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)⊗ 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)⊗ Q(5,2) makes use of a new combinatorial construction of this near hexagon.