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.