Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. Then, would $G$ have $\frac{n}{\omega}$ disjoint maximal cliques, that is $\frac{n}{\omega}$ maximal cliques of order $\omega$ which are mutually vertex disjoint.

The result is immediate if $G$ is complete multipartite, or even if $G$ is a unitary cayley graph. From here, we know that, if $\omega|n$, $G$ would be CIS(clique intersection stable), that is every maximal clique intersects every maximal independent set. I guess it should work in this case, and also for other graphs on abelian groups. Will the result hold even for Cayley graphs of non-abelian groups? Thanks beforehand.