A New Finite Simple Group with Abelian 2-sylow Subgroups.

A 2-Sylow subgroup of J is elementary abelian of order 8 and J has no subgroup of index 2. If r is an involution in J, then C(r) = (r) X K, where K _ A5. Let G be a finite group with the following properties: (a) S2-subgroups of G are abelian; (b) G has no subgroup of index 2; and (c) G contains an involution t such that 0(t) = (t) X F, where F A5. Then G is a (new) simple group isomorphic to J. The order of Gis 11(113-1)(11 + 1). AnS2-subgroup of G is elementary abelian of order 8, and all odd order Sylow subgroups are cyclic (of prime order). Every maximal subgroup of G is conjugate to one of the following groups: (1) 2-Sylow normalizer which is a holomorph of an elementary abelian group of order 8 by a noncyclic group of order 21, (2) 3-Sylow normalizer (which is also a 5-Sylow normalizer), and this is a holomorph of a cyclic group of order 15 by an elementary abelian group of order 4, (3) 7-Sylow normalizer which is a Frobenius group of order 42, (4) 11-Sylow normalizer which is a Frobenius group of order 110, (5) 19-Sylow normalizer which is a Frobenius group of order 114, (6) centralizer (t) X F of an involution t, where F _ A5, and (7) the projective special linear group PSL(2,11). The group G has precisely one conjugate class of subgroups isomorphic to PSL(2,11) and two conjugate classes of subgroups isomorphic to A5. IfH is a subgroup of G isomorphic to PSL(2,11), then two icosahedral subgroups H1 and H2 of H which are nonconjugate in H remain nonconjugate in G. The outer automorphism group of G is trivial. The Schur's multiplicator of G is trivial. The group G has 15 conjugate classes of elements (which are all real) and hence 15 irreducible (complex) characters As, (i = 1,..., 15). We have obtained the full character table of G, and in particular the degrees 1,6 (1) are 1, 56, 56, 76, 76, 77, 77, 77, 120, 120, 120, 133, 133, 133, and 209. The modular representation of G generated by matrices A and B (using G J) is a faithful representation of G of the smallest possible degree in any field. The group G does not have a doubly transitive permutation representation. W. A. Coppel has shown that the group G is a subgroup of the Dickson's simple group E2(11) related to the simple Lie algebra of type (G2).