Suppose $G_i$ are finite groups for $i=1,2$ and G is the direct product of $G_i$. If V is a finite dimensional irreducible representation of $G$, then it is well known that $V$ is a tensor product of $V_i$,$i=1,2$ and each $V_i$ is an irreducible representation of $G_i$.

The question I have is when $V$ is given, is there a canonical way to construct $V_i$ from $V$?