Abstract

Motivated by the World Wide Web Atlas of Finite Group Representations and the re-cent classification of low-dimensional representations of quasisimple groups in cross-characteristic fields by Hiss and Malle, we construct with a computer over 650 rep-resentations of finite simple groups. Explicit matrices for these representations are available on the Internet and are included on an attached CD-ROM. Our main tool is a GAP program for decomposing permutation modules. It uses reduction modulo various primes and rational reconstruction to give an acceptable performance. In addition, we define standard generators for the groups under consideration, and exhibit black box algorithms for finding standard generators and checking whether given elements of the group are standard generators. To my parents Acknowledgements I have benefited greatly from the guidance and support of my supervisor, Professor Robert Wilson. I wish to thank him for his encouragement and enthusiasm in this project. I feel privileged to have been one of his students. I am indebted to my examiners Professor Derek Holt and Dr Paul Flavell for their detailed reading of the text and for pointing out several improvements. I thank Dr John Bray who has very helpfully shared his knowledge of computational group theory with me. He has also provided two interesting representations for inclusion here. I also thank Richard Barraclough for helping me with the Monster group programs and Dr Frank Lübeck for helping me with my questions about CHEVIE. My work has been made greatly easier by the GAP computer algebra system. I thank all the developers for their hard work in producing such a marvellous tool and making it freely available. I am very grateful to Sophie Whyte, Elizabeth Wharton and Marijke van Gans, with whom I have shared an office for the past three years. I wish to thank them for many useful conversations, for helping with the crossword, and for patiently putting up with my annoying habits. I also thank the School of Mathematics at the University of Birmingham for providing a stimulating environment for research.

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