 Conjugation by any element of "G" outside of S6 defines the nontrivial outer automorphism of S6.On cycles, it exchanges permutations of type (12) with (12)(34)(56) (class 21 with class 23), and of type (123) with (145)(263) (class 31 with class 32).The outer automorphism also exchanges permutations of type (12)(345) with (123456) (class 2131 with class 61).