To be more precise, consider $X=Mat(n,m;\mathbb{Z}_k)$ to be the set of $n\times m$ matrices with coefficients in $\mathbb{Z}_k$. Let $G=S_n\times S_m$ act on $X$ by permuting the $n$ rows and the $m$ columns. I want to know the cardinality of the set of the orbits $|X/G|$. I know that by Burnside's Lemma the following hold $$|X/G|=\frac{1}{|G|}\sum\limits_{g\in G}|X^g|$$ where $X^g=\{x\in X\mid gx=x\}$, but I don't know how to compute $|X^g|$ for a general $g\in G$.
Any help would be appreciated.