A new near octagon and the Suzuki tower

TL;DR

This paper introduces a new near octagon of order (2,10), explores its automorphism group, and uses it to geometrically construct certain strongly regular graphs within the Suzuki tower, also discovering a smaller near octagon.

Contribution

It constructs a novel near octagon of order (2,10), analyzes its automorphism group, and provides geometric constructions of the G_2(4)-graph and Suzuki graph.

Findings

Constructed a near octagon of order (2,10) with automorphism group G_2(4):2.

Embedded 416 copies of the Hall-Janko near octagon within it.

Discovered a new near octagon of order (2,4) as a subgeometry.

Abstract

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.