Semi-extraspecial $p$-groups with automorphisms of large order

Abstract

In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $|G:G^{\prime }|-1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\mathrm{SU}}_3(p^{2a})$ for some integer $a$. If $p$ is odd, this is equivalent to saying that $G$ is isomorphic to a Sylow $p$-subgroup of ${\mathrm{SL}}_3(p^a)$.