4 - The Individual Groups

Summary

We now handle individually each sporadic almost simple group G having a faithful pseudo-permutation character of rank at most 5, and list each such character π found by our computer search. The numbering of the irreducible characters χ_i agrees with that in the GAP version of the ATLAS character tables, which agrees with the printed ATLAS [CCNPW85] in the case of simple groups. We examine each of these pseudo-permutation characters in turn to determine if it is the character of some permutation representation of G, and, if so, we describe all such representations (up to permutational isomorphism). Specifically, for each representation of rank up to 5, we provide

• the irreducible constituents of the permutation character, as well as the degrees of these constituents,

• a point stabilizer,

• (for rank > 2) the collapsed adjacency matrices for the nontrivial orbital digraphs, and

• (for rank > 3) the non-complete distance-regular generalized orbital graphs.

Note that the information on rank 2 and 3 graphs which is suppressed in this chapter is given in a general way in Section 2.5.

We also give presentations for many sporadic almost simple groups. Specifically, let K be a sporadic simple group, other than Ru, Ly, and B, such that K has a faithful permutation representation of rank at most 5.