An irreducible representation of a finite group $G$ is a representation that cannot be expressed as the direct sum of $G$-representations. Irreducible representations are useful because of Maschke's theorem, which allows us to decompose arbitrary finite dimensional $G$-representations as direct sums of irreducible representations in which the number of times a summand occurs is independent of the decomposition chosen (provided char $k \nmid |G|$, where $k$ is the underlying field).
My question is, what happens when we try to do all this with the direct sum replaced by the tensor product? Are there finitely many $G$-representations which cannot be decomposed as the tensor product of other $G$-representations? Is there a $G$-representation for some $G$ which can be decomposed into a tensor product of representations atomic with respect to the tensor product in two different ways?