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?