# Orbit sizes for irreducible representations may differ from orbit sizes for conjugacy classes under action of automorphism group

## Statement

It is possible to have a finite group $G$ with the property that under the action of the automorphism group $\operatorname{Aut}(G)$, the orbit sizes for the set $R(G)$ of irreducible linear representations of $G$ are not the same as the orbit sizes for the set $C(G)$ of conjugacy classes of $G$.

A finite group where the orbit sizes are in fact the same is termed a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group.

## Related facts

### Opposite facts

This gives a sufficient condition for being a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group. The following hold for all finite groups:

## Proof

There are examples of groups of order 27.