A non-abelian representation of the dual polar space $DQ(2n,2)$

Abstract

We prove that the dual polar space $DQ\left(2n,2\right)$, $n\ge 3$, of rank $n$ associated with a non-singular parabolic quadric in $PG\left(2n,2\right)$ admits a faithful non-abelian representation in the extraspecial $2$-group $2_{+}^{1+2^n}$. The near $2n$-gon ${\mathbb{I}}_n$ (section 2.4) is a geometric hyperplane of $DQ\left(2n,2\right)$. For $n\ge 3$, we first construct a faithful non-abelian representation of ${\mathbb{I}}_n$ in $2_{+}^{1+2^n}$ and subsequently extend it to a faithful non-abelian representation of $DQ\left(2n,2\right)$ in $2_{+}^{1+2^n}$.