| Résume | Given a subgroup $H$ of $GL(d,p)$, there exists a $d$-generated $p$-group $G$ whose automorphism group induces the linear group $H$ on the Frattini quotient $G/\Phi(G)$. The proof of this fact by Bryant and Kovács is non-constructive. In some sense $G$ is a ``non-linear representation'' of $H$. We consider how to go from a maximal subgroup $H$ of $GL(d,p)$ to a nonabelian $p$-group $G$ with minimal class/exponent/order. This involves understanding the $H$-submodule structure of Weyl modules and some intriguing $H$-homomorphisms. This work was motivated by applications to algorithms for $p$-groups and geometry, and is joint with John Bamberg, Luke Morgan and Alice Niemeyer. |
