Dihedral families of quarks, leptons, and Higgs bosons

We consider finite groups of small order for family symmetry. It is found that the binary dihedral group $Q_6$, along with the assumption that the Higgs sector is of type II, predicts mass matrix of a nearest neighbor interaction type for quarks and leptons. We present a supersymmetric model based on $Q_6$ with spontaneously induced $CP$ phases. The quark sector contains 8 real parameters with one independent phase to describe the quark masses and their mixing. Predictions in the $|V_{ub}|-\overline{\eta }$, $|V_{ub}|-\sin 2\beta ({\varphi }_1)$, and $|V_{ub}|-|V_{td}/V_{ts}|$ planes are given. The lepton sector contains also 9 parameters. A normal as well as an inverted spectrum of neutrino masses is possible, and we compute $V_{e3}$. We find that $|V_{e3}{|}^2>{10}^{-4}$ in the case of a normal spectrum, and $|V_{e3}{|}^2>8\times {10}^{-4}$ in the case of an inverted spectrum. It is also found that $Q_6$ symmetry forbids all Baryon number violating terms of $d=4$, and the contributions to EDMs from the A terms vanish in this model.