The 5-modular characters of the McLaughlin group and its covering group

In the present paper we determine the 5-modular decomposition matrices for the sporadic group 3.G - 3McL, the triple cover of the McLaughlin group. There are only blocks of defect 0 and blocks of maximal defect, since the Sylow 5-subgroup of 3.G is a trivial intersection subgroup. There are three blocks of maximal defect, the principal block and two complex conjugate blocks containing faithful characters. The results were obtained using the computer system MOC for calculating with modular characters which was developed by the authors. The advantage of MOC is the fact that it keeps track of the various steps it has undertaken in finding decomposition numbers. One is therefore able to produce a proof in traditional form which can be verified by hand calculations using only ordinary character tables of the groups involved. These tables can be found in the ATLAS [2] or are available in the Aachen CAS system [7]. On two instances we also use the so called condensation methods of the Meat-Axe. Some features of MOC and condensation are described in [6, Section 3]. A more detailed account of condensation can be found in [8].