B. Fischer [in Representation theory of finite groups and finite-dimensional algebras, Proc. Conf., Bielefeld/Ger. 1991, Prog. Math. 95, 1-16 (1991; Zbl 0804.20003)] presented a powerful method to calculate the irreducible complex character table of a group of the form \(\overline G=G\cdot N\). This method involves the calculation of certain matrices, called Fischer-Clifford matrices for each conjugacy class of the group \(\overline G/N\). Calculating these matrices is important as they allow to compute the whole character table of the group \(\overline G\). Since many maximal subgroups of the simple groups are extensions of normal subgroups, this method can be used to find their character tables. In particular, Moori et. al. have used the method of Fischer to calculate the character table of a maximal subgroup of \(Fi_{24}'\) of the form \(3^7\cdot O'(3)\) [in F. Ali and J. Moori, Represent. Theory 7, 300-321 (2003; Zbl 1065.20020)] and a maximal subgroup of \(F_{22}\) of the form \(2^6\cdot\text{Sp}_6(2)\) [in J. Moori and Z. Mpono, Int. J. Math. Game Theory Algebra 10, No. 1, 1-12 (2000; Zbl 1029.20010)] and a maximal subgroup of the group \(\text{Sp}_8(2)\) of the form \(2^7\cdot\text{Sp}_6(2)\) [in F. Ali and J. Moori, Int. J. Math. Game Theory Algebra 14, No. 2, 101-121 (2004; Zbl 1076.20005)].
In the paper under review the authors use the method of Fischer-Clifford matrices to compute the character table of a group \(B(2,n)\) which is the Weyl group of the Lie algebra of type \(B_n\). This group is of the form \(\overline G=2^n\cdot S_n\) where the extension is split and \(2^n\) denotes an elementary Abelian group of order \(2^n\). In general one can define the group \(B(m,n)\) to be the wreath product of \(\mathbb{Z}_m\) by \(S_n\), which is called the generalized symmetric group. We have \(B(m,n)=m^n\cdot S_n\) where \(m^n\) stands for \(n\) direct copies of the cyclic group \(\mathbb{Z}_m\) and the extension splits, and the method of Fischer-Clifford can be applied to it. The authors develop a series of computer programs written in GAP which give various parameters of the Fischer-Clifford matrices of \(B(m,n)\). One of the computer programs has the ability to form the blocks of the Fischer-Clifford matrices of \(B(2,n)\). For the demonstration of the computer programs the authors present the character table of the group \(2^6\cdot S_6\) in this paper. As the authors indicate, the computer program has the ability to compute the character table of the group \(B(2,25)\) which is in fact a large group.