On $p$-groups with automorphism groups related to the Chevalley group $G_2(p)$

John Bamberg, Saul D. Freedman, Luke Morgan

Published 2017-10-04Version 1

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class $2$, rank $7$ and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) \times G_2(p)$ on the Frattini quotient $P/\Phi(P)$. The constructed group $P$ is the smallest $p$-group satisfying these properties, having order $p^{14}$, and when $p = 3$, our construction gives two non-isomorphic $p$-groups. To show that $P$ does satisfy these properties, we explore the reducibility of the exterior square of each irreducible $7$-dimensional $\mathbb{F}_q[G_2(q)]$-module, where $q$ is a power of $p$. We also give a general description of the automorphism group of a $p$-group of nilpotency class $2$ and exponent $p$.