## A note on the Hall-Janko near octagon

I recently found out via some computations in my research work that the Hall-Janko near octagon contains suboctagons isomorphic to the generalized octagon of order $(2,1)$. I couldn’t find it mentioned anywhere in the literature but it seemed like something that would be well known.

Anyway, I played around with it and found out that there are $280$ such suboctagons and every pair intersects in $5$ or $15$ points. If you define a graph by taking vertices as these $280$ suboctagons and adjacency defined by intersection in $15$ points then you get an $srg(280,36,8,4)$. Digging in on the internet for this strongly regular graph I found this page which suggested that it might be the same srg as mentioned on that page.  I contacted Andries Brouwer informing him about this discovery and he replied that the existence of this suboctagon was implicitly  known and in fact my graph is equal to the graph he mentioned on the web page. He has updated one of his other webpage related to the Hall-Janko near octagon (or the Cohen-Tits octagon) adding my observation: http://www.win.tue.nl/~aeb/graphs/HJ315.html

(I would try to explain what the Hall-Janko near octagon is in some other post)

EDIT: It has been pointed out to me by Bart that the fact that $GO(2,1)$ is contained inside the Hall-Janko near octagon is already mentioned in the paper “On the geometry of the Hall-Janko group on 315 points” by Satoshi Yoshiara.