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

## Statement

### For two groups

Suppose and are finite groups.Suppose are the sizes of the conjugacy classes of (with repetitions, i.e., if a particular conjugacy class size occurs for multiple conjugacy classes, it appears that many times on the list) and are the sizes of the conjugacy classes of .

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

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

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