Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c_0, c_1, ..., c_{\ell}$ be a set of conjugacy class representatives for $G$. In the case where $\gamma$ is faithful, I recently obtained the formula $$ \frac{1}{|G|} \prod_{i=1}^{\ell} (n-\chi(c_i))$$ as the size of a certain abelian group associated with $\gamma$ (see Theorem 3 in this paper if interested). In particular, this implies that this quantity is always an integer (if $\gamma$ is not faithful, then the product is 0).

Is there a nice way to see that this is an integer just using basic character theory?