Template:Irrep degrees automorphism group action facts to check against

FACTS TO CHECK AGAINST for action of automorphism group on irreducible representations and degrees of irreducible representations
number of orbits of irreducible representations equals number of orbits under automorphism group | number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group
cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group