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2-subgroup of M22 abelian of type alternating group automorphism group Burgoyne and Fong characterization of M12 classes of elements commutator subgroup computation conjugate classes Coxeter and Moser cyclic group cyclic of order denoted Description Dickson digits 22 discussion element of class element of order elementary abelian subgroup factor group fixes the digits Frattini subgroup Frobenius 1904 Frobenius group Garbe and Mennicke given in Appendix group M11 group of order group on four group PSL 3,4 izer Janko lattice of subgroups linear group Mathieu group M22 maximal subgroup modulo Nagao non-abelian non-Sylow subgroups non-trivial normal subgroup normalizer order order eight order four order sixteen order three self-centralizing simple group split extension Steiner systems subgroup contains subgroup is given subgroup of order Sylow 2-sub Sylow 2-subgroup Sylow's third theorem symmetric group transitive extension transitive group two-fold transitive upper bound algorithm Whitelaw Witt writer has shown