5,6},{2,3,-1,5,6,-4},{3,-1,-2,6,-4,-5},{4,5,6,1,2,3},{5,6,-4,2,3,-1},{6,-4,-5,3,-1,-2}},{a-b+c-d+e-f,a-b+c+d-e+f,((a+b-d-e)^2+(b+c-e-f)^2+(a-c-d+f)^2)/2,((a+b+d+e)^2+(a-c+d-f)^2+(b+c+e+f)^2)/2},,,{1,-3,-2,4,-6,-5,1,1,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};

Hoop["C3Kr"]={6,7,{{1,2,3,4,5,6},{2,-3,1,5,-6,4},{3,1,-2,6,4,-5},{4,5,6,1,2,3},
{5,-6,4,2,-3,1},{6,4,-5,3,1,-2}},{a-b-c-d+e+f,a-b-c+d-e-f,
((a+b-d-e)^2+(a+c-d-f)^2+(b-c-e+f)^2)/2,
((a+b+d+e)^2+(b-c+e-f)^2+(a+c+d+f)^2)/2},,,{1,3,2,4,6,5,1,1,1,1},{1,1,1,0.,1,0.},
{a,c,b,-d,-e,-f}};

Hoop["C3C4c"]:=(*10/1/6*){6,8,
{{1,2,3,4,5,6},{2,3,1,5,6,4},{3,1,2,6,4,5},{4,5,6,-1,-2,-3},{5,6,4,-2,-3,-1},{6,4,5,-3,-1,-2}},
{(a+b+c)^2+(d+e+f)^2,        3*((a-b+(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]-d+e)^2)/4,        3*((a-b-(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]+d-e)^2)/            4(*,((a-b)^2+(b-c)^2+(c-a)^2+(d-e)^2+(e-f)^2+(f-d)^2)^2/
4-3*(c*(d-e)+a*(e-f)+b*(f-d))^2*)},
{(a+b+c)^2+(d+e+f)^2,
If[d+e+f==0,If[a+b+c<0,Pi,0],ArcTan[a+b+c,d+e+f]],        3*((a-b+(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]-d+e)^2)/4,
If[(a+b-2*c)/Sqrt[3]-d+e==0,If[a-b+(d+e-2*f)/Sqrt[3]<0,Pi,0],            ArcTan[a-b+(d+e-2*f)/Sqrt[3],(a+b-2*c)/Sqrt[3]-d+e]]-Pi/6,        3*((a-b-(d+e-2*f)/Sqrt[3])^2+((a+b-2*c)/Sqrt[3]+d-e)^2)/4,
If[(a+b-2*c)/Sqrt[3]+d-e==0,
If[a-b-(d+e-2*f)/Sqrt[3]<0,Pi,0],            ArcTan[a-b-(d+e-2*f)/Sqrt[3],(a+b-2*c)/Sqrt[3]+d-e]]-Pi/6},      {Sqrt[a/9]*Cos[b]+Sqrt[c/12]*Cos[d+Pi/6]+Sqrt[c/36]*Sin[d+Pi/6]+Sqrt[e/12]*Cos[f+Pi/6]+Sqrt[e/36]*Sin[f+Pi/6],        Sqrt[a/9]*Cos[b]-Sqrt[c/12]*Cos[d+Pi/6]+Sqrt[c/36]*Sin[d+Pi/6]-Sqrt[e/12]*Cos[f+Pi/6]+Sqrt[e/36]*Sin[f+Pi/6],        Sqrt[a/9]*Cos[b]-Sqrt[c/9]*Sin[d+Pi/6]-Sqrt[e/9]*Sin[f+Pi/6],        Sqrt[a/9]*Sin[b]+Sqrt[c/36]*Cos[d+Pi/6]-Sqrt[c/12]*Sin[d+Pi/6]-Sqrt[e/36]*Cos[f+Pi/6]+Sqrt[e/12]*Sin[f+Pi/6],        Sqrt[a/9]*Sin[b]+Sqrt[c/36]*Cos[d+Pi/6]+Sqrt[c/12]*Sin[d+Pi/6]-Sqrt[e/36]*Cos[f+Pi/6]-Sqrt[e/12]*Sin[f+Pi/6],        Sqrt[a/9]*Sin[b]-Sqrt[c/9]*Cos[d+Pi/6]+Sqrt[e/9]*Cos[f+Pi/6]},
{1,3,2,-4,-6,-5,1,1,1,0},{1,0.,1,0.,1,0.},{a,b,c,-d,-e,-f}};

Hoop["C3C4c"] :=   
{6, 
          8, {{1, 2, 3, 4, 5, 6}, {2, 3, 1, 5, 6, 4}, {3, 1, 2, 6, 4, 5}, {4, 
              5, 6, -1, -2, -3}, {5, 6, 
              4, -2, -3, -1}, {6, 4, 
              5, -3, -1, -2}}, 
{(a + b + 
                    c)^2 + (d + e + f)^2, 
            3*((a - b + (d + e - 2*f)/
                            Sqrt[3])^2 + ((a + b - 2*c)/Sqrt[3] - d + 
                          e)^2)/4, 
            3*((a - b - (d + e - 2*f)/
                            Sqrt[3])^2 + ((a + b - 2*c)/Sqrt[3] + d - 
                          e)^2)/
                4  }, {(a + b + c)^2 + (d + 
                    e + f)^2, 
            If[d + e + f == 0, If[a + b + c < 0, Pi, 0], 
              ArcTan[a + b + c, d + e + f]], 
            3*((a - b + (d + e - 2*f)/
                            Sqrt[3])^2 + ((a + b - 2*c)/Sqrt[3] - d + 
                          e)^2)/4, 
            If[(a + b - 2*c)/Sqrt[3] - d + e == 0, 
                If[a - b + (d + e - 2*f)/Sqrt[3] < 0, Pi, 0], 
                ArcTan[a - 
                    b + (d + e - 2*f)/Sqrt[3], (a + b - 2*c)/
                      Sqrt[3] - d + e]] - Pi/6, 
            3*((a - b - (d + e - 2*f)/
                            Sqrt[3])^2 + ((a + b - 2*c)/Sqrt[3] + d - 
                          e)^2)/4, 
            If[(a + b - 2*c)/Sqrt[3] + d - e == 0, 
                If[a - b - (d + e - 2*f)/Sqrt[3] < 0, Pi, 0], 
                ArcTan[a - 
                    b - (d + e - 2*f)/Sqrt[3], (a + b - 2*c)/
                      Sqrt[3] + d - e]] - 
              Pi/6}, 
{Sqrt[a/9]*Cos[b] + 
              Sqrt[c/12]*Cos[d + Pi/6] + Sqrt[c/36]*Sin[d + Pi/6] + 
              Sqrt[e/12]*Cos[f + Pi/6] + 
              Sqrt[e/36]*Sin[f + Pi/6], 
Sqrt[a/9]*Cos[b] - 
              Sqrt[c/12]*Cos[d + Pi/6] + Sqrt[c/36]*Sin[d + Pi/6] - 
              Sqrt[e/12]*Cos[f + Pi/6] + 
              Sqrt[e/36]*Sin[f + Pi/6], 
Sqrt[a/9]*Cos[b] - 
              Sqrt[c/9]*Sin[d + Pi/6] - 
              Sqrt[e/9]*Sin[f + Pi/6], 
Sqrt[a/9]*Sin[b] + 
              Sqrt[c/36]*Cos[d + Pi/6] - Sqrt[c/12]*Sin[d + Pi/6] - 
              Sqrt[e/36]*Cos[f + Pi/6] + 
              Sqrt[e/12]*Sin[f + Pi/6], 
Sqrt[a/9]*Sin[b] + 
              Sqrt[c/36]*Cos[d + Pi/6] + Sqrt[c/12]*Sin[d + Pi/6] - 
              Sqrt[e/36]*Cos[f + Pi/6] - 
              Sqrt[e/12]*Sin[f + Pi/6], 
Sqrt[a/9]*Sin[b] - 
              Sqrt[c/9]*Cos[d + Pi/6] + 
              Sqrt[e/9]*Cos[f + Pi/6]}, 
{1, 3, 
            2, -4, -6, -5, 1, 1, 1, 0}, {1, 0. , 1, 0. , 1, 
            0. }, {a, b, c, -d, -e, -f}};

Hoop["D3C2c"]={6,9,{{1,2,3,4,5,6},{2,3,1,5,-6,-4},{3,1,2,-6,4,-5},{4,-6,5,1,3,-2},{5,4,-6,2,1,-3},{6,-5,-4,-3,-2,1}},{a+b+c+d+e-f,a+b+c-d-e+f,((a-b)^2+(a-c)^2+(b-c)^2-(d-e)^2-(d+f)^2-(e+f)^2)/2},,,{1,3,2,4,5,6,1,1,2},{1,1,1,0.,1,0.},{a,c,b,-d,-e,-f}};

Hoop["Clm3r"]={6,10,{{1,2,3,4,5,6},{2,-3,1,5,-6,4},{3,1,-2,6,4,-5},{4,5,6,-1,-2,-3},{5,-6,4,-2,3,-1},{6,4,-5,-3,-1,2}},{(a-b-c)^2+(d-e-f)^2,-3*(-b*d+c*d+a*e+c*e-a*f-b*f)^2+((a+b)^2+(b-c)^2+(a+c)^2+(d+e)^2+(e-f)^2+(d+f)^2)^2/4},,,{1,3,2,-4,-6,-5,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};

Hoop["Cl3r"]={6,11,{{1,2,3,4,5,6},{2,3,-1,5,6,-4},{3,-1,-2,6,-4,-5},{4,5,6,-1,-2,-3},{5,6,-4,-2,-3,1},{6,-4,-5,-3,1,2}},{(a-b+c)^2+(d-e+f)^2,-3*(-b*d-c*d+a*e-c*e+a*f+b*f)^2+((a+b)^2+(a-c)^2+(b+c)^2+(d+e)^2+(-d+f)^2+(e+f)^2)^2/4},,,{1,-3,-2,-4,6,5,1,1},{1,1,1,0.,1,0.},{a,b,c,-d,-e,-f}};

Hoop["g2401c"]={6,12,
{{1,2,3,4,5,6},{2,3,1,6,4,5},{3,1,2,5,6,4},{4,5,6,I,2*I,3*I},{5,6,4,3*I,I,2*I},{6,4,5,2*I,3*I,I}},{((a-b)^2+(a-c)^2+(b-c)^2-I*((d-e)^2+(d-f)^2+(e-f)^2))/2,(a+b+c)^2-I*(d+e+f)^2},,,{1,3,2,-4*I,-5*I,-6*I,2,1},{1,1,1,0.,1,0.},{a,c,b,-d,-e,-f}};

Hoop["C8"]={8,1,{{1,2,3,4,5,6,7,8},{2,3,4,5,6,7,8,1},{3,4,5,6,7,8,1,2},
{4,5,6,7,8,1,2,3},{5,6,7,8,1,2,3,4},{6,7,8,1,2,3,4,5},
{7,8,1,2,3,4,5,6},{8,1,2,3,4,5,6,7}},{a-b+c-d+e-f+g-h,
a+b+c+d+e+f+g+h,(a-c+e-g)^2+(b-d+f-h)^2,
((a-e)^2+(c-g)^2)^2+4*((-b+f)*(c-g)+(a-e)*(d-h))*
((a-e)*(b-f)+(c-g)*(d-h))+((b-f)^2+(d-h)^2)^2},,,
{1,8,7,6,5,4,3,2,1,1,1,1},{1,1,1,1,0.,1,0.},{a,-b,c,-d,e,-f,g,-h}};

Hoop["C4C2"]={8,2,{{1,2,3,4,5,6,7,8},{2,3,4,1,6,7,8,5},{3,4,1,2,7,8,5,6},
{4,1,2,3,8,5,6,7},{5,6,7,8,1,2,3,4},{6,7,8,5,2,3,4,1},
{7,8,5,6,3,4,1,2},{8,5,6,7,4,1,2,3}},{a+b+c+d-e-f-g-h,
a-b+c-d+e-f+g-h,a-b+c-d-e+f-g+h,a+b+c+d+e+f+g+h,
(a-c-e+g)^2+(b-d-f+h)^2,(a-c+e-g)^2+(b-d+f-h)^2},
{a+b+c+d+e+f+g+h,a-b+c-d+e-f+g-h,a+b+c+d-e-f-g-h,
a-b+c-d-e+f-g+h,(a-c-e+g)^2+(b-d-f+h)^2,
ArcTan[a-c-e+g,b-d-f+h],(a-c+e-g)^2+(b-d+f-h)^2,
ArcTan[a-c+e-g,b-d+f-h]},{(a+b+c+d+2*Sqrt[e]*Cos[f]+2*Sqrt[g]*Cos[h])/8,
(a-b+c-d+2*Sqrt[e]*Sin[f]+2*Sqrt[g]*Sin[h])/8,
(a+b+c+d-2*Sqrt[e]*Cos[f]-2*Sqrt[g]*Cos[h])/8,
(a-b+c-d-2*Sqrt[e]*Sin[f]-2*Sqrt[g]*Sin[h])/8,
(a+b-c-d-2*Sqrt[e]*Cos[f]+2*Sqrt[g]*Cos[h])/8,
(a-b-c+d-2*Sqrt[e]*Sin[f]+2*Sqrt[g]*Sin[h])/8,
(a+b-c-d+2*Sqrt[e]*Cos[f]-2*Sqrt[g]*Cos[h])/8,
(a-b-c+d+2*Sqrt[e]*Sin[f]-2*Sqrt[g]*Sin[h])/8},
{1,4,3,2,5,8,7,6,1,1,1,1,1,1},{1,1,1,1,1,0.,1,0.},
{a,-b,c,-d,e,-f,g,-h}};

Hoop["D4"]={8,3,{{1,2,3,4,5,6,7,8},{2,1,8,7,6,5,4,3},{3,4,5,6,7,8,1,2},
{4,3,2,1,8,7,6,5},{5,6,7,8,1,2,3,4},{6,5,4,3,2,1,8,7},
{7,8,1,2,3,4,5,6},{8,7,6,5,4,3,2,1}},{a+b-c-d+e+f-g-h,
a-b+c-d+e-f+g-h,a-b-c+d+e-f-g+h,a+b+c+d+e+f+g+h,
(a-e)^2-(b-f)^2+(c-g)^2-(d-h)^2},{a+b+c+d+e+f+g+h,
a+b-c-d+e+f-g-h,a-b+c-d+e-f+g-h,a-b-c+d+e-f-g+h,
(a-e)^2-(b-f)^2+(c-g)^2-(d-h)^2,ArcTan[a-e,c-g],(b-f)^2+(d-h)^2,ArcTan[b-f,d-h]},{(a+b+c+d+4*Sqrt[e+g]*Cos[f]),
(a+b-c-d+4*Sqrt[g]*Cos[h]),(a-b+c-d+4*Sqrt[e+g]*Sin[f]),
(a-b-c+d+4*Sqrt[g]*Sin[h]),(a+b+c+d-4*Sqrt[e+g]*Cos[f]),
(a+b-c-d-4*Sqrt[g]*Cos[h]),(a-b+c-d-4*Sqrt[e+g]*Sin[f]),
(a-b-c+d-4*Sqrt[g]*Sin[h])}/8,{1,2,7,4,5,6,3,8,1,1,1,1,2},{1,1,1,1,1,0.,1,0.},};

Hoop["Q8"]={8,4,{{1,2,3,4,5,6,7,8},{2,5,8,3,6,1,4,7},{3,4,5,6,7,8,1,2},
{4,7,2,5,8,3,6,1},{5,6,7,8,1,2,3,4},{6,1,4,7,2,5,8,3},
{7,8,1,2,3,4,5,6},{8,3,6,1,4,7,2,5}},{a+b-c-d+e+f-g-h,
a-b+c-d+e-f+g-h,a-b-c+d+e-f-g+h,a+b+c+d+e+f+g+h,
(a-e)^2+(b-f)^2+(c-g)^2+(d-h)^2},{a+b+c+d+e+f+g+h,
a-b+c-d+e-f+g-h,a-b-c+d+e-f-g+h,a+b-c-d+e+f-g-h,
(a-e)^2+(b-f)^2,ArcTan[a-e,b-f],(c-g)^2+(d-h)^2,ArcTan[c-g,d-h]},
{(a+b+c+d+4*Sqrt[e]*Cos[f]),(a-b-c+d+4*Sqrt[e]*Sin[f]),
(a+b-c-d+4*Sqrt[g]*Cos[h]),(a-b+c-d+4*Sqrt[g]*Sin[h]),
(a+b+c+d-4*Sqrt[e]*Cos[f]),(a-b-c+d-4*Sqrt[e]*Sin[f]),
(a+b-c-d-4*Sqrt[g]*Cos[h]),(a-b+c-d-4*Sqrt[g]*Sin[h])}/8,{1,6,7,8,5,2,3,4,1,1,1,1,2},{1,1,1,1,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};

Hoop["KC2"]={8,5,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,7,8,5,6},
{4,3,2,1,8,7,6,5},{5,6,7,8,1,2,3,4},{6,5,8,7,2,1,4,3},
{7,8,5,6,3,4,1,2},{8,7,6,5,4,3,2,1}},{a+b+c+d-e-f-g-h,
a+b-c-d+e+f-g-h,a-b+c-d+e-f+g-h,a-b-c+d-e+f+g-h,
a-b-c+d+e-f-g+h,a-b+c-d-e+f-g+h,a+b-c-d-e-f+g+h,
a+b+c+d+e+f+g+h},{a+b+c+d-e-f-g-h,a+b-c-d+e+f-g-h,
a-b+c-d+e-f+g-h,a-b-c+d-e+f+g-h,a-b-c+d+e-f-g+h,
a-b+c-d-e+f-g+h,a+b-c-d-e-f+g+h,a+b+c+d+e+f+g+h},
{(a+b+c+d+e+f+g+h)/8,(a+b-c-d-e-f+g+h)/8,
(a-b+c-d-e+f-g+h)/8,(a-b-c+d+e-f-g+h)/8,
(-a+b+c-d+e-f-g+h)/8,(-a+b-c+d-e+f-g+h)/8,
(-a-b+c+d-e-f+g+h)/8,(-a-b-c-d+e+f+g+h)/8},
{1,2,3,4,5,6,7,8,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1},
{a,b,c,d,-e,-f,-g,-h}};

Hoop["C4C4c"]={8,8,{{1,2,3,4,5,6,7,8},{2,3,4,1,6,7,8,5},{3,4,1,2,7,8,5,6},{4,1,2,3,8,5,6,7},{5,6,7,8,-1,-2,-3,-4},{6,7,8,5,-2,-3,-4,-1},{7,8,5,6,-3,-4,-1,-2},{8,5,6,7,-4,-1,-2,-3}},{(b-d+e-g)^2+(a-c-f+h)^2,(b-d-e+g)^2+(a-c+f-h)^2,(a-b+c-d)^2+(e-f+g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2},
{(b-d+e-g)^2+(a-c-f+h)^2,Which[b-d+e-g==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]],(b-d-e+g)^2+(a-c+f-h)^2,Which[b-d-e+g==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]],(a-b+c-d)^2+(e-f+g-h)^2,Which[e-f+g-h==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]],(a+b+c+d)^2+(e+f+g+h)^2,Which[e+f+g+h==0,0,aa$66==0,(Pi/2)*Sign[bb$66],True,ArcTan[aa$66,bb$66]]},
{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]),(-(Sqrt[e]*Cos[f])+Sqrt[g]*Cos[h]+Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]),(-(Sqrt[a]*Cos[b])-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]),(-(Sqrt[e]*Cos[f])+Sqrt[g]*Cos[h]-Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]),(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]),(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]),(-(Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]),(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])}/4,{1,4,3,2,-5,-8,-7,-6,1,1,1,1},{1,0.,1,0.,1,0.,1,0.},{a,-b,c,-d,-e,f,-g,h}};

Hoop["KiC4c"]={8,9,{{1,2,3,4,5,6,7,8},{2,1,4,3,8,7,6,5},{3,4,1,2,7,8,5,6},{4,3,2,1,6,5,8,7},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-4,-3,-2,-1},{7,8,5,6,-3,-4,-1,-2},{8,7,6,5,-2,-1,-4,-3}},{(a-b+c-d)^2+(e-f+g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2,(a-c)^2-(b-d)^2+(e-g)^2-(f-h)^2},{(a-c)^2-(b-d)^2+(e-g)^2-(f-h)^2,If[e==g,0,ArcTan[a-c,e-g]],(b-d)^2+(f-h)^2,If[f==h,0,ArcTan[b-d,f-h]],(a-b+c-d)^2+(e-f+g-h)^2,If[e+g==f+h,0,ArcTan[a-b+c-d,e-f+g-h]],(a+b+c+d)^2+(e+f+g+h)^2,If[e+f==g+h,0,ArcTan[a+b+c+d,e+f+g+h]]},{(Sqrt[a+c]*Cos[b])/2+(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,(Sqrt[c]*Cos[d])/2-(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,-(Sqrt[a+c]*Cos[b])/2+(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,
-(Sqrt[c]*Cos[d])/2-(Sqrt[e]*Cos[f])/4+(Sqrt[g]*Cos[h])/4,(Sqrt[a+c]*Sin[b])/2+(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4,(Sqrt[c]*Sin[d])/2-(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4,-(Sqrt[a+c]*Sin[b])/2+(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4,-(Sqrt[c]*Sin[d])/2-(Sqrt[e]*Sin[f])/4+(Sqrt[g]*Sin[h])/4},{1,2,3,4,-5,-8,-7,-6,1,1,2},{1,0.,1,0.,1,0,1,0.},{a,b,c,d,-e,-f,-g,-h}};

Hoop["C4iC4c"]={8,10,{{1,2,3,4,5,6,7,8},{2,3,4,1,8,5,6,7},{3,4,1,2,7,8,5,6},{4,1,2,3,6,7,8,5},{5,6,7,8,-1,-2,-3,-4},{6,7,8,5,-4,-1,-2,-3},{7,8,5,6,-3,-4,-1,-2},{8,5,6,7,-2,-3,-4,-1}},{(a-c)^2+(b-d)^2+(e-g)^2+(f-h)^2,(a-b+c-d)^2+(e-f+g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2},,,{1,4,3,2,-5,-6,-7,-8,2,1,1},{1,1,2,1,0.,1,0.},{a,-b,c,-d,-e,-f,-g,-h}};

Hoop["QC2r"]={8,11,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,-1,-2,-7,-8,5,6},{4,3,-2,-1,-8,-7,6,5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,-5,-6,3,4,-1,-2},{8,7,-6,-5,4,3,-2,-1}},{(a-b)^2+(c-d)^2+(e-f)^2+(g-h)^2,(a+b)^2+(c+d)^2+(e+f)^2+(g+h)^2},,,{1,2,-3,-4,-5,-6,-7,-8,2,2},{1,1,1,1,0.,1,0.},{a,b,-c,-d,-e,-f,-g,-h}};

Hoop["Q8C2r"]={8,12,{{1,2,3,4,5,6,7,8},{2,-5,1,8,-3,-4,6,-7},{3,1,-5,-6,-2,7,-8,4},
{4,-6,8,-5,7,-3,-1,2},{5,-3,-2,7,1,8,4,6},{6,7,-4,-2,8,-5,3,-1},
{7,-8,6,-1,4,2,-5,-3},{8,4,-7,3,6,-1,-2,-5}},
{a-b-c-d-e-f+g+h,a+b+c-d-e+f+g-h,a+b+c+d-e-f-g+h,
a-b-c+d-e+f-g-h,(b-c)^2+(a+e)^2+(d+g)^2+(f+h)^2},
{a-b-c-d-e-f+g+h,a+b+c-d-e+f+g-h,a+b+c+d-e-f-g+h,
a-b-c+d-e+f-g-h,(b-c)^2+(a+e)^2,ArcTan[a+e,b-c],(d+g)^2+(f+h)^2,ArcTan[d+g,f+h]},{(a+b+c+d+4*Sqrt[e]*Cos[f]),
(-a+b+c-d+4*Sqrt[e]*Sin[f]),(-a+b+c-d-4*Sqrt[e]*Sin[f]),
(-a-b+c+d+4*Sqrt[g]*Cos[h]),(-a-b-c-d+4*Sqrt[e]*Cos[f]),
(-a+b-c+d+4*Sqrt[g]*Sin[h]),(a+b-c-d+4*Sqrt[g]*Cos[h]),
(a-b+c-d+4*Sqrt[g]*Sin[h])}/8,{1,3,2,-7,5,-8,-4,-6,1,1,1,1,2},{1,1,1,1,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};

Hoop["O4"]={8,13,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,7,8,5,6},{4,3,2,1,8,7,6,5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,5,6,-3,-4,-1,-2},{8,7,6,5,-4,-3,-2,-1}},{(a-b-c+d)^2+(e-f-g+h)^2,(a-b+c-d)^2+(e-f+g-h)^2,(a+b-c-d)^2+(e+f-g-h)^2,(a+b+c+d)^2+(e+f+g+h)^2},
{(a-b-c+d)^2+(e-f-g+h)^2,If[e+h==f+g,0,ArcTan[a-b-c+d,e-f-g+h]],(a-b+c-d)^2+(e-f+g-h)^2,If[e+g==f+h,0,ArcTan[a-b+c-d,e-f+g-h]],(a+b-c-d)^2+(e+f-g-h)^2,If[e+f==g+h,0,ArcTan[a+b-c-d,e+f-g-h]],(a+b+c+d)^2+(e+f+g+h)^2,If[e+f+g+h==0,0,ArcTan[a+b+c+d,e+f+g+h]]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(-(Sqrt[a]*Cos[b])-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(-(Sqrt[a]*Cos[b])+Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h])/4,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4,(-(Sqrt[a]*Sin[b])-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4,(-(Sqrt[a]*Sin[b])+Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h])/4},{1,2,3,4,-5,-6,-7,-8,1,1,1,1},{1,0.,1,0.,1,0.,1,0.},{a,b,c,d,-e,-f,-g,-h}};

Hoop["P8"]={8,16,{{1,2,3,4,5,6,7,8},{2,1,-4*I,3*I,6,5,-8*I,7*I},{3,4*I,1,-2*I,7,8*I,5,-6*I},{4,-3*I,2*I,1,8,-7*I,6*I,5},{5,6,7,8,1,2,3,4},{6,5,-8*I,7*I,2,1,-4*I,3*I},{7,8*I,5,-6*I,3,4*I,1,-2*I},{8,-7*I,6*I,5,4,-3*I,2*I,1}},{(a+e)^2-(b+f)^2-(c+g)^2-(d+h)^2,(a-e)^2-(b-f)^2-(c-g)^2-(d-h)^2},{(a+e)^2-(b+f)^2,arcTanh[a+e,b+f],(c+g)^2+(d+h)^2,ArcTan[c+g,d+h],(a-e)^2-(b-f)^2,arcTanh[a-e,b-f],(c-g)^2+(d-h)^2,ArcTan[c-g,d-h]},{(Sqrt[a]*Cosh[b]+Sqrt[e]*Cosh[f])/2,(Sqrt[a]*Sinh[b]+Sqrt[e]*Sinh[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[a]*Cosh[b]-Sqrt[e]*Cosh[f])/2,(Sqrt[a]*Sinh[b]-Sqrt[e]*Sinh[f])/2,(Sqrt[c]*Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]-Sqrt[g]*Sin[h])/2},{1,2,3,4,5,6,7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};

Hoop["C2D4r"]={8,18,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,4,1,2,-7,-8,-5,-6},{4,3,2,1,-8,-7,-6,-5},{5,6,7,8,-1,-2,-3,-4},{6,5,8,7,-2,-1,-4,-3},{7,8,5,6,3,4,1,2},{8,7,6,5,4,3,2,1}},{(a+b)^2-(c+d)^2+(e+f)^2-(g+h)^2,(a-b)^2-(c-d)^2+(e-f)^2-(g-h)^2},{(a+b)^2+(e+f)^2,If[e==-f,0,ArcTan[a+b,e+f]],(c+d)^2+(g+h)^2,If[g==-h,0,ArcTan[c+d,g+h]],(a-b)^2+(e-f)^2,If[e==f,0,ArcTan[a-b,e-f]],(c-d)^2+(g-h)^2,If[g==h,0,ArcTan[c-d,g-h]]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(Sqrt[a]*Cos[b]-Sqrt[e]*Cos[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(Sqrt[a]*Sin[b]-Sqrt[e]*Sin[f])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[c]*Sin[d]-Sqrt[g]*Sin[h])/2},{1,2,3,4,-5,-6,7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,b,-c,-d,-e,-f,-g,-h}};

Hoop["CL3"]={8,19,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,8,7},{3,-4,1,-2,7,-8,5,-6},{4,-3,2,-1,8,-7,6,-5},{5,-6,-7,8,1,-2,-3,4},{6,-5,-8,7,2,-1,-4,3},{7,8,-5,-6,3,4,-1,-2},{8,7,-6,-5,4,3,-2,-1}},{4*(-(d*e)+c*f-b*g+a*h)^2+(a^2-b^2-c^2+d^2-e^2+f^2+g^2-h^2)^2},,,{1,2,3,-4,5,-6,-7,-8,2},{2,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,h}};

Hoop["Octr"]=(*Octonion Algebra*)
{8,21,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,-8,7},{3,-4,-1,2,7,8,-5,-6},{4,3,-2,-1,8,-7,6,-5},{5,-6,-7,-8,-1,2,3,4},{6,5,-8,7,-2,-1,-4,3},{7,8,5,-6,-3,4,-1,-2},{8,-7,6,5,-4,-3,2,-1}},{a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2},{a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2,ArcTan[a,b],c^2+d^2,ArcTan[c,d],e^2+f^2,ArcTan[e,f],g^2+h^2,ArcTan[g,h]},{Sqrt[a-c-e-g]*Cos[b],Sqrt[a-c-e-g]*Sin[b],Sqrt[c]*Cos[d],Sqrt[c]*Sin[d],Sqrt[e]*Cos[f],Sqrt[e]*Sin[f],Sqrt[g]*Cos[h],Sqrt[g]*Sin[h]},{1,-2,-3,-4,-5,-6,-7,-8,4},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,-h}};

Hoop["KiC4r"]={8,22,{{1,2,3,4,5,6,7,8},{2,1,-8,-7,6,5,-4,-3},{3,-4,-1,2,7,-8,-5,6},{4,-3,6,-5,8,-7,2,-1},{5,6,7,8,1,2,3,4},{6,5,-4,-3,2,1,-8,-7},{7,-8,-5,6,3,-4,-1,2},{8,-7,2,-1,4,-3,6,-5}},{(a-e)^2-(b-f)^2+(c-g)^2-(d-h)^2,(a+b+e+f)^2+(c-d+g-h)^2,(a-b+e-f)^2+(c+d+g+h)^2},,,{1,2,-3,-8,5,6,-7,-4,2,1,1},{1,1,1,1,0.,1,0.},{a,-b,-c,-d,e,-f,-g,-h}};

Hoop["CL21"]={8,27,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,8,-7},{3,-4,1,-2,7,-8,5,-6},{4,3,2,1,8,7,6,5},{5,-6,-7,8,1,-2,-3,4},{6,5,-8,-7,2,1,-4,-3},{7,8,-5,-6,3,4,-1,-2},{8,-7,-6,5,4,-3,-2,1}},{-(d-e)^2-(c+f)^2+(b+g)^2+(a-h)^2,-(d+e)^2-(c-f)^2+(b-g)^2+(a+h)^2},{(b+g)^2+(a-h)^2,ArcTan[a-h,b+g],(d-e)^2+(c+f)^2,ArcTan[c+f,d-e],(b-g)^2+(a+h)^2,ArcTan[a+h,b-g],(d+e)^2+(c-f)^2,ArcTan[c-f,d+e]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(-(Sqrt[c]*Sin[d])+Sqrt[g]*Sin[h])/2,(Sqrt[c]*Cos[d]-Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]-Sqrt[e]*Sin[f])/2,(-(Sqrt[a]*Cos[b])+Sqrt[e]*Cos[f])/2},{1,-2,3,4,5,6,-7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,h}};

Hoop["CL03"]={8,28,{{1,2,3,4,5,6,7,8},{2,-1,4,-3,6,-5,8,-7},{3,-4,-1,2,7,-8,-5,6},{4,3,-2,-1,8,7,-6,-5},{5,-6,-7,8,-1,2,3,-4},{6,5,-8,-7,-2,-1,4,3},{7,8,5,6,-3,-4,-1,-2},{8,-7,6,-5,-4,3,-2,1}},{(d+e)^2+(c-f)^2+(b+g)^2+(a-h)^2,(d-e)^2+(c+f)^2+(b-g)^2+(a+h)^2},{(b+g)^2+(a-h)^2,ArcTan[a-h,b+g],(d+e)^2+(c-f)^2,ArcTan[c-f,d+e],(b-g)^2+(a+h)^2,ArcTan[a+h,b-g],(d-e)^2+(c+f)^2,ArcTan[c+f,d-e]},{(Sqrt[a]*Cos[b]+Sqrt[e]*Cos[f])/2,(Sqrt[a]*Sin[b]+Sqrt[e]*Sin[f])/2,(Sqrt[c]*Cos[d]+Sqrt[g]*Cos[h])/2,(Sqrt[c]*Sin[d]+Sqrt[g]*Sin[h])/2,(Sqrt[c]*Sin[d]-Sqrt[g]*Sin[h])/2,(-(Sqrt[c]*Cos[d])+Sqrt[g]*Cos[h])/2,(Sqrt[a]*Sin[b]-Sqrt[e]*Sin[f])/2,(-(Sqrt[a]*Cos[b])+Sqrt[e]*Cos[f])/2},{1,-2,-3,-4,-5,-6,-7,8,2,2},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,h}};

Hoop["Octi"]=(*Split-Octonion*){8,30,{{1,2,3,4,5,6,7,8},{2,1,4,3,6,5,-8,-7},{3,-4,-1,2,7,8,-5,-6},{4,-3,-2,1,8,7,6,5},{5,-6,-7,-8,-1,2,3,4},{6,-5,-8,-7,-2,1,-4,-3},{7,8,5,-6,-3,4,-1,-2},{8,7,6,-5,-4,3,2,1}},{a^2-b^2+c^2-d^2+e^2-f^2+g^2-h^2},{a^2+c^2,ArcTan[a,c],b^2+d^2,ArcTan[b,d],e^2+g^2,
ArcTan[e,g],f^2+h^2,ArcTan[f,h]},{Sqrt[a]*Cos[b],Sqrt[c]*Cos[d],Sqrt[a]*Sin[b],Sqrt[c]*Sin[d],Sqrt[e]*Cos[f],Sqrt[g]*Cos[h],Sqrt[e]*Sin[f],Sqrt[g]*Sin[h]},{1,2,-3,4,-5,6,-7,8,4},{1,0.,1,0.,1,0.,1,0.},{a,-b,-c,-d,-e,-f,-g,-h}};

Hoop["C3C3"]={9,2,{{1,2,3,4,5,6,7,8,9},{2,3,1,5,6,4,8,9,7},{3,1,2,6,4,5,9,7,8},{4,5,6,7,8,9,1,2,3},{5,6,4,8,9,7,2,3,1},{6,4,5,9,7,8,3,1,2},{7,8,9,1,2,3,4,5,6},{8,9,7,2,3,1,5,6,4},{9,7,8,3,1,2,6,4,5}},{a+b+c+d+e+f+g+h+i,((-a+c+e-f+g-h)^2+(a-b-d+f+h-i)^2+(b-c+d-e-g+i)^2)/2,((b-c-d+f+g-h)^2+(a-b+e-f-g+i)^2+(a-c-d+e-h+i)^2)/2,((a-b+d-e+g-h)^2+(b-c+e-f+h-i)^2+(-a+c-d+f-g+i)^2)/2,((a+b+c-d-e-f)^2+(a+b+c-g-h-i)^2+(d+e+f-g-h-i)^2)/2},{a+b+c+d+e+f+g+h+i,((-a+c+e-f+g-h)^2+(a-b-d+f+h-i)^2+(b-c+d-e-g+i)^2)/2,ArcTan[2*a-b-c-d-e+2*f-g+2*h-i,-(Sqrt[3]*(b-c+d-e-g+i))],((b-c-d+f+g-h)^2+(a-b+e-f-g+i)^2+(a-c-d+e-h+i)^2)/2,ArcTan[2*a-b-c-d+2*e-f-g-h+2*i,-(Sqrt[3]*(b-c-d+f+g-h))],((a-b+d-e+g-h)^2+(b-c+e-f+h-i)^2+(-a+c-d+f-g+i)^2)/2,ArcTan[2*a-b-c+2*d-e-f+2*g-h-i,-(Sqrt[3]*(b-c+e-f+h-i))],((a+b+c-d-e-f)^2+(a+b+c-g-h-i)^2+(d+e+f-g-h-i)^2)/2,ArcTan[2*a+2*b+2*c-d-e-f-g-h-i,-(Sqrt[3]*(d+e+f-g-h-i))]},{(a+2*Sqrt[b]*Cos[c]+2*Sqrt[d]*Cos[e]+2*Sqrt[f]*Cos[g]+2*Sqrt[h]*Cos[i])/9,(a+2*Sqrt[h]*Cos[i]+2*Sqrt[b]*Cos[c+(2*Pi)/3]+2*Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(2*Pi)/3])/9,(a+2*Sqrt[h]*Cos[i]+2*Sqrt[b]*Cos[c+(4*Pi)/3]+2*Sqrt[d]*Cos[e+(4*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3])/9,(a+2*Sqrt[f]*Cos[g]+2*Sqrt[b]*Cos[c+(2*Pi)/3]+2*Sqrt[h]*Cos[i+(2*Pi)/3]+2*Sqrt[d]*Cos[e+(4*Pi)/3])/9,(a+2*Sqrt[d]*Cos[e]+2*Sqrt[f]*Cos[g+(2*Pi)/3]+2*Sqrt[h]*Cos[i+(2*Pi)/3]+2*Sqrt[b]*Cos[c+(4*Pi)/3])/9,(a+2*Sqrt[b]*Cos[c]+2*Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[h]*Cos[i+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3])/9,(a+2*Sqrt[f]*Cos[g]+2*Sqrt[d]*Cos[e+(2*Pi)/3]+2*Sqrt[b]*Cos[c+(4*Pi)/3]+2*Sqrt[h]*Cos[i+(4*Pi)/3])/9,(a+2*Sqrt[b]*Cos[c]+2*Sqrt[f]*Cos[g+(2*Pi)/3]+2*Sqrt[d]*Cos[e+(4*Pi)/3]+2*Sqrt[h]*Cos[i+(4*Pi)/3])/9,(a+2*Sqrt[d]*Cos[e]+2*Sqrt[b]*Cos[c+(2*Pi)/3]+2*Sqrt[f]*Cos[g+(4*Pi)/3]+2*Sqrt[h]*Cos[i+(4*Pi)/3])/9},{1,3,2,7,9,8,4,6,5,1,1,1,1,1},{1,1,0.,1,0.,1,0.,1,0.},{a,c,b,g,i,h,d,f,e}};

\(\(Hoop["Q12"] = {12, 
          1, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              6, 5, 7, 9, 8, 10, 12, 11, 1, 3, 2}, {5, 4, 6, 8, 7, 9, 11, 10, 
              12, 2, 1, 3}, {6, 5, 4, 9, 8, 7, 12, 11, 10, 3, 2, 1}, {7, 8, 
              9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 
              1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 12, 
              11, 1, 3, 2, 4, 6, 5, 7, 9, 8}, {11, 10, 12, 2, 1, 3, 5, 4, 6, 
              8, 7, 9}, {12, 11, 10, 3, 2, 1, 6, 5, 4, 9, 8, 7}}, {a + b + 
              c + d + e + f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - 
              l, \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2, \ \((\((a - b + g - h)\)\^2 + \((a - c + g - i)\)\^2 + \((\(-b\) + c - h + i)\)\^2 - \((d - e + j - k)\)\^2 - \((d - f + j - l)\)\^2 - \((\(-e\) + f - k + l)\)\^2)\)/
              2, \ \((\((a - b - g + h)\)\^2 + \((\(-b\) + c + h - i)\)\^2 + \((a - c - g + i)\)\^2 + \((d - e - j + k)\)\^2 + \((\(-e\) + f + k - l)\)\^2 + \((d - f - j + l)\)\^2)\)/2}, {a + b + c + d + e + f + g + h + i + j + k + 
              l, a + b + c - d - e - f + g + h + i - j - k - 
              l, \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2, ArcTan[a + b + c - g - h - i, 
              d + e + f - j - k - 
                l], \ \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \((\(-a\) + c - g + i)\)\^2)\)/2, 
            ArcTan[2\ a - b - c + 2\ g - h - 
                i, \(-\@3\)\ \((b - c + h - 
                    i)\)], \ \((\((d - e + j - k)\)\^2 + \((e - f + k - l)\)\^2 + \((\(-d\) + f - j + l)\)\^2)\)/2, 
            ArcTan[2\ d - e - f + 2\ j - k - 
                l, \(-\@3\)\ \((e - f + k - 
                    l)\)], \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\)\^2 + \((b - c - h + i)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ g + h + 
                i, \(-\@3\)\ \((b - c - h + 
                    i)\)], \ \((\((d - e - j + k)\)\^2 + \((\(-d\) + f + j - l)\)\^2 + \((e - f - k + l)\)\^2)\)/2, 
            ArcTan[2\ d - e - f - 2\ j + k + 
                l, \(-\@3\)\ \((e - f - k + l)\)]}, {\ \((a + b + 
                    2\ \@c\ Cos[d] + 4\ \@e\ Cos[f] + 4\ \@i\ Cos[j])\)/
                2, \ \((a + b + 2\ \@c\ Cos[d] + 
                  4\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  4\ \@i\ Cos[j + \(2\ π\)\/3])\), \ \((a + b + 
                  2\ \@c\ Cos[d] + 4\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  4\ \@i\ Cos[j - \(2\ π\)\/3])\), \ \((a - b + 
                  4\ \@g\ Cos[h] + 4\ \@k\ Cos[l] + 
                  2\ \@c\ Sin[d])\), \ \((a - b + 
                  4\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  4\ \@k\ Cos[l + \(2\ π\)\/3] + 
                  2\ \@c\ Sin[d])\), \ \((a - b + 
                  4\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  4\ \@k\ Cos[l - \(2\ π\)\/3] + 
                  2\ \@c\ Sin[d])\), \ \((a + b - 2\ \@c\ Cos[d] + 
                  4\ \@e\ Cos[f] - 4\ \@i\ Cos[j])\), \ \((a + b - 
                  2\ \@c\ Cos[d] + 4\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  4\ \@i\ Cos[j + \(2\ π\)\/3])\), \ \((a + b - 
                  2\ \@c\ Cos[d] + 4\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  4\ \@i\ Cos[j - \(2\ π\)\/3])\), \ \((a - b + 
                  4\ \@g\ Cos[h] - 4\ \@k\ Cos[l] - 
                  2\ \@c\ Sin[d])\), \ \((a - b + 
                  4\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  4\ \@k\ Cos[l + \(2\ π\)\/3] - 
                  2\ \@c\ Sin[d])\), \ \((a - b + 
                  4\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  4\ \@k\ Cos[l - \(2\ π\)\/3] - 2\ \@c\ Sin[d])\)}/
            12, {1, 3, 2, 10, 11, 12, 7, 9, 8, 4, 5, 6, 1, 1, 1, 2, 2}, {1, 
            1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }, };\)\)

\(\(Hoop["C3C4"] = {12, 
          2, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3}, {5, 6, 4, 8, 9, 7, 11, 12, 
              10, 2, 3, 1}, {6, 4, 5, 9, 7, 8, 12, 10, 11, 3, 1, 2}, {7, 8, 
              9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 
              1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 11, 
              12, 1, 2, 3, 4, 5, 6, 7, 8, 9}, {11, 12, 10, 2, 3, 1, 5, 6, 4, 
              8, 9, 7}, {12, 10, 11, 3, 1, 2, 6, 4, 5, 9, 7, 8}}, {a + b + 
              c + d + e + f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - 
              l, \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2, \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/
              2, \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/
              2, \((a - b - g + h)\)\^2 + \((a - b - g + h)\)\ \((b - c - h + 
                    i)\) + \((b - c - h + i)\)\^2 + \((d - e - j + k)\)\^2 + \((d - e - j + k)\)\ \((e - f - k + 
                    l)\) + \((e - f - k + l)\)\^2 + \@3\ \((\((b - c - h + 
                          i)\)\ \((d - e - j + k)\) - \((a - b - g + 
                          h)\)\ \((e - f - k + 
                          l)\))\), \((a - b - g + h)\)\^2 + \((a - b - g + 
                    h)\)\ \((b - c - h + 
                    i)\) + \((b - c - h + i)\)\^2 + \((d - e - j + k)\)\^2 + \((d - e - j + k)\)\ \((e - f - k + 
                    l)\) + \((e - f - k + l)\)\^2 - \@3\ \((\((b - c - h + 
                          i)\)\ \((d - e - j + k)\) - \((a - b - g + 
                          h)\)\ \((e - f - k + l)\))\)}, {a + b + c + d + e + 
              f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - 
              l, \((a + b + c - g - h - i)\)\^2 + \((d + e + f - j - k - l)\)\^2, ArcTan[a + b + c - g - h - i, 
              d + e + f - j - k - 
                l], \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/2, 
            ArcTan[
              2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - 
                l, \(-\@3\)\ \((b - c + e - f + h - i + k - 
                    l)\)], \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + 
                l, \(-\@3\)\ \((b - c - e + f + h - i - k + 
                    l)\)], \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + 
                l, \(-\@3\)\ \((b - c + e - f - h + i - k + 
                    l)\)], \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - 
                l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}, {\ \((a + 
                  b + 2\ \@c\ Cos[d] + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] + 
                  2\ \@i\ Cos[j] + 2\ \@k\ Cos[l])\), \ \((a + b + 
                  2\ \@c\ Cos[d] + 2\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a + b + 
                  2\ \@c\ Cos[d] + 2\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\), \ \((a - b + 
                  2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] - 
                  2\ \@k\ Cos[l] + 2\ \@c\ Sin[d])\), \ \((a - b + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l + \(2\ π\)\/3] + 
                  2\ \@c\ Sin[d])\), \ \((a - b + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l - \(2\ π\)\/3] + 
                  2\ \@c\ Sin[d])\), \ \((a + b - 2\ \@c\ Cos[d] + 
                  2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] - 
                  2\ \@k\ Cos[l])\), \ \((a + b - 2\ \@c\ Cos[d] + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a + b - 
                  2\ \@c\ Cos[d] + 2\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\), \((a - b + 
                  2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] + 
                  2\ \@k\ Cos[l] - 2\ \@c\ Sin[d])\), \ \((a - b + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l + \(2\ π\)\/3] - 
                  2\ \@c\ Sin[d])\), \((a - b + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l - \(2\ π\)\/3] - 2\ \@c\ Sin[d])\)}/
            12, {1, 3, 2, 10, 12, 11, 7, 9, 8, 4, 6, 5, 1, 1, 1, 1, 1, 1, 1, 
            1}, {1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 0. , 1, 0. }, {a, c, b, d, 
            f, e, g, i, h, j, l, k}};\)\)

\(\(Hoop["A4"] = {12, 
          3, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              11, 9, 1, 8, 12, 10, 5, 3, 7, 2, 6}, {5, 12, 7, 2, 9, 10, 11, 
              6, 1, 8, 3, 4}, {6, 10, 8, 3, 7, 11, 12, 4, 2, 9, 1, 5}, {7, 5, 
              12, 10, 2, 9, 1, 11, 6, 4, 8, 3}, {8, 6, 10, 11, 3, 7, 2, 12, 
              4, 5, 9, 1}, {9, 4, 11, 12, 1, 8, 3, 10, 5, 6, 7, 2}, {10, 8, 
              6, 7, 11, 3, 4, 2, 12, 1, 5, 9}, {11, 9, 4, 8, 12, 1, 5, 3, 10, 
              2, 6, 7}, {12, 7, 5, 9, 10, 2, 6, 1, 11, 3, 4, 8}}, {a + b + 
              c + d + e + f + g + h + i + j + k + 
              l, \ \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/
              2, \((a + d - g - j)\)\ \((a - d + g - j)\)\ \((a - d - g + 
                    j)\) + \((b + e - h - k)\)\ \((b - e + h - k)\)\ \((b - 
                    e - h + k)\) - \((a - d + g - j)\)\ \((b - e - h + 
                    k)\)\ \((c + f - i - l)\) - \((a - d - g + j)\)\ \((b + 
                    e - h - k)\)\ \((c - f + i - l)\) - \((a + d - g - 
                    j)\)\ \((b - e + h - k)\)\ \((c - f - i + l)\) + \((c + 
                    f - i - l)\)\ \((c - f + i - l)\)\ \((c - f - i + 
                    l)\)}, , , {1, 3, 2, 4, 9, 11, 7, 12, 5, 10, 6, 8, 1, 1, 
            3}, , };\)\)

\(\(Hoop["D3C2"] = {12, 
          4, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 
              3, 1, 5, 6, 4, 8, 9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 
              8, 12, 10, 11}, {4, 6, 5, 1, 3, 2, 10, 12, 11, 7, 9, 8}, {5, 4, 
              6, 2, 1, 3, 11, 10, 12, 8, 7, 9}, {6, 5, 4, 3, 2, 1, 12, 11, 
              10, 9, 8, 7}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 
              7, 11, 12, 10, 2, 3, 1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 
              2, 6, 4, 5}, {10, 12, 11, 7, 9, 8, 4, 6, 5, 1, 3, 2}, {11, 10, 
              12, 8, 7, 9, 5, 4, 6, 2, 1, 3}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 
              3, 2, 1}}, {a + b + c + d + e + f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - l, 
            a + b + c + d + e + f - g - h - i - j - k - l, 
            a + b + c - d - e - f - g - h - i + j + k + 
              l, \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \((\(-a\) + c - g + i)\)\^2 - \((\(-d\) + e - j + k)\)\^2 - \((d - f + j - l)\)\^2 - \((\(-e\) + f - k + l)\)\^2)\)/
              2, \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\)\^2 + \((b - c - h + i)\)\^2 - \((\(-d\) + e + j - k)\)\^2 - \((\(-e\) + f + k - l)\)\^2 - \((d - f - j + l)\)\^2)\)/2}, {a + b + c + d + e + f + g + h + i + 
              j + k + l, a + b + c - d - e - f + g + h + i - j - k - l, 
            a + b + c + d + e + f - g - h - i - j - k - l, 
            a + b + c - d - e - f - g - h - i + j + k + 
              l, \((\((a - b + g - h)\)\^2 + \((b - c + h - i)\)\^2 + \((\(-a\) + c - g + i)\)\^2)\)/2, 
            ArcTan[2\ a - b - c + 2\ g - h - 
                i, \(-\@3\)\ \((b - c + h - 
                    i)\)], \((\((\(-d\) + e - j + k)\)\^2 + \((d - f + j - l)\)\^2 + \((\(-e\) + f - k + l)\)\^2)\)/2, 
            ArcTan[\(-2\)\ d + e + f - 2\ j + k + 
                l, \(-\@3\)\ \((\(-e\) + f - k + 
                    l)\)], \((\((a - b - g + h)\)\^2 + \((\(-a\) + c + g - i)\)\^2 + \((b - c - h + i)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ g + h + 
                i, \(-\@3\)\ \((b - c - h + 
                    i)\)], \((\((\(-d\) + e + j - k)\)\^2 + \((\(-e\) + f + k - l)\)\^2 + \((d - f - j + l)\)\^2)\)/2, 
            ArcTan[\(-2\)\ d + e + f + 2\ j - k - 
                l, \(-\@3\)\ \((\(-e\) + f + k - l)\)]}, {\ \((a + b + c + 
                  d + 4\ \@e\ Cos[f] + 4\ \@i\ Cos[j])\), \ \((a + b + c + 
                  d + 4\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  4\ \@i\ Cos[j + \(2\ π\)\/3])\), \ \((a + b + c + d + 
                  4\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  4\ \@i\ Cos[j - \(2\ π\)\/3])\), \ \((a - b + c - d - 
                  4\ \@g\ Cos[h] - 4\ \@k\ Cos[l])\), \ \((a - b + c - d - 
                  4\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  4\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a - b + c - d - 
                  4\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  4\ \@k\ Cos[l - \(2\ π\)\/3])\), \ \((a + b - c - d + 
                  4\ \@e\ Cos[f] - 4\ \@i\ Cos[j])\), \ \((a + b - c - d + 
                  4\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  4\ \@i\ Cos[j + \(2\ π\)\/3])\), \ \((a + b - c - d + 
                  4\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  4\ \@i\ Cos[j - \(2\ π\)\/3])\), \ \((a - b - c + d - 
                  4\ \@g\ Cos[h] + 4\ \@k\ Cos[l])\), \ \((a - b - c + d - 
                  4\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  4\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a - b - c + d - 
                  4\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  4\ \@k\ Cos[l - \(2\ π\)\/3])\)}/12, {1, 3, 2, 4, 5, 6, 
            7, 9, 8, 10, 11, 12, 1, 1, 1, 1, 2, 2}, {1, 1, 1, 1, 1, 0. , 1, 
            0. , 1, 0. , 1, 0. }, };\)\)

\(\(Hoop["C3K"] = {12, 
          5, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              5, 6, 1, 2, 3, 10, 11, 12, 7, 8, 9}, {5, 6, 4, 2, 3, 1, 11, 12, 
              10, 8, 9, 7}, {6, 4, 5, 3, 1, 2, 12, 10, 11, 9, 7, 8}, {7, 8, 
              9, 10, 11, 12, 1, 2, 3, 4, 5, 6}, {8, 9, 7, 11, 12, 10, 2, 3, 
              1, 5, 6, 4}, {9, 7, 8, 12, 10, 11, 3, 1, 2, 6, 4, 5}, {10, 11, 
              12, 7, 8, 9, 4, 5, 6, 1, 2, 3}, {11, 12, 10, 8, 9, 7, 5, 6, 4, 
              2, 3, 1}, {12, 10, 11, 9, 7, 8, 6, 4, 5, 3, 1, 2}}, {a + b + 
              c + d + e + f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - l, 
            a + b + c + d + e + f - g - h - i - j - k - l, 
            a + b + c - d - e - f - g - h - i + j + k + 
              l, \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/
              2, \ \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/
              2, \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)/
              2, \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + l)\)\^2)\)/
              2}, {a + b + c + d + e + f + g + h + i + j + k + l, 
            a + b + c - d - e - f + g + h + i - j - k - l, 
            a + b + c + d + e + f - g - h - i - j - k - l, 
            a + b + c - d - e - f - g - h - i + j + k + 
              l, \((\((a - b + d - e + g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f - g + i - j + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c + 2\ d - e - f + 2\ g - h - i + 2\ j - k - 
                l, \(-\@3\)\ \((b - c + e - f + h - i + k - 
                    l)\)], \((\((a - b - d + e + g - h - j + k)\)\^2 + \((\(-a\) + c + d - f - g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ d + e + f + 2\ g - h - i - 2\ j + k + 
                l, \(-\@3\)\ \((b - c - e + f + h - i - k + 
                    l)\)], \((\((a - b + d - e - g + h - j + k)\)\^2 + \((\(-a\) + c - d + f + g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c + 2\ d - e - f - 2\ g + h + i - 2\ j + k + 
                l, \(-\@3\)\ \((b - c + e - f - h + i - k + 
                    l)\)], \((\((a - b - d + e - g + h + j - k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f + g - i - j + l)\)\^2)\)/2, 
            ArcTan[2\ a - b - c - 2\ d + e + f - 2\ g + h + i + 2\ j - k - 
                l, \(-\@3\)\ \((b - c - e + f - h + i + k - l)\)]}, {\((a + 
                  b + c + d + 2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] + 
                  2\ \@i\ Cos[j] + 2\ \@k\ Cos[l])\), \ \((a + b + c + d + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a + b + c + d + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\), \ \((a - b + c - d + 
                  2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] + 2\ \@i\ Cos[j] - 
                  2\ \@k\ Cos[l])\), \ \((a - b + c - d + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a - b + c - d + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] + 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\), \ \((a + b - c - d + 
                  2\ \@e\ Cos[f] + 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] - 
                  2\ \@k\ Cos[l])\), \ \((a + b - c - d + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a + b - c - d + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] + 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] - 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\), \ \((a - b - c + d + 
                  2\ \@e\ Cos[f] - 2\ \@g\ Cos[h] - 2\ \@i\ Cos[j] + 
                  2\ \@k\ Cos[l])\), \ \((a - b - c + d + 
                  2\ \@e\ Cos[f + \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h + \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j + \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l + \(2\ π\)\/3])\), \ \((a - b - c + d + 
                  2\ \@e\ Cos[f - \(2\ π\)\/3] - 
                  2\ \@g\ Cos[h - \(2\ π\)\/3] - 
                  2\ \@i\ Cos[j - \(2\ π\)\/3] + 
                  2\ \@k\ Cos[l - \(2\ π\)\/3])\)}/12, {1, 3, 2, 4, 6, 5, 
            7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 
            0.`, 1, 0.`, 1, 0.`, 1, 0.`}, {a, b, c, \(-d\), \(-e\), \(-f\), 
            g, h, i, \(-j\), \(-k\), \(-l\)}};\)\)

\(\(Hoop["C3Qr"] = {12, 
          7, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              5, 6, \(-1\), \(-2\), \(-3\), \(-10\), \(-11\), \(-12\), 7, 8, 
              9}, {5, 6, 
              4, \(-2\), \(-3\), \(-1\), \(-11\), \(-12\), \(-10\), 8, 9, 
              7}, {6, 4, 
              5, \(-3\), \(-1\), \(-2\), \(-12\), \(-10\), \(-11\), 9, 7, 
              8}, {7, 8, 9, 10, 11, 
              12, \(-1\), \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, 7, 
              11, 12, 10, \(-2\), \(-3\), \(-1\), \(-5\), \(-6\), \(-4\)}, {9, 7, 8, 12, 10, 11, \(-3\), \(-1\), \(-2\), \(-6\), \(-4\), \(-5\)}, {10, 
              11, 12, \(-7\), \(-8\), \(-9\), 4, 5, 
              6, \(-1\), \(-2\), \(-3\)}, {11, 12, 
              10, \(-8\), \(-9\), \(-7\), 5, 6, 
              4, \(-2\), \(-3\), \(-1\)}, {12, 10, 
              11, \(-9\), \(-7\), \(-8\), 6, 4, 
              5, \(-3\), \(-1\), \(-2\)}}, {\((a + b + c)\)\^2 + \((d + e + f)\)\^2 + \((g + h + i)\)\^2 + \((j + k + l)\)\^2, \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 + \((d - e)\)\^2 + \((e - f)\)\^2 + \((\(-d\) + f)\)\^2 + \((g - h)\)\^2 + \((h - i)\)\^2 + \((\(-g\) + i)\)\^2 + \((j - k)\)\^2 + \((k - l)\)\^2 + \((\(-j\) + l)\)\^2)\)\^2/4 - 
              3\ \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((b\ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 + \((e\ g - f\ g - d\ h + f\ h + d\ i - e\ i)\)\^2 + \((b\ j - c\ j - a\ k + c\ k + a\ l - b\ l)\)\^2 + \((\(-e\)\ j + f\ j + d\ k - f\ k - d\ l + e\ l)\)\^2 + \((h\ j - i\ j - g\ k + i\ k + g\ l - h\ l)\)\^2)\)}, , , {1, 3, 
            2, \(-4\), \(-6\), \(-5\), \(-7\), \(-9\), \(-8\), \(-10\), \(-12\), \(-11\), 2, 2}, , {a, c, 
            b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \(-l\), \(-k\)}};\)\)

\(\(Hoop["C3P4"] = {12, 
          8, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 4, 8, 
              9, 7, 11, 12, 10}, {3, 1, 2, 6, 4, 5, 9, 7, 8, 12, 10, 11}, {4, 
              5, 6, 1, 2, 3, 10\ I, 11\ I, 
              12\ I, \(-7\)\ I, \(-8\)\ I, \(-9\)\ I}, {5, 6, 4, 2, 3, 1, 11\ I, 
              12\ I, 
              10\ I, \(-8\)\ I, \(-9\)\ I, \(-7\)\ I}, {6, 4, 5, 3, 1, 2, 12\ I, 
              10\ I, 
              11\ I, \(-9\)\ I, \(-7\)\ I, \(-8\)\ I}, {7, 8, 
              9, \(-10\)\ I, \(-11\)\ I, \(-12\)\ I, 1, 2, 3, 4\ I, 5\ I, 
              6\ I}, {8, 9, 
              7, \(-11\)\ I, \(-12\)\ I, \(-10\)\ I, 2, 3, 1, 5\ I, 6\ I, 
              4\ I}, {9, 7, 
              8, \(-12\)\ I, \(-10\)\ I, \(-11\)\ I, 3, 1, 2, 6\ I, 4\ I, 
              5\ I}, {10, 11, 12, 7\ I, 
              8\ I, 
              9\ I, \(-4\)\ I, \(-5\)\ I, \(-6\)\ I, 1, 2, 3}, {11, 12, 10, 8\ I, 
              9\ I, 
              7\ I, \(-5\)\ I, \(-6\)\ I, \(-4\)\ I, 2, 3, 1}, {12, 10, 11, 9\ I, 
              7\ I, 
              8\ I, \(-6\)\ I, \(-4\)\ I, \(-5\)\ I, 3, 1, 
              2}}, {\((a + b + c)\)\^2 - \((d + e + f)\)\^2 - \((g + h + i)\)\^2 - \((j + k + l)\)\^2, \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((\(-d\) + f)\)\^2 - \((g - h)\)\^2 - \((h - i)\)\^2 - \((\(-g\) + i)\)\^2 - \((j - k)\)\^2 - \((k - l)\)\^2 - \((\(-j\) + l)\)\^2)\)\^2/4 + 
              3\ \((\((b\ d - c\ d - a\ e + c\ e + a\ f - b\ f)\)\^2 + \((b\ g - c\ g - a\ h + c\ h + a\ i - b\ i)\)\^2 - \((e\ g - f\ g - d\ h + f\ h + d\ i - e\ i)\)\^2 + \((b\ j - c\ j - a\ k + c\ k + a\ l - b\ l)\)\^2 - \((\(-e\)\ j + f\ j + d\ k - f\ k - d\ l + e\ l)\)\^2 - \((h\ j - i\ j - g\ k + i\ k + g\ l - h\ l)\)\^2)\)}, , , {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 2, 
            2}, {a, c, 
            b, \(-d\), \(-f\), \(-e\), \(-g\), \(-i\), \(-h\), \(-j\), \(-l\), \(-k\)}, };\)\)

\(\(Hoop["Q12C2c"] = {12, 
          14, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 4, 5, 6, 1, 
              12, 7, 8, 9, 10, 11}, {3, 4, 5, 6, 1, 2, 11, 12, 7, 8, 9, 
              10}, {4, 5, 6, 1, 2, 3, 10, 11, 12, 7, 8, 9}, {5, 6, 1, 2, 3, 
              4, 9, 10, 11, 12, 7, 8}, {6, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 
              7}, {7, 8, 9, 10, 11, 
              12, \(-1\), \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, 10, 
              11, 12, 7, \(-6\), \(-1\), \(-2\), \(-3\), \(-4\), \(-5\)}, {9, 
              10, 11, 12, 7, 
              8, \(-5\), \(-6\), \(-1\), \(-2\), \(-3\), \(-4\)}, {10, 11, 
              12, 7, 8, 
              9, \(-4\), \(-5\), \(-6\), \(-1\), \(-2\), \(-3\)}, {11, 12, 7, 
              8, 9, 10, \(-3\), \(-4\), \(-5\), \(-6\), \(-1\), \(-2\)}, {12, 
              7, 8, 9, 10, 
              11, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-1\)}}, {\((a - b + c - d + e - f)\)\^2 + \((g - h + i - j + k - l)\)\^2, 
            1\/2\ \((\((a - b + d - e)\)\^2 + \((a - c + d - f)\)\^2 + \((b - c + e - f)\)\^2 + \((g - h + j - k)\)\^2 + \((g - i + j - l)\)\^2 + \((h - i + k - l)\)\^2)\), 
            1\/2\ \((\((a + b - d - e)\)\^2 + \((b + c - e - f)\)\^2 + \((a - c - d + f)\)\^2 + \((g + h - j - k)\)\^2 + \((h + i - k - l)\)\^2 + \((g - i - j + l)\)\^2)\), \((a + b + c + d + e + f)\)\^2 + \((g + h + i + j + k + l)\)\^2}, , , {1, 6, 5, 4, 3, 
            2, \(-7\), \(-8\), \(-9\), \(-10\), \(-11\), \(-12\), 1, 2, 2, 
            1}, , };\)\)

\(\(Hoop["C3KC2r"] = {12, 
          19, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 3, 1, 5, 6, 
              4, \(-8\), 9, \(-7\), 11, 12, 10}, {3, 1, 2, 6, 4, 
              5, \(-9\), \(-7\), 8, 12, 10, 11}, {4, 5, 6, 1, 2, 3, \(-10\), 
              11, 12, \(-7\), 8, 9}, {5, 6, 4, 2, 3, 1, \(-11\), 12, 10, 8, 
              9, \(-7\)}, {6, 4, 5, 3, 1, 2, \(-12\), 10, 11, 9, \(-7\), 
              8}, {7, \(-8\), \(-9\), \(-10\), \(-11\), \(-12\), 
              1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\)}, {8, 9, \(-7\), 11, 
              12, 10, \(-2\), 3, 1, 5, 6, 4}, {9, \(-7\), 8, 12, 10, 
              11, \(-3\), 1, 2, 6, 4, 5}, {10, 11, 12, \(-7\), 8, 9, \(-4\), 
              5, 6, 1, 2, 3}, {11, 12, 10, 8, 9, \(-7\), \(-5\), 6, 4, 2, 3, 
              1}, {12, 10, 11, 9, \(-7\), 8, \(-6\), 4, 5, 3, 1, 2}}, {a + 
              b + c + d + e + f + g - h - i - j - k - l, 
            a + b + c - d - e - f - g + h + i - j - k - l, 
            a + b + c - d - e - f + g - h - i + j + k + l, 
            a + b + c + d + e + f - g + h + i + j + k + l, 
            1\/2\ \((\((a - b + d - e - g - h + j - k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((\(-a\) + c - d + f + g + i - j + l)\)\^2)\), 
            1\/2\ \((\((a - b - d + e + g + h + j - k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((\(-a\) + c + d - f - g - i - j + l)\)\^2)\), 
            1\/2\ \((\((a - b - d + e - g - h - j + k)\)\^2 + \((\(-a\) + c + d - f + g + i + j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), 
            1\/2\ \((\((a - b + d - e + g + h - j + k)\)\^2 + \((\(-a\) + c - d + f - g - i + j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)}, , , {1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1}, , {a, c, b, 
            d, f, e, g, i, h, j, l, k}};\)\)

\(\(Hoop["C6Kr"] = {12, 
          20, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, \(-3\), 1, 
              5, \(-6\), 4, 8, \(-9\), 7, 11, \(-12\), 10}, {3, 1, \(-2\), 6, 
              4, \(-5\), 9, 7, \(-8\), 12, 10, \(-11\)}, {4, 5, 6, 1, 2, 3, 
              10, 11, 12, 7, 8, 9}, {5, \(-6\), 4, 2, \(-3\), 1, 11, \(-12\), 
              10, 8, \(-9\), 7}, {6, 4, \(-5\), 3, 1, \(-2\), 12, 
              10, \(-11\), 9, 7, \(-8\)}, {7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 
              5, 6}, {8, \(-9\), 7, 11, \(-12\), 10, 2, \(-3\), 1, 5, \(-6\), 
              4}, {9, 7, \(-8\), 12, 10, \(-11\), 3, 1, \(-2\), 6, 
              4, \(-5\)}, {10, 11, 12, 7, 8, 9, 4, 5, 6, 1, 2, 
              3}, {11, \(-12\), 10, 8, \(-9\), 7, 5, \(-6\), 4, 2, \(-3\), 
              1}, {12, 10, \(-11\), 9, 7, \(-8\), 6, 4, \(-5\), 3, 
              1, \(-2\)}}, {a - b - c + d - e - f + g - h - i + j - k - l, 
            a - b - c - d + e + f - g + h + i + j - k - l, 
            a - b - c - d + e + f + g - h - i - j + k + l, 
            a - b - c + d - e - f - g + h + i - j + k + l, 
            1\/2\ \((\((a + b - d - e - g - h + j + k)\)\^2 + \((b - c - e + f - h + i + k - l)\)\^2 + \((a + c - d - f - g - i + j + l)\)\^2)\), 
            1\/2\ \((\((a + b - d - e + g + h - j - k)\)\^2 + \((a + c - d - f + g + i - j - l)\)\^2 + \((b - c - e + f + h - i - k + l)\)\^2)\), 
            1\/2\ \((\((a + b + d + e + g + h + j + k)\)\^2 + \((b - c + e - f + h - i + k - l)\)\^2 + \((a + c + d + f + g + i + j + l)\)\^2)\), 
            1\/2\ \((\((a + b + d + e - g - h - j - k)\)\^2 + \((a + c + d + f - g - i - j - l)\)\^2 + \((b - c + e - f - h + i - k + l)\)\^2)\)}, , , {1, 
            3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 1, 1, 1, 1, 1, 1, 1, 
            1}, , {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)

\(\(Hoop["C4C4"] = {16, 
          2, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 3, 
              4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13}, {3, 4, 1, 2, 
              7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14}, {4, 1, 2, 3, 8, 5, 
              6, 7, 12, 9, 10, 11, 16, 13, 14, 15}, {5, 6, 7, 8, 9, 10, 11, 
              12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 7, 8, 5, 10, 11, 12, 9, 
              14, 15, 16, 13, 2, 3, 4, 1}, {7, 8, 5, 6, 11, 12, 9, 10, 15, 
              16, 13, 14, 3, 4, 1, 2}, {8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 
              14, 15, 4, 1, 2, 3}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 
              4, 5, 6, 7, 8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 
              7, 8, 5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 
              6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 
              7}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 
              12}, {14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 
              9}, {15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 
              10}, {16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 
              11}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((e + f + g + h - m - n - o - p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((e - f + g - h - m + n - o + p)\)\^2, \((a - c + e - g + i - k + m - o)\)\^2 + \((b - d + f - h + j - l + n - p)\)\^2, \((a - c - e + g + i - k - m + o)\)\^2 + \((b - d - f + h + j - l - n + p)\)\^2, \((b - d - e + g - j + l + m - o)\)\^2 + \((a - c + f - h - i + k - n + p)\)\^2, \((b - d + e - g - j + l - m + o)\)\^2 + \((a - c - f + h - i + k + n - p)\)\^2}, , , {1, 4, 3, 2, 13, 
            16, 15, 14, 9, 12, 11, 10, 5, 8, 7, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
            1}, , };\)\)

\(\(Hoop["KiC4"] = {16, 
          3, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 
              16, 15}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 
              14}, {4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 
              13}, {5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 4, 3, 
              2}, {6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 
              1}, {7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 3, 2, 1, 
              4}, {8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 
              3}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 
              7}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 
              6}, {12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 
              5}, {13, 16, 15, 14, 1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 
              10}, {14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 
              9}, {15, 14, 13, 16, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 
              12}, {16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 
              11}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((\(-e\) - f - g - h + m + n + o + p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((\(-e\) + f - g + h + m - n + o - p)\)\^2, \((a - c + i - k)\)\^2 - \((b - d + j - l)\)\^2 - \((e - g + m - o)\)\^2 + \((f - h + n - p)\)\^2, \((a - c - i + k)\)\^2 - \((b - d - j + l)\)\^2 + \((e - g - m + o)\)\^2 - \((f - h - n + p)\)\^2}, , , {1, 2, 3, 4, 13, 16, 15, 14, 9, 10, 11, 12, 5, 8, 7, 6, 1, 1, 1, 
            1, 1, 1, 2, 2}, , };\)\)

\(\(Hoop["C4iC4"] = {16, 
          4, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 
              16, 13}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 
              14}, {4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 
              15}, {5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 4, 3, 
              2}, {6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 
              3}, {7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 3, 2, 1, 
              4}, {8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 
              1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 
              5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 
              6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 
              7}, {13, 16, 15, 14, 1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 
              10}, {14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 
              11}, {15, 14, 13, 16, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 
              12}, {16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 
              9}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, \((a + b + c + d - i - j - k - l)\)\^2 + \((\(-e\) - f - g - h + m + n + o + p)\)\^2, \((a - b + c - d - i + j - k + l)\)\^2 + \((\(-e\) + f - g + h + m - n + o - p)\)\^2, \((a - c + i - k)\)\^2 + \((b - d + j - l)\)\^2 - \((e - g + m - o)\)\^2 - \((f - h + n - p)\)\^2, \((a - c - i + k)\)\^2 + \((b - d - j + l)\)\^2 + \((e - g - m + o)\)\^2 + \((f - h - n + p)\)\^2}, , , {1, 4, 3, 2, 13, 14, 15, 16, 9, 12, 11, 10, 5, 6, 7, 8, 1, 1, 1, 
            1, 1, 1, 2, 2}, , };\)\)

\(\(Hoop["D8"] = {16, 
          7, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 1, 
              16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3}, {3, 4, 5, 6, 
              7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2}, {4, 3, 2, 1, 16, 
              15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5}, {5, 6, 7, 8, 9, 10, 11, 
              12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 5, 4, 3, 2, 1, 16, 15, 14, 
              13, 12, 11, 10, 9, 8, 7}, {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 
              1, 2, 3, 4, 5, 6}, {8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 
              11, 10, 9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 
              7, 8}, {10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 
              11}, {11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 
              10}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 
              13}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 
              12}, {14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 
              15}, {15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14}, {16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 
              1}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, \((a - e + i - m)\)\^2 - \((b - f + j - n)\)\^2 + \((c - g + k - o)\)\^2 - \((d - h + l - p)\)\^2, \(-2\)\ \((\((a - i)\)\ \((c - k)\) - \((b - j)\)\ \((d - l)\) + \((c - k)\)\ \((e - m)\) - \((d - l)\)\ \((f - n)\) - \((a - i)\)\ \((g - o)\) + \((e - m)\)\ \((g - o)\) + \((b - j)\)\ \((h - p)\) - \((f - n)\)\ \((h - p)\))\)\^2 + \((\((a - i)\)\^2 - \((b - j)\)\^2 + \((c - k)\)\^2 - \((d - l)\)\^2 + \((e - m)\)\^2 - \((f - n)\)\^2 + \((g - o)\)\^2 - \((h - p)\)\^2)\)\^2}, , , {1, 2, 15, 4, 13, 6, 11, 8, 9, 
            10, 7, 12, 5, 14, 3, 16, 1, 1, 1, 1, 2, 2}, gp, };\)\)

\(\(Hoop["Q16"] = {16, 
          9, {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {2, 9, 
              16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11}, {3, 4, 5, 6, 
              7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2}, {4, 11, 2, 9, 16, 
              7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13}, {5, 6, 7, 8, 9, 10, 11, 
              12, 13, 14, 15, 16, 1, 2, 3, 4}, {6, 13, 4, 11, 2, 9, 16, 7, 
              14, 5, 12, 3, 10, 1, 8, 15}, {7, 8, 9, 10, 11, 12, 13, 14, 15, 
              16, 1, 2, 3, 4, 5, 6}, {8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 
              5, 12, 3, 10, 1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 
              5, 6, 7, 8}, {10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 5, 
              12, 3}, {11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 
              10}, {12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 
              5}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 
              12}, {14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 
              7}, {15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14}, {16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 
              9}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, \((a - e + i - m)\)\^2 - \((b - f + j - n)\)\^2 + \((c - g + k - o)\)\^2 - \((d - h + l - p)\)\^2, \((a - i)\)\^4 + 
              2\ \((a - i)\)\^2\ \((b - j)\)\^2 + \((b - j)\)\^4 + 
              2\ \((b - j)\)\^2\ \((c - k)\)\^2 + \((c - k)\)\^4 - 
              4\ \((a - i)\)\ \((b - j)\)\ \((c - k)\)\ \((d - l)\) + 
              2\ \((a - i)\)\^2\ \((d - l)\)\^2 + 
              2\ \((c - k)\)\^2\ \((d - l)\)\^2 + \((d - l)\)\^4 - 
              4\ \((a - i)\)\ \((c - k)\)\^2\ \((e - m)\) - 
              4\ \((b - j)\)\ \((c - k)\)\ \((d - l)\)\ \((e - m)\) + 
              2\ \((a - i)\)\^2\ \((e - m)\)\^2 + 
              2\ \((b - j)\)\^2\ \((e - m)\)\^2 + 
              2\ \((d - l)\)\^2\ \((e - m)\)\^2 + \((e - m)\)\^4 - 
              4\ \((a - i)\)\ \((c - k)\)\ \((d - l)\)\ \((f - n)\) - 
              4\ \((b - j)\)\ \((d - l)\)\^2\ \((f - n)\) - 
              4\ \((c - k)\)\ \((d - l)\)\ \((e - m)\)\ \((f - n)\) + 
              2\ \((a - i)\)\^2\ \((f - n)\)\^2 + 
              2\ \((b - j)\)\^2\ \((f - n)\)\^2 + 
              2\ \((c - k)\)\^2\ \((f - n)\)\^2 + 
              2\ \((e - m)\)\^2\ \((f - n)\)\^2 + \((f - n)\)\^4 + 
              4\ \((a - i)\)\^2\ \((c - k)\)\ \((g - o)\) + 
              4\ \((a - i)\)\ \((b - j)\)\ \((d - l)\)\ \((g - o)\) - 
              4\ \((b - j)\)\ \((d - l)\)\ \((e - m)\)\ \((g - o)\) - 
              4\ \((c - k)\)\ \((e - m)\)\^2\ \((g - o)\) + 
              4\ \((a - i)\)\ \((d - l)\)\ \((f - n)\)\ \((g - o)\) - 
              4\ \((d - l)\)\ \((e - m)\)\ \((f - n)\)\ \((g - o)\) + 
              2\ \((b - j)\)\^2\ \((g - o)\)\^2 + 
              2\ \((c - k)\)\^2\ \((g - o)\)\^2 + 
              2\ \((d - l)\)\^2\ \((g - o)\)\^2 + 
              4\ \((a - i)\)\ \((e - m)\)\ \((g - o)\)\^2 + 
              2\ \((f - n)\)\^2\ \((g - o)\)\^2 + \((g - o)\)\^4 + 
              4\ \((a - i)\)\ \((b - j)\)\ \((c - k)\)\ \((h - p)\) + 
              4\ \((b - j)\)\^2\ \((d - l)\)\ \((h - p)\) + 
              4\ \((b - j)\)\ \((c - k)\)\ \((e - m)\)\ \((h - p)\) - 
              4\ \((a - i)\)\ \((c - k)\)\ \((f - n)\)\ \((h - p)\) - 
              4\ \((c - k)\)\ \((e - m)\)\ \((f - n)\)\ \((h - p)\) - 
              4\ \((d - l)\)\ \((f - n)\)\^2\ \((h - p)\) - 
              4\ \((a - i)\)\ \((b - j)\)\ \((g - o)\)\ \((h - p)\) + 
              4\ \((b - j)\)\ \((e - m)\)\ \((g - o)\)\ \((h - p)\) + 
              4\ \((a - i)\)\ \((f - n)\)\ \((g - o)\)\ \((h - p)\) - 
              4\ \((e - m)\)\ \((f - n)\)\ \((g - o)\)\ \((h - p)\) + 
              2\ \((a - i)\)\^2\ \((h - p)\)\^2 + 
              2\ \((c - k)\)\^2\ \((h - p)\)\^2 + 
              2\ \((d - l)\)\^2\ \((h - p)\)\^2 + 
              2\ \((e - m)\)\^2\ \((h - p)\)\^2 + 
              4\ \((b - j)\)\ \((f - n)\)\ \((h - p)\)\^2 + 
              2\ \((g - o)\)\^2\ \((h - p)\)\^2 + \((h - p)\)\^4}, , , {1, 
            10, 15, 12, 13, 14, 11, 16, 9, 2, 7, 4, 5, 6, 3, 8, 1, 1, 1, 1, 
            2, 2}, gp, };\)\)

\(\(Hoop["C4K"] = {16, 
          10, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 3, 4, 1, 6, 7, 8, 5, 10, 11, 12, 9, 14, 15, 
              16, 13}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 
              14}, {4, 1, 2, 3, 8, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 
              15}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 
              12}, {6, 7, 8, 5, 2, 3, 4, 1, 14, 15, 16, 13, 10, 11, 12, 
              9}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 11, 12, 9, 
              10}, {8, 5, 6, 7, 4, 1, 2, 3, 16, 13, 14, 15, 12, 9, 10, 
              11}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 11, 12, 9, 14, 15, 16, 13, 2, 3, 4, 1, 6, 7, 8, 
              5}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 
              6}, {12, 9, 10, 11, 16, 13, 14, 15, 4, 1, 2, 3, 8, 5, 6, 
              7}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 
              4}, {14, 15, 16, 13, 10, 11, 12, 9, 6, 7, 8, 5, 2, 3, 4, 
              1}, {15, 16, 13, 14, 11, 12, 9, 10, 7, 8, 5, 6, 3, 4, 1, 
              2}, {16, 13, 14, 15, 12, 9, 10, 11, 8, 5, 6, 7, 4, 1, 2, 
              3}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a + b + c + d - e - f - g - h - i - j - k - l + m + n + 
              o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - 
              o + p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a - b + c - d - e + f - g + h - i + j - k + l + m - n + 
              o - p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - 
              o - p, \((a - c - e + g - i + k + m - o)\)\^2 + \((b - d - f + h - j + l + n - p)\)\^2, \((a - c + e - g - i + k - m + o)\)\^2 + \((b - d + f - h - j + l - n + p)\)\^2, \((a - c - e + g + i - k - m + o)\)\^2 + \((b - d - f + h + j - l - n + p)\)\^2, \((a - c + e - g + i - k + m - o)\)\^2 + \((b - d + f - h + j - l + n - p)\)\^2}, {a + b + c + d + e + f + g + h + i + 
              j + k + l + m + n + o + p, 
            a + b + c + d - e - f - g - h - i - j - k - l + m + n + o + p, 
            a - b + c - d + e - f + g - h - i + j - k + l - m + n - o + p, 
            a - b + c - d - e + f - g + h + i - j + k - l - m + n - o + p, 
            a - b + c - d - e + f - g + h - i + j - k + l + m - n + o - p, 
            a - b + c - d + e - f + g - h + i - j + k - l + m - n + o - p, 
            a + b + c + d - e - f - g - h + i + j + k + l - m - n - o - p, 
            a + b + c + d + e + f + g + h - i - j - k - l - m - n - o - 
              p, 
 (*ij*) \((a - c - e + g - i + k + m - o)\)\^2 + \((b - d - f + h - j + l + n - p)\)\^2, 
            If[b - d - f + h - j + l + n - p == 0, 0, 
              ArcTan[a - c - e + g - i + k + m - o, 
                b - d - f + h - j + l + n - 
                  p]], 
 (*kl*) \((a - c + e - g - i + k - m + o)\)\^2 + \((b - d + f - h - j + l - n + p)\)\^2, 
            If[b - d + f - h - j + l - n + p == 0, 0, 
              ArcTan[a - c + e - g - i + k - m + o, 
                b - d + f - h - j + l - n + 
                  p]], 
 (*mn*) \((a - c - e + g + i - k - m + o)\)\^2 + \((b - d - f + h + j - l - n + p)\)\^2, 
            If[b - d - f + h + j - l - n + p == 0, 0, 
              ArcTan[a - c - e + g + i - k - m + o, 
                b - d - f + h + j - l - n + 
                  p]], 
 (*op*) \((a - c + e - g + i - k + m - o)\)\^2 + \((b - d + f - h + j - l + n - p)\)\^2, 
            If[b - d + f - h + j - l + n - p == 0, 0, 
              ArcTan[a - c + e - g + i - k + m - o, 
                b - d + f - h + j - l + n - p]]}, { (*a*) a + b + c + d + e + 
                f + g + h + 2  Sqrt[i]\ Cos[j] + 2  Sqrt[k]\ Cos[l] + 
                2  Sqrt[m]\ Cos[n] + 
                2  Sqrt[o]\ Cos[p], 
 (*b*) a + b - c - d - 
                e - f + g + h + 2  Sqrt[i]\ Sin[j] + 2  Sqrt[k]\ Sin[l] + 
                2  Sqrt[m]\ Sin[n] + 
                2  Sqrt[o]\ Sin[p], 
 (*c*) a + b + c + d + 
                e + f + g + h - 2  Sqrt[i]\ Cos[j] - 2  Sqrt[k]\ Cos[l] - 
                2  Sqrt[m]\ Cos[n] - 
                2  Sqrt[o]\ Cos[p], 
 (*d*) a + b - c - d - 
                e - f + g + h - 2  Sqrt[i]\ Sin[j] - 2  Sqrt[k]\ Sin[l] - 
                2  Sqrt[m]\ Sin[n] - 
                2  Sqrt[o]\ Sin[p], 
 (*e*) a - b + c - d - 
                e + f - g + h - 2  Sqrt[i]\ Cos[j] + 2  Sqrt[k]\ Cos[l] - 
                2  Sqrt[m]\ Cos[n] + 
                2  Sqrt[o]\ Cos[p], 
 (*f*) a - b - c + d + 
                e - f - g + h - 2  Sqrt[i]\ Sin[j] + 2  Sqrt[k]\ Sin[l] - 
                2  Sqrt[m]\ Sin[n] + 
                2  Sqrt[o]\ Sin[p], 
 (*g*) a - b + c - d - 
                e + f - g + h + 2  Sqrt[i]\ Cos[j] - 2  Sqrt[k]\ Cos[l] + 
                2  Sqrt[m]\ Cos[n] - 
                2  Sqrt[o]\ Cos[p], 
 (*h*) a - b - c + d + 
                e - f - g + h + 2  Sqrt[i]\ Sin[j] - 2  Sqrt[k]\ Sin[l] + 
                2  Sqrt[m]\ Sin[n] - 
                2  Sqrt[o]\ Sin[p], 
 (*i*) a - b - c + d - 
                e + f + g - h - 2  Sqrt[i]\ Cos[j] - 2  Sqrt[k]\ Cos[l] + 
                2  Sqrt[m]\ Cos[n] + 
                2  Sqrt[o]\ Cos[p], 
 (*j*) a - b + c - d + 
                e - f + g - h - 2  Sqrt[i]\ Sin[j] - 2  Sqrt[k]\ Sin[l] + 
                2  Sqrt[m]\ Sin[n] + 
                2  Sqrt[o]\ Sin[p], 
 (*k*) a - b - c + d - 
                e + f + g - h + 2  Sqrt[i]\ Cos[j] + 2  Sqrt[k]\ Cos[l] - 
                2  Sqrt[m]\ Cos[n] - 
                2  Sqrt[o]\ Cos[p], 
 (*l*) a - b + c - d + 
                e - f + g - h + 2  Sqrt[i]\ Sin[j] + 2  Sqrt[k]\ Sin[l] - 
                2  Sqrt[m]\ Sin[n] - 
                2  Sqrt[o]\ Sin[p], 
 (*m*) a + b - c - d + 
                e + f - g - h + 2  Sqrt[i]\ Cos[j] - 2  Sqrt[k]\ Cos[l] - 
                2  Sqrt[m]\ Cos[n] + 
                2  Sqrt[o]\ Cos[p], 
 (*n*) a + b + c + d - 
                e - f - g - h + 2  Sqrt[i]\ Sin[j] - 2  Sqrt[k]\ Sin[l] - 
                2  Sqrt[m]\ Sin[n] + 
                2  Sqrt[o]\ Sin[p], 
 (*o*) a + b - c - d + 
                e + f - g - h - 2  Sqrt[i]\ Cos[j] + 2  Sqrt[k]\ Cos[l] + 
                2  Sqrt[m]\ Cos[n] - 
                2  Sqrt[o]\ Cos[p], 
 (*p*) a + b + c + d - 
                e - f - g - h - 2  Sqrt[i]\ Sin[j] + 2  Sqrt[k]\ Sin[l] + 
                2  Sqrt[m]\ Sin[n] - 2  Sqrt[o]\ Sin[p]}/16, {1, 4, 3, 2, 5, 
            8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 1, 1, 1, 1, 1, 1, 1, 1, 
            1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 0. , 1, 0. , 1, 0. , 1, 
            0. }, };\)\)

\(\(Hoop["D4C2"] = {16, 
          11, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 
              16, 15}, {3, 8, 5, 2, 7, 4, 1, 6, 11, 16, 13, 10, 15, 12, 9, 
              14}, {4, 7, 6, 1, 8, 3, 2, 5, 12, 15, 14, 9, 16, 11, 10, 
              13}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 
              12}, {6, 5, 8, 7, 2, 1, 4, 3, 14, 13, 16, 15, 10, 9, 12, 
              11}, {7, 4, 1, 6, 3, 8, 5, 2, 15, 12, 9, 14, 11, 16, 13, 
              10}, {8, 3, 2, 5, 4, 7, 6, 1, 16, 11, 10, 13, 12, 15, 14, 
              9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 
              7}, {11, 16, 13, 10, 15, 12, 9, 14, 3, 8, 5, 2, 7, 4, 1, 
              6}, {12, 15, 14, 9, 16, 11, 10, 13, 4, 7, 6, 1, 8, 3, 2, 
              5}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 
              4}, {14, 13, 16, 15, 10, 9, 12, 11, 6, 5, 8, 7, 2, 1, 4, 
              3}, {15, 12, 9, 14, 11, 16, 13, 10, 7, 4, 1, 6, 3, 8, 5, 
              2}, {16, 11, 10, 13, 12, 15, 14, 9, 8, 3, 2, 5, 4, 7, 6, 
              1}}, {a + b + c + d + e + f + g + h - i - j - k - l - m - n - 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + 
              o - p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - 
              o + p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + 
              o + p, a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, \((a - e + i - m)\)\^2 - \((b - f + j - n)\)\^2 + \((c - g + k - o)\)\^2 - \((d - h + l - p)\)\^2, \((a - e - i + m)\)\^2 - \((b - f - j + n)\)\^2 + \((c - g - k + o)\)\^2 - \((d - h - l + p)\)\^2}, , , {1, 2, 7, 
            4, 5, 6, 3, 8, 9, 10, 15, 12, 13, 14, 11, 16, 1, 1, 1, 1, 1, 1, 
            1, 1, 2, 2}, , {a, c, b, d, f, e, g, i, h, j, l, 
            k}};\)\)

\(\(Hoop["Q8C2"] = {16, 
          12, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 5, 4, 7, 6, 1, 8, 3, 10, 13, 12, 15, 14, 9, 
              16, 11}, {3, 8, 5, 2, 7, 4, 1, 6, 11, 16, 13, 10, 15, 12, 9, 
              14}, {4, 3, 6, 5, 8, 7, 2, 1, 12, 11, 14, 13, 16, 15, 10, 
              9}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 
              12}, {6, 1, 8, 3, 2, 5, 4, 7, 14, 9, 16, 11, 10, 13, 12, 
              15}, {7, 4, 1, 6, 3, 8, 5, 2, 15, 12, 9, 14, 11, 16, 13, 
              10}, {8, 7, 2, 1, 4, 3, 6, 5, 16, 15, 10, 9, 12, 11, 14, 
              13}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 13, 12, 15, 14, 9, 16, 11, 2, 5, 4, 7, 6, 1, 8, 
              3}, {11, 16, 13, 10, 15, 12, 9, 14, 3, 8, 5, 2, 7, 4, 1, 
              6}, {12, 11, 14, 13, 16, 15, 10, 9, 4, 3, 6, 5, 8, 7, 2, 
              1}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 
              4}, {14, 9, 16, 11, 10, 13, 12, 15, 6, 1, 8, 3, 2, 5, 4, 
              7}, {15, 12, 9, 14, 11, 16, 13, 10, 7, 4, 1, 6, 3, 8, 5, 
              2}, {16, 15, 10, 9, 12, 11, 14, 13, 8, 7, 2, 1, 4, 3, 6, 
              5}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - 
              o - p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + 
              o + p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + 
              o - p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - 
              o + p, \((a - e + i - m)\)\^2 + \((b - f + j - n)\)\^2 + \((c - g + k - o)\)\^2 + \((d - h + l - p)\)\^2, \((a - e - i + m)\)\^2 + \((b - f - j + n)\)\^2 + \((c - g - k + o)\)\^2 + \((d - h - l + p)\)\^2}, , , {1, 6, 7, 
            8, 5, 2, 3, 4, 9, 14, 15, 16, 13, 10, 11, 12, 1, 1, 1, 1, 1, 1, 
            1, 1, 2, 2}, 
          gp, {a, c, b, d, f, e, g, i, h, j, l, k}};\)\)

Hoop["P16"]={16,13,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,8,15,6,5,12,3,10,9,16,7,14,13,4,11},{3,16,1,6,7,4,5,10,11,8,9,14,15,12,13,2},{4,7,14,1,8,11,2,5,12,15,6,9,16,3,10,13},{5,6,7,8,9,10,11,12,13,14,15,16,1,2,3,4},{6,5,12,3,10,9,16,7,14,13,4,11,2,1,8,15},{7,4,5,10,11,8,9,14,15,12,13,2,3,16,1,6},{8,11,2,5,12,15,6,9,16,3,10,13,4,7,14,1},{9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8},{10,9,16,7,14,13,4,11,2,1,8,15,6,5,12,3},{11,8,9,14,15,12,13,2,3,16,1,6,7,4,5,10},{12,15,6,9,16,3,10,13,4,7,14,1,8,11,2,5},{13,14,15,16,1,2,3,4,5,6,7,8,9,10,11,12},{14,13,4,11,2,1,8,15,6,5,12,3,10,9,16,7},{15,12,13,2,3,16,1,6,7,4,5,10,11,8,9,14},{16,3,10,13,4,7,14,1,8,11,2,5,12,15,6,9}},{a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p,a-b+c-d+e-f+g-h+i-j+k-l+m-n+o-p,a+b-c-d+e+f-g-h+i+j-k-l+m+n-o-p,a-b-c+d+e-f-g+h+i-j-k+l+m-n-o+p,a+b+c-d-e-f-g+h+i+j+k-l-m-n-o+p,a+b-c+d-e-f+g-h+i+j-k+l-m-n+o-p,a-b+c+d-e+f-g-h+i-j+k+l-m+n-o-p,a-b-c-d-e+f+g+h+i-j-k-l-m+n+o+p,4*((a-i)*(e-m)-(b-j)*(f-n)-(c-k)*(g-o)-(d-l)*(h-p))^2+((a-i)^2-(b-j)^2-(c-k)^2-(d-l)^2-(e-m)^2+(f-n)^2+(g-o)^2+(h-p)^2)^2},,,{1,2,3,4,13,14,15,16,9,10,11,12,5,6,7,8,1,1,1,1,1,1,1,1,2},gp,{a,b,c,-d,e,-f,-g,-h,i,-j,-k,-l,-m,-n,-o,p}};

Hoop["KK"] = {16, 
          14, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 
              16, 15}, {3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 
              14}, {4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 
              13}, {5, 6, 7, 8, 1, 2, 3, 4, 13, 14, 15, 16, 9, 10, 11, 
              12}, {6, 5, 8, 7, 2, 1, 4, 3, 14, 13, 16, 15, 10, 9, 12, 
              11}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 11, 12, 9, 
              10}, {8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 
              9}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 
              7}, {11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 
              6}, {12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 
              5}, {13, 14, 15, 16, 9, 10, 11, 12, 5, 6, 7, 8, 1, 2, 3, 
              4}, {14, 13, 16, 15, 10, 9, 12, 11, 6, 5, 8, 7, 2, 1, 4, 
              3}, {15, 16, 13, 14, 11, 12, 9, 10, 7, 8, 5, 6, 3, 4, 1, 
              2}, {16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 
              1}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a + b - c - d - e - f + g + h + i + j - k - l - m - n + 
              o + p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, a - b - c + d - e + f + g - h + i - j - k + l - m + n + 
              o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - 
              o - p, a + b + c + d - e - f - g - h - i - j - k - l + m + n + 
              o + p, a + b - c - d - e - f + g + h - i - j + k + l + m + n - 
              o - p, a - b + c - d - e + f - g + h - i + j - k + l + m - n + 
              o - p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + 
              o + p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + 
              o - p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - 
              o + p, 
            a - b - c + d - e + f + g - h - i + j + k - l + m - n - o + 
              p}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a + b - c - d - e - f + g + h + i + j - k - l - m - n + 
              o + p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, a - b - c + d - e + f + g - h + i - j - k + l - m + n + 
              o - p, a + b + c + d + e + f + g + h - i - j - k - l - m - n - 
              o - p, a + b + c + d - e - f - g - h - i - j - k - l + m + n + 
              o + p, a + b - c - d - e - f + g + h - i - j + k + l + m + n - 
              o - p, a - b + c - d - e + f - g + h - i + j - k + l + m - n + 
              o - p, a + b - c - d + e + f - g - h - i - j + k + l - m - n + 
              o + p, a - b - c + d + e - f - g + h - i + j + k - l - m + n + 
              o - p, a - b + c - d + e - f + g - h - i + j - k + l - m + n - 
              o + p, a - b - c + d - e + f + g - h - i + j + k - l + m - n - 
              o + p}, { (a + b + c + d + e + f + g + h + i + j + k + l + 
                  m + n + o + p), (a - b - c + d - e + f + g - h + i + 
                  j + k - l + m - n - o - p),  (a - b + c - d + e - f + 
                  g - h + i + j - k + l - m - n + o - p),  (a + b - c - 
                  d - e - f + g + h + i + j - k - l - m + n - o + 
                  p),  (a + b + c + d - e - f - g - h + i - j - k - l + 
                  m + n + o - p),  (a - b - c + d + e - f - g + h + i - 
                  j - k + l + m - n - o + p), (a - b + c - d - e + f - 
                  g + h + i - j + k - l - m - n + o + p),  (a + b - c - 
                  d + e + f - g - h + i - j + k + l - m + n - o - 
                  p), (a + b + c + d + e + f + g + h - i - j - k - l - 
                  m - n - o - p), (a - b - c + d - e + f + g - h - i - 
                  j - k + l - m + n + o + p), (a - b + c - d + e - f + 
                  g - h - i - j + k - l + m + n - o + p), (a + b - c - 
                  d - e - f + g + h - i - j + k + l + m - n + o - 
                  p),  (a + b + c + d - e - f - g - h - i + j + k + l - 
                  m - n - o + p),  (a - b - c + d + e - f - g + h - i + 
                  j + k - l - m + n + o - p),  (a - b + c - d - e + f - 
                  g + h - i + j - k + l + m + n - o - p),  (a + b - c - 
                  d + e + f - g - h - i + j - k - l + m - n + o + p)}/
            16, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 1, 
            1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 
            1, 1, 1, 1, 1, 1, 1, 1, 1}, {a, c, b, d, f, e, g, i, h, j, l, 
            k}};

\(\(Hoop["Oct"] = {16, 
          15, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 9, 12, 3, 14, 5, 8, 15, 10, 1, 4, 11, 6, 13, 
              16, 7}, {3, 4, 9, 10, 15, 16, 5, 6, 11, 12, 1, 2, 7, 8, 13, 
              14}, {4, 11, 2, 9, 16, 7, 14, 5, 12, 3, 10, 1, 8, 15, 6, 
              13}, {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 
              4}, {6, 13, 8, 15, 2, 9, 4, 11, 14, 5, 16, 7, 10, 1, 12, 
              3}, {7, 16, 13, 6, 3, 12, 9, 2, 15, 8, 5, 14, 11, 4, 1, 
              10}, {8, 7, 14, 13, 4, 3, 10, 9, 16, 15, 6, 5, 12, 11, 2, 
              1}, {9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 
              8}, {10, 1, 4, 11, 6, 13, 16, 7, 2, 9, 12, 3, 14, 5, 8, 
              15}, {11, 12, 1, 2, 7, 8, 13, 14, 3, 4, 9, 10, 15, 16, 5, 
              6}, {12, 3, 10, 1, 8, 15, 6, 13, 4, 11, 2, 9, 16, 7, 14, 
              5}, {13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 
              12}, {14, 5, 16, 7, 10, 1, 12, 3, 6, 13, 8, 15, 2, 9, 4, 
              11}, {15, 8, 5, 14, 11, 4, 1, 10, 7, 16, 13, 6, 3, 12, 9, 
              2}, {16, 15, 6, 5, 12, 11, 2, 1, 8, 7, 14, 13, 4, 3, 10, 
              9}}, {a + b + c + d + e + f + g + h + i + j + k + l + m + n + 
              o + p, a - b - c + d + e - f - g + h + i - j - k + l + m - n - 
              o + p, a - b + c - d + e - f + g - h + i - j + k - l + m - n + 
              o - p, a + b - c - d + e + f - g - h + i + j - k - l + m + n - 
              o - p, a - b + c - d - e + f - g + h + i - j + k - l - m + n - 
              o + p, a + b - c - d - e - f + g + h + i + j - k - l - m - n + 
              o + p, a + b + c + d - e - f - g - h + i + j + k + l - m - n - 
              o - p, a - b - c + d - e + f + g - h + i - j - k + l - m + n + 
              o - p, \((a - i)\)\^2 + \((b - j)\)\^2 + \((c - k)\)\^2 + \((d - l)\)\^2 + \((e - m)\)\^2 + \((f - n)\)\^2 + \((g - o)\)\^2 + \((h - p)\)\^2}, , , {1, 10, 11, 12, 13, 14, 15, 16, 9, 2, 3, 4, 5, 6, 7, 8, 1, 1, 
            1, 1, 1, 1, 1, 1, 4}, , {a, c, b, d, f, e, g, i, h, j, l, 
            k}};\)\)

Hoop["r31r"]={16,17,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15},{3,-4,1,-2,7,-8,5,-6,11,-12,9,-10,15,-16,13,-14},{4,-3,2,-1,8,-7,6,-5,12,-11,10,-9,16,-15,14,-13},{5,-6,-7,8,-1,2,3,-4,13,-14,-15,16,-9,10,11,-12},{6,-5,-8,7,-2,1,4,-3,14,-13,-16,15,-10,9,12,-11},{7,8,-5,-6,-3,-4,1,2,15,16,-13,-14,-11,-12,9,10},{8,7,-6,-5,-4,-3,2,1,16,15,-14,-13,-12,-11,10,9},{9,-10,-11,12,-13,14,15,-16,1,-2,-3,4,-5,6,7,-8},{10,-9,-12,11,-14,13,16,-15,2,-1,-4,3,-6,5,8,-7},{11,12,-9,-10,-15,-16,13,14,3,4,-1,-2,-7,-8,5,6},{12,11,-10,-9,-16,-15,14,13,4,3,-2,-1,-8,-7,6,5},{13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4},{14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3},{15,-16,13,-14,11,-12,9,-10,7,-8,5,-6,3,-4,1,-2},{16,-15,14,-13,12,-11,10,-9,8,-7,6,-5,4,-3,2,-1}},{(a^2+b^2+c^2+d^2-e^2-f^2-g^2-h^2+i^2+j^2+k^2+l^2-m^2-n^2-o^2-p^2)^2-4*(-(h*i-g*j+f*k-e*l-d*m+c*n-b*o+a*p)^2+(a*i-b*j-c*k+d*l+e*m-f*n-g*o+h*p)^2-(a*e-b*f-c*g+d*h+i*m-j*n-k*o+l*p)^2+(a*c-b*d-e*g+f*h+i*k-j*l-m*o+n*p)^2+(a*b+c*d-e*f-g*h+i*j+k*l-m*n-o*p)^2)},,,{1,2,3,-4,-5,6,7,8,9,-10,-11,-12,13,14,15,-16,4},gp,{a,b,c,-d,e,-f,-g,-h,i,-j,-k,-l,-m,-n,-o,p}};

\(\(Hoop["CL4"] = {16, 
          21, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 
              16, 15}, {3, \(-4\), 1, \(-2\), 7, \(-8\), 5, \(-6\), 
              11, \(-12\), 9, \(-10\), 15, \(-16\), 13, \(-14\)}, {4, \(-3\), 
              2, \(-1\), 8, \(-7\), 6, \(-5\), 12, \(-11\), 10, \(-9\), 
              16, \(-15\), 14, \(-13\)}, {5, \(-6\), \(-7\), 8, 
              1, \(-2\), \(-3\), 4, 13, \(-14\), \(-15\), 16, 
              9, \(-10\), \(-11\), 12}, {6, \(-5\), \(-8\), 7, 
              2, \(-1\), \(-4\), 3, 14, \(-13\), \(-16\), 15, 
              10, \(-9\), \(-12\), 11}, {7, 8, \(-5\), \(-6\), 3, 
              4, \(-1\), \(-2\), 15, 16, \(-13\), \(-14\), 11, 
              12, \(-9\), \(-10\)}, {8, 7, \(-6\), \(-5\), 4, 
              3, \(-2\), \(-1\), 16, 15, \(-14\), \(-13\), 12, 
              11, \(-10\), \(-9\)}, {9, \(-10\), \(-11\), 12, \(-13\), 14, 
              15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 
              7, \(-8\)}, {10, \(-9\), \(-12\), 11, \(-14\), 13, 16, \(-15\), 
              2, \(-1\), \(-4\), 3, \(-6\), 5, 8, \(-7\)}, {11, 
              12, \(-9\), \(-10\), \(-15\), \(-16\), 13, 14, 3, 
              4, \(-1\), \(-2\), \(-7\), \(-8\), 5, 6}, {12, 
              11, \(-10\), \(-9\), \(-16\), \(-15\), 14, 13, 4, 
              3, \(-2\), \(-1\), \(-8\), \(-7\), 6, 5}, {13, 14, 15, 
              16, \(-9\), \(-10\), \(-11\), \(-12\), 5, 6, 7, 
              8, \(-1\), \(-2\), \(-3\), \(-4\)}, {14, 13, 16, 
              15, \(-10\), \(-9\), \(-12\), \(-11\), 6, 5, 8, 
              7, \(-2\), \(-1\), \(-4\), \(-3\)}, {15, \(-16\), 
              13, \(-14\), \(-11\), 12, \(-9\), 10, 7, \(-8\), 
              5, \(-6\), \(-3\), 4, \(-1\), 2}, {16, \(-15\), 
              14, \(-13\), \(-12\), 11, \(-10\), 9, 8, \(-7\), 
              6, \(-5\), \(-4\), 3, \(-2\), 
              1}}, {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\^2 - 
              4\ \((\((h\ i - g\ j + f\ k - e\ l - d\ m + c\ n - b\ o + a\ p)\)\^2 + \((a\ i - b\ j - c\ k + d\ l - e\ m + f\ n + g\ o - h\ p)\)\^2 + \((a\ e - b\ f - c\ g + d\ h + i\ m - j\ n - k\ o + l\ p)\)\^2 + \((a\ c - b\ d + e\ g - f\ h + i\ k - j\ l + m\ o - n\ p)\)\^2 + \((a\ b + c\ d + e\ f + g\ h + i\ j + k\ l + m\ n + o\ p)\)\^2)\)}, , , {1, 2, 3, \(-4\), 
            5, \(-6\), \(-7\), \(-8\), 
            9, \(-10\), \(-11\), \(-12\), \(-13\), \(-14\), \(-15\), 16, 
            4}, , };\)\)

Hoop["CL31"]={16,22,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,-1,4,-3,6,-5,8,-7,10,-9,12,-11,14,-13,16,-15},{3,-4,1,-2,7,-8,5,-6,11,-12,9,-10,15,-16,13,-14},{4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13},{5,-6,-7,8,1,-2,-3,4,13,-14,-15,16,9,-10,-11,12},{6,5,-8,-7,2,1,-4,-3,14,13,-16,-15,10,9,-12,-11},{7,8,-5,-6,3,4,-1,-2,15,16,-13,-14,11,12,-9,-10},{8,-7,-6,5,4,-3,-2,1,16,-15,-14,13,12,-11,-10,9},{9,-10,-11,12,-13,14,15,-16,1,-2,-3,4,-5,6,7,-8},{10,9,-12,-11,-14,-13,16,15,2,1,-4,-3,-6,-5,8,7},{11,12,-9,-10,-15,-16,13,14,3,4,-1,-2,-7,
-8,5,6},{12,-11,-10,9,-16,15,14,-13,4,-3,-2,1,-8,7,6,-5},{13,14,15,16,-9,-10,-11,-12,5,6,7,8,-1,-2,-3,-4},{14,-13,16,-15,-10,9,-12,11,6,-5,8,-7,-2,1,-4,3},{15,-16,13,-14,-11,12,-9,10,7,-8,5,-6,-3,4,-1,2},{16,15,14,13,-12,-11,-10,-9,8,7,6,5,-4,-3,-2,-1}},{(a^2+b^2-c^2-d^2-e^2-f^2+g^2+h^2-i^2-j^2+k^2+l^2+m^2+n^2-o^2-p^2)^2-4*(-(-(h*i)-g*j+f*k+e*l-d*m-c*n+b*o+a*p)^2-(-(g*i)+h*j+e*k-f*l-c*m+d*n+a*o-b*p)^2+(-(f*i)+e*j+h*k-g*l-b*m+a*n+d*o-c*p)^2+(-(d*i)+c*j-b*k+a*l-h*m+g*n-f*o+e*p)^2+(-(d*e)+c*f-b*g+a*h+l*m-k*n+j*o-i*p)^2)},,,{1,-2,3,4,5,6,-7,8,9,10,-11,12,-13,14,-15,-16,4},gp,{a,c,b,-d,-e,-f}};

\(\(Hoop["CL22"] = {16, 
          23, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, \(-1\), 4, \(-3\), 6, \(-5\), 8, \(-7\), 
              10, \(-9\), 12, \(-11\), 14, \(-13\), 
              16, \(-15\)}, {3, \(-4\), \(-1\), 2, 7, \(-8\), \(-5\), 6, 
              11, \(-12\), \(-9\), 10, 15, \(-16\), \(-13\), 14}, {4, 
              3, \(-2\), \(-1\), 8, 7, \(-6\), \(-5\), 12, 
              11, \(-10\), \(-9\), 16, 
              15, \(-14\), \(-13\)}, {5, \(-6\), \(-7\), 8, 
              1, \(-2\), \(-3\), 4, 13, \(-14\), \(-15\), 16, 
              9, \(-10\), \(-11\), 12}, {6, 5, \(-8\), \(-7\), 2, 
              1, \(-4\), \(-3\), 14, 13, \(-16\), \(-15\), 10, 
              9, \(-12\), \(-11\)}, {7, 8, 5, 6, 3, 4, 1, 2, 15, 16, 13, 14, 
              11, 12, 9, 10}, {8, \(-7\), 6, \(-5\), 4, \(-3\), 2, \(-1\), 
              16, \(-15\), 14, \(-13\), 12, \(-11\), 
              10, \(-9\)}, {9, \(-10\), \(-11\), 12, \(-13\), 14, 
              15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 7, \(-8\)}, {10, 
              9, \(-12\), \(-11\), \(-14\), \(-13\), 16, 15, 2, 
              1, \(-4\), \(-3\), \(-6\), \(-5\), 8, 7}, {11, 12, 9, 
              10, \(-15\), \(-16\), \(-13\), \(-14\), 3, 4, 1, 
              2, \(-7\), \(-8\), \(-5\), \(-6\)}, {12, \(-11\), 
              10, \(-9\), \(-16\), 15, \(-14\), 13, 4, \(-3\), 
              2, \(-1\), \(-8\), 7, \(-6\), 5}, {13, 14, 15, 
              16, \(-9\), \(-10\), \(-11\), \(-12\), 5, 6, 7, 
              8, \(-1\), \(-2\), \(-3\), \(-4\)}, {14, \(-13\), 
              16, \(-15\), \(-10\), 9, \(-12\), 11, 6, \(-5\), 
              8, \(-7\), \(-2\), 1, \(-4\), 3}, {15, \(-16\), \(-13\), 
              14, \(-11\), 12, 9, \(-10\), 7, \(-8\), \(-5\), 6, \(-3\), 4, 
              1, \(-2\)}, {16, 15, \(-14\), \(-13\), \(-12\), \(-11\), 10, 9, 
              8, 7, \(-6\), \(-5\), \(-4\), \(-3\), 2, 
              1}}, {\((a\^2 + b\^2 + c\^2 + d\^2 - e\^2 - f\^2 - g\^2 - h\^2 - i\^2 - j\^2 - k\^2 - l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\^2 - 
              4\ \((\((\(-h\)\ i - g\ j + f\ k + e\ l - d\ m - c\ n + b\ o + a\ p)\)\^2 + \((\(-g\)\ i + h\ j + e\ k - f\ l - c\ m + d\ n + a\ o - b\ p)\)\^2 + \((\(-f\)\ i + e\ j - h\ k + g\ l - b\ m + a\ n - d\ o + c\ p)\)\^2 - \((\(-d\)\ i + c\ j - b\ k + a\ l - h\ m + g\ n - f\ o + e\ p)\)\^2 - \((\(-d\)\ e + c\ f - b\ g + a\ h + l\ m - k\ n + j\ o - i\ p)\)\^2)\)}, , , {1, \(-2\), \(-3\), \(-4\), 5, 6, 7, \(-8\), 9, 10, 11, \(-12\), \(-13\), 14, 15, 
            16, 4}, , };\)\)

\(\(Hoop["CL13"] = {16, 
          24, 
{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 
              14, 15, 16}, {2, \(-1\), 4, \(-3\), 6, \(-5\), 8, \(-7\), 
              10, \(-9\), 12, \(-11\), 14, \(-13\), 
              16, \(-15\)}, {3, \(-4\), \(-1\), 2, 7, \(-8\), \(-5\), 6, 
              11, \(-12\), \(-9\), 10, 15, \(-16\), \(-13\), 14}, {4, 
              3, \(-2\), \(-1\), 8, 7, \(-6\), \(-5\), 12, 
              11, \(-10\), \(-9\), 16, 
              15, \(-14\), \(-13\)}, {5, \(-6\), \(-7\), 8, \(-1\), 2, 
              3, \(-4\), 13, \(-14\), \(-15\), 16, \(-9\), 10, 
              11, \(-12\)}, {6, 5, \(-8\), \(-7\), \(-2\), \(-1\), 4, 3, 14, 
              13, \(-16\), \(-15\), \(-10\), \(-9\), 12, 11}, {7, 8, 5, 
              6, \(-3\), \(-4\), \(-1\), \(-2\), 15, 16, 13, 
              14, \(-11\), \(-12\), \(-9\), \(-10\)}, {8, \(-7\), 
              6, \(-5\), \(-4\), 3, \(-2\), 1, 16, \(-15\), 
              14, \(-13\), \(-12\), 11, \(-10\), 9}, {9, \(-10\), \(-11\), 
              12, \(-13\), 14, 15, \(-16\), 1, \(-2\), \(-3\), 4, \(-5\), 6, 
              7, \(-8\)}, {10, 9, \(-12\), \(-11\), \(-14\), \(-13\), 16, 15, 
              2, 1, \(-4\), \(-3\), \(-6\), \(-5\), 8, 7}, {11, 12, 9, 
              10, \(-15\), \(-16\), \(-13\), \(-14\), 3, 4, 1, 
              2, \(-7\), \(-8\), \(-5\), \(-6\)}, {12, \(-11\), 
              10, \(-9\), \(-16\), 15, \(-14\), 13, 4, \(-3\), 
              2, \(-1\), \(-8\), 7, \(-6\), 5}, {13, 14, 15, 16, 9, 10, 11, 
              12, 5, 6, 7, 8, 1, 2, 3, 4}, {14, \(-13\), 16, \(-15\), 
              10, \(-9\), 12, \(-11\), 6, \(-5\), 8, \(-7\), 2, \(-1\), 
              4, \(-3\)}, {15, \(-16\), \(-13\), 14, 11, \(-12\), \(-9\), 10, 
              7, \(-8\), \(-5\), 6, 3, \(-4\), \(-1\), 2}, {16, 
              15, \(-14\), \(-13\), 12, 11, \(-10\), \(-9\), 8, 
              7, \(-6\), \(-5\), 4, 
              3, \(-2\), \(-1\)}}, {\((a\^2 + b\^2 + c\^2 + d\^2 + e\^2 + f\^2 + g\^2 + h\^2 + i\^2 + j\^2 + k\^2 + l\^2 + m\^2 + n\^2 + o\^2 + p\^2)\)\^2 - 4\ \((\((e\ i + f\ j + g\ k + h\ l + a\ m + b\ n + c\ o + d\ p)\)\^2 + \((c\ i + d\ j + a\ k + b\ l - g\ m - h\ n - e\ o - f\ p)\)\^2 + \((b\ i + a\ j - d\ k - c\ l - f\ m - e\ n + h\ o + g\ p)\)\^2 + \((a\ i - b\ j - c\ k + d\ l - e\ m + f\ n + g\ o - h\ p)\)\^2 + \((\(-d\)\ e + c\ f - b\ g + a\ h - l\ m + k\ n - j\ o + i\ p)\)\^2)\)}, , , {1, \(-2\), \(-3\), \(-4\), \(-5\), \(-6\), \(-7\), 8, 9, 10, 11, \(-12\), 13, \(-14\), \(-15\), \(-16\), 
            4}, , };\)\)

Hoop["CL04"]={16,25,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,-1,4,-3,6,-5,8,-7,10,-9,12,-11,14,-13,16,-15},{3,-4,-1,2,7,-8,-5,6,11,-12,-9,10,15,-16,-13,14},{4,3,-2,-1,8,7,-6,-5,12,11,-10,-9,16,15,-14,-13},{5,-6,-7,8,-1,2,3,-4,13,-14,-15,16,-9,10,11,-12},{6,5,-8,-7,-2,-1,4,3,14,13,-16,-15,-10,-9,12,11},{7,8,5,6,-3,-4,-1,-2,15,16,13,14,-11,-12,-9,-10},{8,-7,6,-5,-4,3,-2,1,16,-15,14,-13,-12,11,-10,9},{9,-10,-11,12,-13,14,15,-16,-1,2,3,-4,5,-6,-7,8},{10,9,-12,-11,-14,-13,16,15,-2,-1,4,3,6,5,-8,-7},{11,12,9,10,-15,-16,-13,-14,-3,-4,-1,-2,7,8,5,6},{12,-11,10,-9,-16,15,-14,13,-4,3,-2,1,8,-7,6,-5},{13,14,15,16,9,10,11,12,-5,-6,-7,-8,-1,-2,-3,-4},{14,-13,16,-15,10,-9,12,-11,-6,5,-8,7,-2,1,-4,3},{15,-16,-13,14,11,-12,-9,10,-7,8,5,-6,-3,4,1,-2},{16,15,-14,-13,12,11,-10,-9,-8,-7,6,5,-4,-3,2,1}},
{(a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2+j^2+k^2+l^2+m^2+n^2+o^2+p^2)^2-4*((-(h*i)-g*j+f*k+e*l-d*m-c*n+b*o+a*p)^2+(-(g*i)+h*j+e*k-f*l-c*m+d*n+a*o-b*p)^2+(-(f*i)+e*j-h*k+g*l-b*m+a*n-d*o+c*p)^2+(-(d*i)+c*j-b*k+a*l+h*m-g*n+f*o-e*p)^2+(-(d*e)+c*f-b*g+a*h-l*m+k*n-j*o+i*p)^2)},,,{1,-2,-3,-4,-5,-6,-7,8,-9,-10,-11,12,-13,14,15,16,4},gp,{a,c,b,-d,-e,-f}};

Hoop["O8"]={16,27,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},{2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15},{3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14},{4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13},{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},{6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11},{7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10},{8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9},{9,10,11,12,13,14,15,16,-1,-2,-3,-4,-5,-6,-7,-8},{10,9,12,11,14,13,16,15,-2,-1,-4,-3,-6,-5,-8,-7},{11,12,9,10,15,16,13,14,-3,-4,-1,-2,-7,-8,-5,-6},{12,11,10,9,16,15,14,13,-4,-3,-2,-1,-8,-7,-6,-5},{13,14,15,16,9,10,11,12,-5,-6,-7,-8,-1,-2,-3,-4},{14,13,16,15,10,9,12,11,-6,-5,-8,-7,-2,-1,-4,-3},{15,16,13,14,11,12,9,10,-7,-8,-5,-6,-3,-4,-1,-2},{16,15,14,13,12,11,10,9,-8,-7,-6,-5,-4,-3,-2,-1}},{(a+b+c+d+e+f+g+h)^2+(i+j+k+l+m+n+o+p)^2,(a-b-c+d+e-f-g+h)^2+(i-j-k+l+m-n-o+p)^2,(a-b+c-d+e-f+g-h)^2+(i-j+k-l+m-n+o-p)^2,(a+b-c-d+e+f-g-h)^2+(i+j-k-l+m+n-o-p)^2,(a+b+c+d-e-f-g-h)^2+(i+j+k+l-m-n-o-p)^2,(a-b-c+d-e+f+g-h)^2+(i-j-k+l-m+n+o-p)^2,(a-b+c-d-e+f-g+h)^2+(i-j+k-l-m+n-o+p)^2,(a+b-c-d-e-f+g+h)^2+(i+j-k-l-m-n+o+p)^2},{(a+b+c+d+e+f+g+h)^2+(i+j+k+l+m+n+o+p)^2,ArcTan[a+b+c+d+e+f+g+h,i+j+k+l+m+n+o+p],(a-b-c+d+e-f-g+h)^2+(i-j-k+l+m-n-o+p)^2,ArcTan[a-b-c+d+e-f-g+h,i-j-k+l+m-n-o+p],(a-b+c-d+e-f+g-h)^2+(i-j+k-l+m-n+o-p)^2,ArcTan[a-b+c-d+e-f+g-h,i-j+k-l+m-n+o-p],(a+b-c-d+e+f-g-h)^2+(i+j-k-l+m+n-o-p)^2,ArcTan[a+b-c-d+e+f-g-h,i+j-k-l+m+n-o-p],(a+b+c+d-e-f-g-h)^2+(i+j+k+l-m-n-o-p)^2,ArcTan[a+b+c+d-e-f-g-h,i+j+k+l-m-n-o-p],(a-b-c+d-e+f+g-h)^2+(i-j-k+l-m+n+o-p)^2,ArcTan[a-b-c+d-e+f+g-h,i-j-k+l-m+n+o-p],(a-b+c-d-e+f-g+h)^2+(i-j+k-l-m+n-o+p)^2,ArcTan[a-b+c-d-e+f-g+h,i-j+k-l-m+n-o+p],(a+b-c-d-e-f+g+h)^2+(i+j-k-l-m-n+o+p)^2,ArcTan[a+b-c-d-e-f+g+h,i+j-k-l-m-n+o+p]},{(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]+Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]+Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]-Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]-Sqrt[c]*Cos[d]+Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]+Sqrt[k]*Cos[l]-Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Cos[b]+Sqrt[c]*Cos[d]-Sqrt[e]*Cos[f]-Sqrt[g]*Cos[h]-Sqrt[i]*Cos[j]-Sqrt[k]*Cos[l]+Sqrt[m]*Cos[n]+Sqrt[o]*Cos[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]+Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]+Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]-Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]-Sqrt[c]*Sin[d]+Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]+Sqrt[k]*Sin[l]-Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8,(Sqrt[a]*Sin[b]+Sqrt[c]*Sin[d]-Sqrt[e]*Sin[f]-Sqrt[g]*Sin[h]-Sqrt[i]*Sin[j]-Sqrt[k]*Sin[l]+Sqrt[m]*Sin[n]+Sqrt[o]*Sin[p])/8},{1,2,3,4,5,6,7,8,-9,-10,-11,-12,-13,-14,-15,-16,1,1,1,1,1,1,1,1},{1,0.,1,0.,1,0.,1,0.,1,0.,1,0.,1,0.,1,0.},{a,b,c,-d,e,-f,-g,-h,i,-j,-k,-l,-m,-n,-o,p}};

Hoop["Q8M2"]={16,34,
{{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},
{2,5,8,3,6,1,4,7,10,13,12,15,14,9,16,11},
{3,4,5,6,7,8,1,2,11,16,13,10,15,12,9,14},
{4,7,2,5,8,3,6,1,12,11,14,13,16,15,10,9},
{5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12},
{6,1,4,7,2,5,8,3,14,9,16,11,10,13,12,15},
{7,8,1,2,3,4,5,6,15,12,9,14,11,16,13,10},
{8,3,6,1,4,7,2,5,16,15,10,9,12,11,14,13},
{9,14,15,16,13,10,11,12,1,6,7,8,5,2,3,4},
{10,9,12,15,14,13,16,11,2,1,8,3,6,5,4,7},
{11,16,9,10,15,12,13,14,3,4,1,6,7,8,5,2},
{12,11,14,9,16,15,10,13,4,7,2,1,8,3,6,5},
{13,10,11,12,9,14,15,16,5,2,3,4,1,6,7,8},
{14,13,16,11,10,9,12,15,6,5,4,7,2,1,8,3},
{15,12,13,14,11,16,9,10,7,8,5,2,3,4,1,6},
{16,15,10,13,12,11,14,9,8,3,6,5,4,7,2,1}},
{a+b+c+d+e+f+g+h-i-j-k-l-m-n-o-p,
 a+b-c-d+e+f-g-h+i+j-k-l+m+n-o-p,
 a-b+c-d+e-f+g-h+i-j+k-l+m-n+o-p, 
 a-b-c+d+e-f-g+h-i+j+k-l-m+n+o-p,
 a-b-c+d+e-f-g+h+i-j-k+l+m-n-o+p,
 a-b+c-d+e-f+g-h-i+j-k+l-m+n-o+p,
 a+b-c-d+e+f-g-h-i-j+k+l-m-n+o+p,
 a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p,
 (a-e)^2+(b-f)^2+(c-g)^2+(d-h)^2-(j-n)^2 -(k-o)^2-(l-p)^2-(i-m)^2},,,
 {1,6,7,8,5,2,3,4,9,10,11,12,13,14,15,16,1,1,1,1,1,1,1,1,4},,};

Hoop["C3C4C2"] = {24,9,{
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24},
{2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16,20,21,19,23,24,22},
{3,1,2,6,4,5,9,7,8,12,10,11,15,13,14,18,16,17,21,19,20,24,22,23},
{4,5,6,7,8,9,10,11,12,1,2,3,16,17,18,19,20,21,22,23,24,13,14,15},
{5,6,4,8,9,7,11,12,10,2,3,1,17,18,16,20,21,19,23,24,22,14,15,13},
{6,4,5,9,7,8,12,10,11,3,1,2,18,16,17,21,19,20,24,22,23,15,13,14},
{7,8,9,10,11,12,1,2,3,4,5,6,19,20,21,22,23,24,13,14,15,16,17,18},
{8,9,7,11,12,10,2,3,1,5,6,4,20,21,19,23,24,22,14,15,13,17,18,16},
{9,7,8,12,10,11,3,1,2,6,4,5,21,19,20,24,22,23,15,13,14,18,16,17},
{10,11,12,1,2,3,4,5,6,7,8,9,22,23,24,13,14,15,16,17,18,19,20,21},
{11,12,10,2,3,1,5,6,4,8,9,7,23,24,22,14,15,13,17,18,16,20,21,19},
{12,10,11,3,1,2,6,4,5,9,7,8,24,22,23,15,13,14,18,16,17,21,19,20},
{13,14,15,16,17,18,19,20,21,22,23,24,1,2,3,4,5,6,7,8,9,10,11,12},
{14,15,13,17,18,16,20,21,19,23,24,22,2,3,1,5,6,4,8,9,7,11,12,10},
{15,13,14,18,16,17,21,19,20,24,22,23,3,1,2,6,4,5,9,7,8,12,10,11},
{16,17,18,19,20,21,22,23,24,13,14,15,4,5,6,7,8,9,10,11,12,1,2,3},
{17,18,16,20,21,19,23,24,22,14,15,13,5,6,4,8,9,7,11,12,10,2,3,1},
{18,16,17,21,19,20,24,22,23,15,13,14,6,4,5,9,7,8,12,10,11,3,1,2}, 
{19,20,21,22,23,24,13,14,15,16,17,18,7,8,9,10,11,12,1,2,3,4,5,6},
{20,21,19,23,24,22,14,15,13,17,18,16,8,9,7,11,12,10,2,3,1,5,6,4},
{21,19,20,24,22,23,15,13,14,18,16,17,9,7,8,12,10,11,3,1,2,6,4,5},
{22,23,24,13,14,15,16,17,18,19,20,21,10,11,12,1,2,3,4,5,6,7,8,9},
{23,24,22,14,15,13,17,18,16,20,21,19,11,12,10,2,3,1,5,6,4,8,9,7},
{24,22,23,15,13,14,18,16,17,21,19,20,12,10,11,3,1,2,6,4,5,9,7,8}},
{a+b+c+d+e+f+g+h+i+j+k+l+m+n+o+p+q+r+s+t+u+v+w+x,
 a+b+c-d-e-f+g+h+i-j-k-l+m+n+o-p-q-r+s+t+u-v-w-x,
 a+b+c+d+e+f+g+h+i+j+k+l-m-n-o-p-q-r-s-t-u-v-w-x,
 a+b+c-d-e-f+g+h+i-j-k-l-m-n-o+p+q+r-s-t-u+v+w+x,
 (a+b+c-g-h-i-m-n-o+s+t+u)^2+(d+e+f-j-k-l-p-q-r+v+w+x)^2,
 (a+b+c-g-h-i+m+n+o-s-t-u)^2+(d+e+f-j-k-l+p+q+r-v-w-x)^2,
((a-b+d-e+g-h+j-k+m-n+p-q+s-t+v-w)^2+(b-c+e-f+h-i+k-l+n-o+q-r+t-u+w-x)^2+
(-a+c-d+f-g+i-j+l-m+o-p+r-s+u-v+x)^2)/2, 
((a-b+d-e+g-h+j-k-m+n-p+q-s+t-v+w)^2+(-a+c-d+f-g+i-j+l+m-o+p-r+s-u+v-x)^2+ (b-c+e-f+h-i+k-l-n+o-q+r-t+u-w+x)^2)/2,
((a-b-d+e+g-h-j+k+m-n-p+q+s-t-v+w)^2+(-a+c+d-f-g+i+j-l-m+o+p-r-s+u+v-x)^2+
(b-c-e+f+h-i-k+l+n-o-q+r+t-u-w+x)^2)/2,
((a-b-d+e+g-h-j+k-m+n+p-q-s+t+v-w)^2+(b-c-e+f+h-i-k+l-n+o+q-r-t+u+w-x)^2+
(-a+c+d-f-g+i+j-l+m-o-p+r+s-u-v+x)^2)/2,
(1/4)*((a-b-d+e-g+h+j-k+m-n-p+q-s+t+v-w)^4+
(a-b+d-e-g+h-j+k+m-n+p-q-s+t-v+w)^4+(-a+c-d+f+g-i+j-l-m+o-p+r+s-u+v-x)^4+
(b-c-e+f-h+i+k-l+n-o-q+r-t+u+w-x)^4+(-a+c+d-f+g-i-j+l-m+o+p-r+s-u-v+x)^4+
(b-c+e-f-h+i-k+l+n-o+q-r-t+u-w+x)^4)-
((a-b-g+h+m-n-s+t)^2+(-a+c+g-i-m+o+s-u)^2+(b-c-h+i+n-o-t+u)^2)*
((d-e-j+k+p-q-v+w)^2+(-d+f+j-l-p+r+v-x)^2+(e-f-k+l+q-r-w+x)^2),
(1/4)*((a-b+d-e-g+h-j+k-m+n-p+q+s-t+v-w)^4+
(a-b-d+e-g+h+j-k-m+n+p-q+s-t-v+w)^4+(-a+c+d-f+g-i-j+l+m-o-p+r-s+u+v-x)^4+
(b-c+e-f-h+i-k+l-n+o-q+r+t-u+w-x)^4+(-a+c-d+f+g-i+j-l+m-o+p-r-s+u-v+x)^4+
(b-c-e+f-h+i+k-l-n+o+q-r+t-u-w+x)^4)-
((a-b-g+h-m+n+s-t)^2+(b-c-h+i-n+o+t-u)^2+(-a+c+g-i+m-o-s+u)^2)*
((d-e-j+k-p+q+v-w)^2+(e-f-k+l-q+r+w-x)^2+(-d+f+j-l+p-r-v+x)^2)},,, {1,3,2,10,12,11,7,9,8,4,6,5,13,15,14,22,24,23,19,21,20,16,18,17,1,1,1,1,1,1,1,1,1,1,1,1},,};

Five non-Moufang tables are appended. They do not have the Frobenius property. They are not Hoops, as indicated by the final "n" in their names.

Hoop["Q4n"] := {mm = 4, nn = 14, 
          hoopTbl = {{1, 2, 3, 4}, {2, 1, 4, -3}, {3, -4, 1, 2}, {4, 
                3, -2, 1}}, 
          sh = {(a^2) - (b^2) - (c^2) - (d^2), (a^2) + (b^2) + (c^2) + (d^2)}, , , 
          gi = {1, 2, 3, 4, 1, 1}, , 
          plex = {a, -b, -c, -d}};

\(\(Hoop["C6n"] := {mm = 6, nn = 13, 
          hoopTbl = {{1, 2, 3, 4, 5, 6}, {2, 3, 1, 6, 4, 5}, {3, 1, 2, 5, 6, 
                4}, {4, 5, 6, 2, 1, 3}, {5, 6, 4, 1, 3, 2}, {6, 4, 5, 3, 2, 
                1}}, sh = {a + b + c - d - e - f, a + b + c + d + e + f, 
              1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 - \((d - e)\)\^2 - \((e - f)\)\^2 - \((\(-d\) + f)\)\^2)\), 
              1\/2\ \((\((a - b)\)\^2 + \((b - c)\)\^2 + \((\(-a\) + c)\)\^2 + \((d - e)\)\^2 + \((e - f)\)\^2 + \((\(-d\) + f)\)\^2)\)}, , , 
          gi = {1, 3, 2, 5, 4, 6, 1, 1, 1, 1}, , 
          a + b + c - d - e - f};\)\)

\(\(Hoop["Q8n"] := {mm = 8, nn = 36, 
          hoopTbl = {{1, 2, 3, 4, 5, 6, 7, 8}, {2, 1, 4, 7, 6, 5, 8, 3}, {3, 
                8, 1, 2, 7, 4, 5, 6}, {4, 3, 6, 1, 8, 7, 2, 5}, {5, 6, 7, 8, 
                1, 2, 3, 4}, {6, 5, 8, 3, 2, 1, 4, 7}, {7, 4, 5, 6, 3, 8, 1, 
                2}, {8, 7, 2, 5, 4, 3, 6, 1}}, 
          sh = {a + b - c - d + e + f - g - h, a - b + c - d + e - f + g - h, 
              a - b - c + d + e - f - g + h, 
              a + b + c + d + e + f + g + 
                h, \((a - e)\)\^2 - \((b - f)\)\^2 - \((c - g)\)\^2 - \((d - h)\)\^2, \((a - e)\)\^2 + \((b - f)\)\^2 + \((c - g)\)\^2 + \((d - h)\)\^2}, , , gi = {1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1}, , };\)\)

The Alt8n table passes the alternativeQ and moufangQ test (i.e. up to a sign) but is not conservative.

Hoop["Alt8n"] = {mm = 8, nn = 33, 
          hoopTbl = {{1, 2, 3, 4, 5, 6, 7, 8}, {2, -1, 4, -3, 
                6, -5, 8, -7}, {3, 4, 1, 2, 7, 8, 5, 6}, {4, -3, 
                2, -1, 8, -7, 
                6, -5}, {5, -6, -7, -8, -1, 2, 3, 4}, {6, 
                5, -8, 7, -2, -1, 
                4, -3}, {7, -8, -5, -6, -3, 4, 1, 2}, {8, 
                7, -6, 5, -4, -3, 2, -1}}, , , , 
          gi = {1, -2, 3, -4, -5, -6, 7, -8, 
              1}, , };

The D3Mn loop is the smallest noncommutative Moufang loop, created by Chein doubling of the D3 group. It is not conservative.

id["D3Mn"]

gd[]

glo // tf

Hoop["D3Mn"] = {mm = 12, nn = 28, 
          hoopTbl = {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {2, 1, 6, 5, 4, 
                3, 8, 7, 10, 9, 12, 11}, {3, 4, 5, 6, 1, 2, 9, 12, 11, 8, 7, 
                10}, {4, 3, 2, 1, 6, 5, 10, 11, 12, 7, 8, 9}, {5, 6, 1, 2, 3, 
                4, 11, 10, 7, 12, 9, 8}, {6, 5, 4, 3, 2, 1, 12, 9, 8, 11, 10, 
                7}, {7, 8, 11, 10, 9, 12, 1, 2, 5, 4, 3, 6}, {8, 7, 10, 11, 
                12, 9, 2, 1, 6, 3, 4, 5}, {9, 10, 7, 12, 11, 8, 3, 6, 1, 2, 
                5, 4}, {10, 9, 12, 7, 8, 11, 4, 5, 2, 1, 6, 3}, {11, 12, 9, 
                8, 7, 10, 5, 4, 3, 6, 1, 2}, {12, 11, 8, 9, 10, 7, 6, 3, 4, 
                5, 2, 1}}, , , , 
          gi = {1, 5, 4, 3, 6.7, 8, 9, 10, 11, 12}, , };

9. Exported functions.

9.1. Main Procedures;  as, hoopTimes, hoopInverse, hoopPower, Use.

Symbolic functions such as topol. tovec are given new values by e.g. topol/.as[{3.,1.,4.}].

as[A_] :=   
Table[
        alph[[ii]] -> A[[ii]], {ii, Min[mm, Length[A]]}]; 
    as[A_, B_] :=   
Table[
        B[[ii]] -> A[[ii]], {ii, 
          Min[mm, Length[A], Length[B]]}];

hoopTimes multiplies two vectors and (if c==0) sets the remainders Rl & Rr. Lists of zeroes t, Rl, Rr are initialised. A & B are padded with zeroes and truncated to the correct length.
	The product t is built up by adding each (signed) product to the appropriate element. If disparate zero-sizes occur, some sizes will be lost from the product AB and AB/A will not recover B; left and right remainders are created so that sizes are conserved and B=AB/A+Rr & A=AB/B+Rl.

hoopTimes[AA_, BB_, c_:  0] :=   
Module[{t = Table[0, {mm}], R, A, B}, 
        Rl = t; Rr = t; R = t; A = Take[Flatten[Append[AA, t]], mm]; 
        B = Take[Flatten[Append[BB, t]], mm]; 
Do[
          Do[t[[Abs[
                  hoopTbl[[ll, kk]]]]] += 
              A[[
                  ll]] B[[
                  kk]] Sign[hoopTbl[[ll, kk]]] , {kk, 
              mm}], {ll, 
            mm}];   
If[
          c == 0, 
R = 
            Chop[Simplify[B - hoopTimes[hoopInverse[A], t, 1]], hmin]; 
          Rl = Chop[Simplify[A - hoopTimes[t, hoopInverse[B], 1]], hmin], ]; 
        Rr = R; t]

Division is effected by the same procedure, hoopTimes, as multiplication, using the multiplicative inverse of the divisor. The inverse always exists because all known hoops have the Moufang property; it is calculated by hoopInverse so that Ainv.A={1,0,..}. 
	inv is set up as a table of zeroes, and then filled with symbolic partial-fractions D[sh[[jj]],var]/sh[[jj]] with sizes (defined in sh) as denominators and their derivations as numerators. gi contains the signed list of locations of 1's in the Cayley table; this identifies the differentiation variable var. In the jj loop, a substitution rule as attempts to calculate the numeric value of each shape for A. Any that differ from zero by less than hmin (arbitrarily set at 4 times the "Machine Epsilon") are ignored -  the calculations continue in a constrained sub-algebra by omitting the "infinite" partial fractions. gi also contains gi[[mm+jj]], the multiplicity of each size; this is a multiplier for the signed partial fraction. Finally inv is divided by mm, and the symbols are replaced by the elements of A. In effect, Cramer's method has been used, so each term of the result has the determinant as a divisor. If the determinant factorizes, each term is expressed as a sum of partial fraction with factors (sizes) as denominators.

hoopInverse[
        A_] :=   
Module[{inv = Table[0, {ii, mm}], ls = Length[sh], var, 
          t}, 
Do[  ss = 
            Sign[gi[[ii]]]; t = 0; 
          var = alph[[gi[[ii]]/ss]];   
Do[  If[
              Chop[Simplify[sh[[jj]] /.  as[A]], hmin] === 0, 
              Null  , 
              t += ss*gi[[mm + jj]]*(D[sh[[jj]], var]/
                      sh[[jj]])], {jj, 
              ls}]; 
inv[[ii]] += t, {ii, mm}]; 
        Simplify[inv/mm /.  as[A]]]

hoopPower calculates (A^p), giving (Sqrt[A]) if p is not specified. The polar-vector interconversions topol, tovec are needed for the algebra (not all are available, many may not exist). Sizes are raised to the power (indicated by 1 in gp) with signed sizes needing special treatment, and angles (indicated by 0. in gp) are multiplied by the power. Finally, tovec is used to obtain the vector form. Negative powers are handled via the inverse. Non-abelian hoops give "pseudo-powers", unrelated to repeated products, because their angles do not add on vector multiplication; this may introduce uncertainty.

hoopPower[A_, r_:  1/2] :=   
Module[{res = {}, lg = Length[gp], 
          p = topol /.  as[A]}, 
        If[r < 0, hoopPower[hoopInverse[A], -r], 
          Do[AppendTo[res, 
              If[IntegerQ[gp[[ii]]], 
                If[Head[p[[ii]]] === Complex, 
                  p[[ii]]^r, Abs[p[[ii]]]^r Sign[p[[ii]]]], 
                r p[[ii]]]
  ], {ii, lg}]]; 
        tovec /.  as[res /.  as[A]]]

A Hoop Algebra "H" is selected by Use["H"]; this sets the relevant expressions mm,nn, hoopTbl, sh, topol, tovec, gi, gp, plex that are needed by the other procedures, until another Use instruction is implemented.

Use[
          H_] :=   
Module[{h = Hoop[H]}, mm = h[[1]]; 
          nn = h[[2]]; hoopTbl = h[[3]]; sh = h[[4]]; 
          topol = h[[5]]; tovec = h[[6]]; gi = h[[7]]; 
          gp = h[[8]]; plex = h[[9]]];

9.2. Constants. J as a sign. BoldFace.

Three constants are set up, and the symbol J is given properties that make it work as a sign with J^3=1 in the same way as I is a sign with I^4=1. bf allows printed results to be shown in boldface.

r3 = Sqrt[3. ]; 
      r6 = 2  r3;   
hmin = 
        4  $MachineEpsilon   ;
  
js=(J^(any_))->(J^(Mod[any, 3]));NumberQ[J] = True;Positive[J] = False;Positive[-J] = False;Abs[J] = 1;Sign[J] = J;Re[J] = 0;Im[J] = 0;Re[   a_ J^b_] := 0;Im[  a_  J^b_] := 0;Re[ J^b_] := 0;Im[J^b_] := 0;Re[a_  J] = 0;Im[a_ J] = 0;
bf[any_] = 
        StyleForm[any, FontWeight -> "Bold"];

9.3. Trigonometric extensions; generalized sinusoids, Quadrant-sensitive hyperbolic tangent, Polyhelix.

genCos[x_, j_:  0] := 
        If[mm < 3, 
          Abort[], 
Re[(((-1))^(2 (mm - j)/
                      mm)) (E^(I Im[(((-1))^(2/mm))] x))]];
genSin[x_, j_:  0] := 
        If[mm < 3, 
          Abort[], 
Im[(((-1))^(2 (mm - j)/
                      mm)) (E^(I Im[(((-1))^(2/mm))] x))]];

arcTanh[adj_, 
          opp_] :=   
If[
          (adj^2) == (opp^2), ∞, 
          Chop[Log[(adj + opp)/(Sqrt[(adj^2) - (opp^2)]) // 
                N]]];

PolyHelix Identity. (Included for reference).
  (∑_(n = 0)^(n = m - 1))(((Sin[x + 2  n π/m]))^(2  p))=(4^(-p))m (({{2  p}, {p}})),{p<m if m is even,else p<2  m}.

This states that the sum of even powers of m equally-spaced sinusoids is a constant. J.J.Thwaites (personal communication) provided the expression for the constant. The m=2, p=1 case gives (Cos^2)(x) + (Sin^2)(x) = 1, the solution to (Overscript[x, ¨])=-1 or \(TraditionalForm\`\(x\_1\)\& . \) = (Subscript[x, 2,])\(TraditionalForm\`\(\(\ \)\(\(x\_2\)\& . \)\)\)= - (Subscript[x, 1]).

10. Protect system symbols, end private context. Unprotect some exported symbols, end package context.

Protect[NumberQ, Positive, Abs, Sign, Re, Im];

End[ ];

(*Protect[Evaluate[$Context <> "*"]]*)

EndPackage[ ];