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

Let (Formula presented.) be the finite simply connected version of an exceptional Chevalley group, and let V be a nontrivial irreducible module, of minimal dimension, for (Formula presented.) over its field of definition. We explore the overgroup structure of (Formula presented.) in (Formula presented.) and the submodule structure of the exterior square (and sometimes the third Lie power) of V. When (Formula presented.) 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 (Formula presented.) on (Formula presented.) is either (Formula presented.) or its normalizer in (Formula presented.).