On p-groups with automorphism groups related to the exceptional Chevalley groups

Let (G) over cap be the finite simply connected version of an exceptional Chevalley group, and let V be a nontrivial irreducible module, of minimal dimension, for (G) over cap over its field of definition. We explore the overgroup structure of (G) over cap in GL(V), and the submodule structure of the exterior square (and sometimes the third Lie power) of V. When (G) over cap is defined over a field of odd prime order p, this allows us to construct the smallest (with respect to certain properties) p-groups P such that the group induced by Auto(P) on P/Phi(P) is either (G) over cap or its normalizer in GL(V):