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In this paper we obtain a classification of those subgroups of the finite general linear group GLd (q) with orders divisible by a primitive prime divisor of qe − 1 for some e>12d. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross‐characteristic representations of some of the finite classical groups. 1991 Mathematics Subject Classification: primary 20G40; secondary 20C20, 20C33, 20C34, 20E99.

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A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'Nan-Scott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2-arc transitive graphs, and a new construction is given for a family of such graphs.

We give a self-contained proof of the O'Nan-Scott Theorem for finite primitive permutation groups.