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

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.