Minimal eclectic flavor group $Q_8⋊S_3$ and neutrino mixing

We perform a comprehensive analysis of the minimal eclectic flavor group $Q_8⋊S_3$ 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 $Q_8⋊S_3$. The most general forms of Kähler potential and superpotential based on $Q_8⋊S_3$ are discussed, and the corresponding fermion mass matrices are presented. A concrete lepton model invariant under $Q_8⋊S_3$ 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.