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

Statement

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