The center of the Lie group $SU(n)$ is isomorphic to $mathbb{Z}_n$. If $d$ divides $n$, the quotient $SU(n)/mathbb{Z}_d$ is also a Lie group. Such groups are locally isomorphic, and their Weyl groups $W(SU(n)/mathbb{Z}_d)$ are the symmetric group ${sum}_n$. However, the integral representations of the Weyl groups are not equivalent. Under the mod $p$ reductions, we consider the structure of invariant rings $H^*(BT^{n-1};mathbb{F}_p)^W$ for $W=W(SU(n)/mathbb{Z}_d)$. Particularly, we ask if each of them is a polynomial ring. Our results show some polynomial and non-polynomial cases.