Minimal eclectic flavor group Q₈ ⋊ S₃ and neutrino mixing

We perform a comprehensive analysis of the minimal eclectic flavor group Q₈ ⋊ S₃ which is isomorphic to GL(2, 3), and all its irreducible representations are induced from the irreducible representations of Q₈ and S₃. The consistency conditions between EFG and generalized CP (gCP) symmetry are revisited, and we find the gCP symmetry compatible with the minimal EFG Q₈ ⋊ S₃. The most general forms of Kähler potential and superpotential based on Q₈ ⋊ S₃ are discussed, and the corresponding fermion mass matrices are presented. A concrete lepton model invariant under Q₈ ⋊ S₃ and gCP is constructed, in which the experimental data of all six lepton masses and six mixing parameters can be successfully described through seven real input parameters. The model predicts a vanishing effective mass m_ββ in neutrinoless double beta decay.