 Similarly to S6 acting on a set of 6 elements in 2 different ways (having an outer automorphism), M12 acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since "M"12 is itself exceptional, one does not consider this outer automorphism to be exceptional itself.The full automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points.Another way to see that S6 has a nontrivial outer automorphism is to use the fact that A6 is isomorphic to PSL2(9), whose automorphism group is the projective semilinear group PΓL2(9), in which PSL2(9) is of index 4, yielding an outer automorphism group of order 4.