The hyperplanes of the \(M_{24}\) near polygon
Consider the unique Steiner system \(S(5,8,24)\) on a set \(\Omega\) of 24 symbols. Let \(\Gamma = (P,L)\) be the partial linear space (with point set \(P\) and line set \(L\)) obtained by taking the 759 octads as points, and the triples of pairwise disjoint octads as lines. The space \(\Gamma\) is the (unique) regular near hexagon of order \((s,t,t_2) = (2,14,2)\) on 759 points; its automorphism group is isomorphic to the Mathieu group \(M_{24}\) [see for example \textit{A. E. Brouwer, A. M.Cohen} and \textit{A. Neumaier}, Distance-regular graphs. Ergeb. Math., Heidelberg: Springer Verlag (1989; Zbl 0747.05073)]. A geometric hyperplane, or hyperplane for short, of a partial linear space is a subset of the point set which meets each line in one or all points of the line. In the note under review the authors determine the \(M_{24}\)-orbits on the hyperplanes of \(\Gamma\) completing, correcting and simplifying the results of Chapter 4 of the work [\textit{E. W. Lambeck}, Contribution to the theory of distance regular graphs. Thesis, Eindhoven Univ. Technology (1990; Zbl 0758.05047)]. This classification is used in the study of circular extensions of \(\Gamma\) [see \textit{H. Cuypers}, Extended near hexagons and line systems. Eindhoven Univ. Technology (1999) (preprint)].
scientific article; zbMATH DE number 1887505