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.
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