Presentations of the Fischer groups \(D_ 4(2):{\mathbf S}_ 3\) and \(D_ 4(3):{\mathbf S}_ 3\) (Q1340975)

The author studies so called reduced Fischer pairs \((G,D)\), where \(G\) is a finite group generated by a conjugacy class \(D\) of 3-transpositions such that the largest solvable normal subgroup of \(G\) is in \(Z(G)\). Let \(S=\langle S\cap D\rangle\cong S_5\) be a subgroup of \(G\). Then he shows that \((G,D)\) is symmetric or classical if and only if for each \(d\in D\) there is some \(S\) such that \(D_d\cap S\neq\phi\). This suggests that the sporadic groups and the groups of the title are factors of the group \(G_0(p,q)\) generated by \(S\) and an element \(y\) with the relations \(y^2=(dy)^3=1\) for all transpositions \(d\) in \(S_5\) and \((z(25))^p=(z(15))^q\) with \(z=y(4321) y(1234)y\). The main result is that \(G_0(2,3)\cong D_4(2):S_3\), \(G_0(3,3)\cong D_4(3):S_3\). There is a similar result due to \textit{J. I. Hall} and \textit{L. Soicher} [Commun. Algebra 23, 2517-2559 (1995; Zbl 0830.20052)]. But in contrast to that paper the proof given by the author is free of any use of a computer.