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Discrete avour symmetries have been proven successful in explaining the leptonic
avour structure. To account for the observed mixing pattern, the avour symmetry has to be broken to di erent subgroups in the charged and neutral lepton sector. However, cross-couplings via non-trivial contractions in the scalar potential force the group to break to the same subgroup. We present a solution to this problem by extending the avour group in such a way that it preserves the avour structure, but leads to an 'accidental' symmetry in the avon potential. We have searched for symmetry groups up to order 1000, which forbid all dangerous cross-couplings and extend one of the interesting groups A4, T7, S4, T0 or (27). We have found a number of candidate groups and present a model based on one of the smallest extensions of A4, namely Q8 o A4. We show that the most general nonsupersymmetric potential allows for the correct vacuum alignment. We investigate the e ects of higher dimensional operators on the vacuum con guration and mixing angles, and give a see-saw-like UV completion. Finally, we discuss the supersymmetrization of the model. Additionally, we release the Mathematica package Discrete providing various useful tools for model building such as easily calculating invariants of discrete groups and avon potentials.
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