Tri-bimaximal neutrino mixing and the family symmetry Z7⋊Z3

The non-Abelian finite group PSL2(7) is the only simple subgroup of SU(3) with a complex three-dimensional irreducible representation. It has two maximal subgroups, S4 which, along with its own A4 subgroup, has been successfully applied in numerous models of flavor, as well as the 21 element Frobenius group Z7⋊Z3, which has gained much less attention. We show that it can also be used to generate tri-bimaximal mixing in the neutrino sector, while allowing for quark and charged lepton hierarchies.