Outer tensor product establishes bijection between irreducible representations of direct factors and direct product

Statement
Suppose $$G$$ and $$H$$ are groups and $$k$$ is a field. Let $$\operatorname{Irr}(S)$$ denote the set of irreducible representation of the group $$S$$ over $$k$$. Then, there is a natural bijection:

$$\operatorname{Irr}(G) \times \operatorname{Irr}(H) \leftrightarrow \operatorname{Irr}(G \times H)$$.

The bijection is given using the outer tensor product of linear representations, as follows. For irreducible representations $$\alpha,\beta$$ of $$G$$ and $$H$$, to vector spaces $$V$$ and $$W$$, $$\alpha \otimes \beta$$ is defined as a linear representation on the tensor product $$V \otimes W$$, with:

$$(\alpha \otimes \beta)(g,h) = \alpha(g) \otimes \beta(h)$$.

Corollaries

 * Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors
 * Number of conjugacy classes of direct product is product of number of conjugacy classes