LAUSR

We extend the unitary groups beyond the and to determine possible grand unified theories that give rise to three-generational Standard Model fermions without simple repetitions. By demanding asymptotic free theories at short distances, we find gauge groups of , , and , together with their anomaly-free irreducible representations, are such candidates. Two additional gauge groups of and can also achieve the generational structure without asymptotic freedom. We also deduce that these models can solve the Peccei-Quinn (PQ) quality problem, which is intrinsic in the axion models, with the leading PQ-breaking operators determined from the symmetry requirement.