A Supercharacter Theory Based on the Sizes of the Conjugacy Classes

In this article a particular supercharacter theory of a finite group is introduced where the superclasses are the unions of conjugacy classes of the same size. It is shown that semidirect products of two groups, A and B, where both A and B are cyclic groups and at least one is of prime order, always allow this type of supercharacter theory. A comparison between this supercharacter theory and the supercharacter theory where the supercharacters are sums of irreducible characters of the same degree is included. Necessary and sufficient conditions are provided that determine when these two types of supercharacter theories coincide for the family of groups that are semidirect products of two cyclic groups one of which is of prime order.