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

TL;DR

This paper constructs and analyzes a new near octagon related to the group L_3(4), exploring its geometric properties, automorphism group, and connections to known distance-regular graphs.

Contribution

It provides a direct construction of the L_3(4) near octagon via a split extension and investigates its geometric and automorphism properties.

Findings

Derived geometric properties of the L_3(4) near octagon

Determined the full automorphism group of the near octagon

Established a relation to the second subconstituent of a known distance-regular graph

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.