High-quality grand unified theories with three generations

We extend the unitary groups beyond the ${\rm SU}(5)$ and ${\rm SU}(6)$ to look for possible grand unified theories that give rise to three-generational Standard Model fermions without the simple repetition. By demanding asymptotic free theories at short distances, we find gauge groups of ${\rm SU}(7)$, ${\rm SU}(8)$ and ${\rm SU}(9)$ together with their anomaly-free irreducible representations are such candidates. Two additional gauge groups of ${\rm SU}(10)$ and ${\rm SU}(11)$ can also achieve the generational structure without asymptotic freedom. We also find 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.