Enumerating isoclinism classes of semi-extraspecial groups

Abstract

We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups with derived subgroup of order $p^2$. To do this, we enumerate $\text{GL}(2,p)$-orbits of sets of irreducible, monic polynomials in ${\mathbb{F}}_p[x]$. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.