Let $G_p,G_q\subset G$ be abelian subgroups of a finite group $G$, which we can assume is a permutation group $G\subseteq S_N$.
I am looking at, in the group algebra $\mathbb{C}G$, calculations of the form: $$\left(\sum_{t\in G_p}a_tt\right)\left(\sum_{s\in G_q}b_s s\right)=\sum_{t\in G_p,\,s\in G_q}a_tb_s\,ts\qquad (a_t,b_t\in\mathbb{C}).$$
I am interested in when these are non-zero. Is there a Maple implementation for group algebras? Or a clever way of handling such calculations? I can't seem to find any on the web. We have permutations, e.g. $(12)(345)=\texttt{Perm}([[1,2],[3,4,5]])$.