Semiclassical parabolic systems related to \(M_{24}\) (Q1388157)

A parabolic system is of classical type if all the rank 2 residues are rank 2 Lie type groups. In order to include more sporadic groups, one would like to add some more possibilities for these residues. The most common choices correspond to the tilde geometry, the Petersen graph and the complete graph. In the paper under review, one considers so-called semiclassical parabolic systems: all rank 2 residues are rank 2 Lie type groups in characteristic two or the triple cover of the symmetric group \(S_6\) (and this occurs at least once). If the rank of the system is equal to three and the diagram is linear with nontrivial rank 2 residues \(3.S_6\) and \(L_3(2)\), then \textit{S. Heiss} [J. Algebra 142, No. 1, 188-200 (1991; Zbl 0735.20007)] and \textit{P. Rowley} [J. Lond. Math. Soc., II. Ser. 40, No. 1, 40-56 (1989; Zbl 0687.20012)] showed that it corresponds to one of the groups \(He\), \(M_{24}\) and \(3^7Sp_6(2)\). \textit{C. Wiedorn} in her thesis classified semiclassical parabolic systems of rank four, if all rank 3 subsystems (under the condition there is at least one such) of the above type correspond to \(3^7Sp_6(2)\), and if moreover all rank 1 subgroups are isomorphic to \(S_3\). Under the latter hypothesis, the authors classify in the paper under review the remaining rank 4 semiclassical parabolic systems (which are called of type \(M_{24}\)) containing a rank 3 subsystem of above type. Geometries for Conway's group \(.1\) and (covers of) the Fischer groups emerge. The proof uses the amalgam method and the Todd-Coxeter algorithm, and also a clever way of identifying certain groups by using more generators than theoretically necessary.