This paper continues the investigations of this series. Suppose K = GF(r) is a field for a prime r; G is a nilpotent; V is a nonsingular symplectic space with form g; and V is a faithful irreducible K[G]-module where G fixes the form g. This paper describes completely the structure of G and its representation upon V when G is symplectic primitive. This latter condition is described in §4 and is a primitivity condition. Let C be a cyclic p-group for a prime p and Q an extraspecial a-group for a

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... ifferent prime q. Suppose C acts faithfully as automorphisms of Q centralizing Z(Q). Assume also that Q/Z(Q) is a chief factor of the semidirect product CQ. The representation theory of CQ is well known. It was first studied thoroughly in Theorem B of Hall and Higman [10] . In this sequence of papers we look at the basic configurations arising out of Theorem B. In Hall-Higman Type Theorems. I [2] (henceforth referred to as [I] ) two questions related to Theorem B were posed. The second question will concern us here. Suppose AG is a group with normal solvable subgroup G and nilpotent complement A. Assume that S < G is an extraspecial s-subgroup normalized by AG where (\A |, s) = 1. Further, assume that Z(S) < Z(AG), S/Z(S) is an AG-chieî factor, and A is faithful upon 5. Let k be a field of characteristic other than s ; and let V be an irreducible k [A G] -module nontrivial for Z(S). When does V\A contain a regular direct summand? With A = C and G = S = Q this is just the case described in the first paragraph. The possible theorems which give answers to this question can be proved by induction in several steps. First one reduces to the case where k is algebraically closed and V\s is irreducible. Next one is forced to look at the representation of .4G upon S = S/Z(S). The module S is analyzed by group-theoretic induction. That is, S is induced by a certain minimal submodule, S0. In providing answers to our question, then, one must take into Received by the editors October 24, 1972. AMS (MOS) subject classifications (1970). Primary 20C15, 20C20; Secondary 16A64, 20H20. account the structure of S0 and the induction from S0 to S. The whole method being sketched here is fully described in [4, §7]. In this paper we are concerned with the structure of S0. In fact, S0 ■ S0/Z(S) where S0 is an extraspecial subgroup of 5. Further, the stabilizer of 50 is NA G(S0) = A0G0 where A0 <^4 and G0<:G. Further, S0 has certain minimality conditions imposed upon it. We describe these now. The commutator map induces a nonsingular symplectic form upon S0 fixed by A0G0. Further, things are so chosen that one of the following two conditions holds. (1) 50|L is homogeneous for all L normal in i40G0 (i.e. S0 is quasiprimitive).