Conjugacy class sizes of direct product are pairwise products of conjugacy class sizes of direct factors

For two groups
 Suppose $$G_1$$ and $$G_2$$ are finite groups.Suppose $$c_{11},c_{12},\dots,c_{1r}$$ are the sizes of the conjugacy classes of $$G_1$$ (with repetitions, i.e., if a particular conjugacy class size occurs for multiple conjugacy classes, it appears that many times on the list) and $$c_{21},c_{22},\dots,c_{2s}$$ are the sizes of the conjugacy classes of $$G_2$$.

Then, the conjugacy class sizes for the external direct product are given by taking pairwise products between conjugacy class sizes of $$G_1$$ and of $$G_2$$:

$$c_{11}c_{21}, c_{11}c_{22}, \dots,c_{11}c_{2s},c_{12}c_{21},c_{12}c_{22},\dots,c_{12}c_{2s},c_{1r}c_{21},\dots,c_{1r}c_{2s}$$

In particular, this means that the conjugacy class size statistics of $$G_1 \times G_2$$ are completely determined by the conjugacy class size statistics of $$G_1$$ and $$G_2$$, without any further information.

Due to the equivalence of internal and external direct product, this result also applies to internal direct products.

Related facts

 * Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors
 * Number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor