A design and a geometry for the group Fi 22

The Fischer group Fi22 acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it. In this paper we construct and investigate a partial linear space admitting the Fischer group Fi22 and defined combinatorially from a symmetric design also admitting this group. For details about the Fischer group we refer to the ATLAS of Finite Groups [3]. The Fischer group Fi22 has two conjugacy classes of subgroups of index 14080 isomorphic to O7(3). One of these subgroups has orbits of size 1, 3159, and 10920 on its own conjugacy class, and 364, 1080, and 12636 on the other. Construct an incidence structure in which the elements of the two classes are points and blocks respectively, and a point and block are incident if one lies in an orbit of size 364 or 1080 of the other. In this situation, we would expect that the number of blocks incident with two points takes one of two possible values, depending on the orbit of the pair of points. Remarkably, it occurs that the two values are the same, namely 148. Thus, we have a square (or symmetric) 2-(14080,1444,148) design (see [2, Chapter 1] for definitions). (This can be seen from the collapsed adjacency matrices on the Web page [10]. The orbits of the point stabilizer in Fi22 are numbered in order of increasing size; so the relevant fact is that the sum of the