 The Frobenius group of affine transformations of F5 (maps "x" formula_22 "ax" + "b" where "a" ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S5.S5 acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields the action above).Ernst Witt found a copy of Aut(S6) in the Mathieu group M12 (a subgroup "T" isomorphic to S6 and an element "σ" that normalizes "T" and acts by outer automorphism).