The Primitive Distance-Transitive Representations of the Fischer Groups

Research partly funded by the DFG (Germany) in the framework of the joint research project \Algorithmic Number Theory and Algebra", and also by the European Community HCM grant in \Computational Group Theory". We classify the primitive distance-transitive representations of the Fischer sporadic simple groups and their automorphism groups. It turns out that the only primitive distance-transitive representations of these groups are their rank 3 representations. In the process of our work, we also classify and study the primitive multiplicity-free permutation representations of these Fischer groups. Our methods, which we describe in some detail, demonstrate the use of computational and randomized techniques in the classification of distance-transitive graphs and the study of very large permutation representations.