This is a crosspost from math.SE. Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form $$ \bigoplus_i V_i \otimes W_i, $$ where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.