Number of orbits under automorphism group

Definition

Let $G$ be a group. The number of orbits under automorphism group or number of automorphism classes of $G$ is the number of orbits of the set $G$ under the action of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(G)$.

Facts

• Number of orbits of irreducible representations equals number of orbits under automorphism group (true for a finite group and a splitting field thereof)