Enumerating isoclinism classes of semi-extraspecial groups

TL;DR

This paper classifies semi-extraspecial p-groups with a specific derived subgroup size by analyzing polynomial orbits and introduces a new construction method for these groups using Heisenberg groups over local algebras.

Contribution

It provides a complete enumeration of isoclinism classes of certain semi-extraspecial groups and introduces a novel construction approach via central quotients of Heisenberg groups.

Findings

Enumerated isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p^2

Established a correspondence between group classes and GL(2,p)-orbits of polynomial sets

Presented a new construction method for semi-extraspecial groups using Heisenberg groups over local algebras

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.