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