THE ESSENTIAL RANK OF FUSION SYSTEMS OF BLOCKS OF SYMMETRIC GROUPS

Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, bD) be a Sylow B-subgroup of S(n). We consider the fusion system [Formula: see text] and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.