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

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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

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:

Similar facts

Proof

There are examples of groups of order 27.

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