Triple generations of the Lyons sporadic simple group

Abstract

The Lyons group denoted by Ly is a Sporadic Simple Group of order 51765179004000000 = 28 37 56 7 11 31 37 67. It(Ly) has a trivial Schur Multiplier and a trivial Outer Automorphism Group. Its maximal subgroups are G2(5) of order 5859000000 and index 8835156, 3 McL:2 of order 5388768000 and index 9606125, 53 L3(5) of order 46500000 and index 1113229656, 2 A11 of order 29916800 and index 1296826875, 51+4 + :4S6 of order 9000000 and index 5751686556, 35:(2 M11) of order 3849120 and index 13448575000, 32+4:2 A5 D8 of order 699840 and index 73967162500, 67:22 of order 1474 and index 35118846000000 and 37:18 of order 666 and index 77725494000000. Its existence was suggested by Richard Lyons. Lyons characterized its order as the unique possible order of any nite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group of degree 11 by the cyclic group of order 2. Sims proved the existence of this group and its uniqueness using permutations and machine calculations. In this dissertation, we compute the (p; q; t)-generations of the Lyons group for dis- tinct primes p, q and t which divide the order of Ly such that p < q < t. For computations, we made use of the Computer Algebra System GAP