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

John Bamberg, Saul D. Freedman, Luke Morgan

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\m…