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

Then N G (E)/E = W A . Since D = E 8 and T = S 8 are a-stable, g a g~l e N G (S) fl N G (E), and hence maps to an element of W A . LEMMA 1.1. Suppose E s and E h are o-stable G-conjugates of E. Then E 8 and E h are G a -conjugate if and only if the images in W A of g°g~x and h a h~x are W^-conjugate.Proof. This follows from [7, Proposition 1] and the choice of 5.By Lemma 1.1, the G a -classes of a-stable G-conjugates of E are in one-to-one correspondence with the conjugacy classes of W A .We have D = E 8 , T = S 8 . Write w = g a g~x e N(S) n N(E), and write w also for the corresponding element of W. Then (ow) 8 = o, (G aw ) 8 = G a , (S aw ) 8 = T a and (Eon) 8 = D a . Once we have chosen w therefore, we may freely work with ow, G W) S aw , Eon instead of a, G a , T a , D o , respectively. LEMMA 1.2. We have N Gow (E)/E aw = C W >W(A)). (WW(A)). Proof Write U = N Gow (E)/E aw , V = C WASince N G (E)/E = W A , and since a centralizes W, there is a monomorphism from U into V. Let v e V, and pick n e N G (E) mapping to v under this monomorphism. Then n stabilizes the coset Eow. Hence by Lang's theorem, for some e e E, ne centralizes ow. Then neE aw e U and maps to v. Thus U = V. LEMMA 1.3. (i) Suppose L =£ F 4 (2 a ) or G 2 (3 a ). Then Aut(L) = ZJV Aut(L) (£ CTVV ). (ii) Let L = F 4 (2 ) or G 2 (3 a ), and let r be a graph automorphism of L normalizing S. Let r 0 be the permutation of O induced by r, and assume that Ar o = A. Assume also that r fixes the W A -class of wW(A). Then Aut(L) = 3 0 0

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