On Rank 2 Geometries of the Mathieu Group $M_{23}$

Abstract

In this paper we determine rank 2 geometries for the Mathieu group M 24 for which object stabilizers are maximal subgroups.

On primitive geometries of rank two

“…Our programs gave 170 geometries up to isomorphism. This confirms the results obtained by Kilic in [10].…”

“…Some geometries do not appear in that table as a rank two geometry does not necessarily give a design. Entry 28 * is the well known S 1 (4,5,11), that is the Steiner system associated to the Mathieu group M 11 . Table 6 gives the orbit lengths for every primitive permutation representation of M 22 .…”

“…Primitive coset geometries have been studied since the 1990's, first by a team led by Francis Buekenhout in Brussels [4,5] and later on by Peter Rowley and Nayil Kilic (see for instance [8,9,10,11,12]). These geometries may be used to construct codes, designs, graphs, etc.…”