We extend the unitary groups beyond the   $SU(5)$   and   $SU(5)$   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   $SU(5)$  ,   $SU(5)$  , and   $SU(5)$  , together with their anomaly-free irreducible representations, are such candidates. Two additional gauge groups of   $SU(5)$   and   $SU(5)$   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.