[Edited due to YCor's comment]comment:] Given a finite group $G$, under what conditions does $G\times G\times G$ (the direct product of three copies of $G$) admit a faithful group action on a set of size $|G|$? This is equivalent to an injective group homomorphism from $G\times G\times G$ to $S_{|G|}$.
For reference, for every group $G$ with a trivial center, $G\times G$ admits an easy faithful group action on $G$ by $(g_1, g_2): g \rightarrow g_1 g g_2^{-1}$.