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The only 4-transitive groups are the symmetric groups S k for k at least 4, the alternating groups A k for k at least 6, and the Mathieu groups M 24, M 23, M 12, and M 11. ( Cameron 1999 , p. 110) The full proof requires the classification of finite simple groups , but some special cases have been known for much longer.
M 11 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. M 11 is a sharply 4-transitive permutation group on 11 objects.
M 24 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial. The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups.
Its group of units has order 2 11 − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23. The Mathieu group M 23 can be identified with the group of F 2-linear automorphisms of F 2 11 that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M 23 on 23 objects.
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M 12 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873).It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the Schur multiplier of M 12 has order 2 (correcting a mistake in (Burgoyne & Fong 1966) where they incorrectly claimed it has order 1).
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