Non-abelian representations of the slim dense near hexagons on 81 and 243 points

TL;DR

This paper demonstrates non-abelian representations of specific near hexagons using extra-special 2-groups, introducing a new combinatorial construction for one of these geometries.

Contribution

It provides the first known non-abelian representations of certain near hexagons in extra-special 2-groups and introduces a novel combinatorial construction for these geometries.

Findings

Non-abelian representation of $Q(5,2) imes ext{L}_3$ in $2^{1+12}_+$

Non-abelian representation of $Q(5,2) imes Q(5,2)$ in $2^{1+18}_-$

New combinatorial construction for the near hexagon $Q(5,2) imes Q(5,2)$

Abstract

We prove that the near hexagon $Q(5,2) \times \mathbb{L}_3$ has a non-abelian representation in the extra-special 2-group $2^{1+12}_+$ and that the near hexagon $Q(5,2) \otimes Q(5,2)$ has a non-abelian representation in the extra-special 2-group $2^{1+18}_-$. The description of the non-abelian representation of $Q(5,2) \otimes Q(5,2)$ makes use of a new combinatorial construction of this near hexagon.