Construction of Large Permutation Representations for Matrix Groups II

Abstract.

A new method to construct permutation representations from matrix groups is described. Not only the permutations are constructed, but also a base and strong generating system are built. In addition, the faithfulness of the permutation representation is guaranteed. All this can be achieved without actually computing and storing the permutations as vectors of images (which are typically very large). To show the usability and applicability to even large problems, a permutation representation of degree 173 067 389 of Janko's fourth group has been computed here. It is used in another publication for a very short, explicit and self-contained existence proof of Janko's fourth group. Finally, defining relations for J ₄ are obtained.