Computing with matrix groups using permutation representations

Permutation representations constructed from matrix groups defined over finite fields often have very high degree. New techniques are presented for performing effective computations with the resulting permutation group. These techniques are designed to work in an environment in which the degree of the permutation group is considered too large to permit the use of standard permutation algorithms for solving problems such as computing the order of the group and testing simplicity. The theory has been successfully tested on a represent at ion of the sporadic simple group Ly, discovered by Lyons [10]. In [5], a permutation representation was constructed for Ly of degree 9, 606, 125 on a conjugacy class of subgroups of order 3. Using this permutation representation and no specific knowledge of the group, we are able to apply our methods to construct a base of at most four points for the result ing permutation group, compute its order and verify simplicity. Monte Carlo algorithms for group membership presented in [2] are used to improve the performance of these algorithms.