The ${\mathrm{L}}_3(4)$ near octagon

Abstract

In recent work we constructed two new near octagons, one related to the finite simple group ${\mathrm{G}}_2(4)$ and another one as a sub-near-octagon of the former. In the present paper, we give a direct construction of this sub-near-octagon using a split extension of the group ${\mathrm{L}}_3(4)$. We derive several geometric properties of this ${\mathrm{L}}_3(4)$ near octagon, and determine its full automorphism group. We also prove that the ${\mathrm{L}}_3(4)$ near octagon is closely related to the second subconstituent of the distance-regular graph on 486 vertices discovered by Soicher in 1993.

23 pages