# 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 with the property that under the action of the automorphism group , the orbit sizes for the set of irreducible linear representations of are *not* the same as the orbit sizes for the set of conjugacy classes of .

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

The most direct opposite fact is: Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

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:

- Number of irreducible representations equals number of conjugacy classes
- Number of orbits of irreducible representations equals number of orbits under automorphism group
- Number of irreducible representations over reals equals number of equivalence classes under real conjugacy
- Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy
- Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements

### Similar facts

## Proof

There are examples of groups of order 27.