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 χi 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.

We now describe the methods used to classify the transitive representations of rank at most 5 of the sporadic almost simple groups, and to analyse their associated generalized orbital digraphs. The methods for the analysis of these digraphs via their collapsed adjacency matrices are general, and are not restricted to sporadic groups or to low rank representations.

Permutation characters

Let G be a finite group. For a permutation representation of G on a finite set Ω, the permutation character π is the map π : G → ℂ such that, for g ∈ G, π(g) is the number of points of Ω fixed by g. Now π is the character (trace map) of the natural matrix representation for G on the complex vector space ℂΩ (see for example [Isa76, p. 68]), and so π is a sum of complex irreducible characters for G. In particular the multiplicity of the trivial character 1G in π is equal to the number of orbits of G in Ω (see [Isa76, Corollary 5.15]) and in particular π is a transitive permutation character (that is, the permutation character of a transitive permutation representation) precisely when 1G has multiplicity 1 in π. In order to understand our arguments concerning permutation characters, the reader who is not familiar with the basic theory of (permutation) characters for finite groups is urged to consult the excellent reference [Isa76], especially Chapter 5.

In this chapter we give tables summarizing the transitive permutation representations and distance-regular generalized orbital graphs we have studied and classified in this book. In particular, Tables 5.1, 5.2, 5.3, and 5.4 list the primitive representations of respective ranks 2, 3, 4, and 5 of the sporadic almost simple groups, while Tables 5.5, 5.6, and 5.7 list the imprimitive representations of these groups for respective ranks 3, 4, and 5. These tables also give the number of distance-regular generalized orbital graphs for each of these representations. Finally, Table 5.8 gives a summary of the graphs of diameters 3 and 4 having a distance-transitive action by a sporadic almost simple group.

For each table of permutation representations, the column labelled ‘G’ gives the group being represented, the ‘case’ column gives the roman-numbered case for G in Chapter 4, where the representation is described in detail, the column labelled ‘H’ gives the point stabilizer, and ‘degree’ the degree of the representation. Notation of the form K[.2] for G and L[.2] for H denotes two representations of the same rank and degree: one of K.2 on the cosets of L.2, and the restriction of this representation to K on the cosets of L. In this situation there will be two entries in the ‘case’ column, corresponding respectively to K and K.2.

Introduction

Many interesting finite geometries, graphs and designs admit automorphism groups of low rank. In fact, it was a study of the rank 3 case which led to the discoveries and constructions of some of the sporadic simple groups (see [Gor82]). For several classification problems about graphs or designs, the case where the automorphism group is almost simple is of central importance, and many of the examples have a transitive automorphism group of low rank. This is the case, for example, for the classification problems of finite distance-transitive graphs [BCN89, PSY87], and of finite flag-transitive designs [BDD88, BDDKLS90].

This book presents a complete classification, up to conjugacy of the point stabilizers, of the faithful transitive permutation representations of rank at most 5 of the sporadic simple groups and their automorphism groups. These results, summarized in Chapter 5, filled a major gap in the existing classification results for finite, low rank, transitive permutation groups. For each representation classified, we also give the collapsed adjacency matrices (defined in Section 2.3) for all the associated orbital digraphs. We use these collapsed adjacency matrices to classify the vertex-transitive, distance-regular graphs for these low rank representations, and discover some new distance-regular graphs of diameter 2 (but of rank greater than 3) for the O'Nan group O'N, the Conway group Co2, and the Fischer group Fi22.

This book is a contribution to the study of the sporadic simple groups, which are the twenty-six fascinating finite simple groups which do not belong to any of the infinite families of finite simple groups. In particular, we classify and study all graphs admitting a sporadic simple group or its automorphism group as a vertex-transitive group of automorphisms of rank at most 5. We do this by first finding all the representations of these sporadic groups as transitive permutation groups of rank at most 5. The classification, construction and analysis of these representations and graphs involve both theoretical arguments about permutation groups and characters, and computational methods involving the use of various computer systems for group theory, character theory and graph theory. The ATLAS of Finite Groups was an invaluable resource throughout all our work.

We have tried to make most of our techniques accessible to a beginning graduate student who is willing to study some basic computational group theory. In particular, the construction and analysis of collapsed adjacency matrices are spelled out in detail. For the theoretical analysis of certain permutation representions, some knowledge of permutation group theory is assumed.

Using the presentations for sporadic groups which are given in this book, and the generating sets for the point stabilizers, the reader should be able to construct and study most of the representations and graphs described in this book.