Constructions of Fischer's Baby Monster over fields of characteristic not \(2\) (Q1577623)

The authors give a description of the computer construction of \(4371\)-dimensional representations over the fields \(\text{GF}(3)\) and \(\text{GF}(5)\) for Fischer's Baby Monster group \(B\). As an application they also obtain \(3\)-modular and \(5\)-modular representations for the sporadic simple Thompson group which in turn will assist in determining the corresponding Brauer character tables. The strategy employed is to construct the appropriate representation for a maximal \(2\)-local subgroup \(H\cong 2^{1+22}.Co_2\) first, then restrict to a subgroup \(D\cong 2^{2+10+20}.M_{22}\) of \(H\), and finally adjoin an outer automorphism \(\alpha\) of order \(3\) to \(D\) in order to get the relevant representation for \(M=D:\langle\alpha\rangle\). In this way, \(B\) is constructed as a finite completion of the free amalgamated product \(H*_DM\) of \(H\) and \(M\) over \(D\).