Quaternion family symmetry of quarks and leptons

To a first approximation, the quark mixing matrix has θ13q=θ23q=0, whereas the lepton mixing matrix has θ23l=π/4. We show how this structure may be understood if the family symmetry is Q8, the quaternion group of eight elements. We find three viable scenarios for the Majorana neutrino mass matrix, each depending on four parameters and predicting a specific mass spectrum. The phenomenology of the two Higgs doublets which generate the Yukawa sector is analyzed and testable predictions are derived. We discuss also the closely related model based on D4, the symmetry group of the square