Minimal eclectic flavor group $Q_{8}\rtimes S_3$ and neutrino mixing

We perform a comprehensive analysis of the minimal eclectic flavor group $Q_{8}\rtimes S_3$ which is isomorphic to $GL(2,3)$, and all its irreducible representations are induced from the irreducible representations of $Q_{8}$ and $S_{3}$. The consistency conditions between EFG and generalized CP (gCP) symmetry are revisited, and we find the gCP symmetry compatible with the minimal EFG $Q_{8}\rtimes S_3$. The most general forms of K\"ahler potential and superpotential based on $Q_{8}\rtimes S_3$ are discussed, and the corresponding fermion mass matrices are presented. A concrete lepton model invariant under $Q_{8}\rtimes S_3$ 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 $m_{\beta\beta}$ in neutrinoless double beta decay.