Hyperbolic plane reflections and the Hall-Janko group (Q1970970)

The author introduces the notion of hyperbolic plane reflections in a symplectic space \(V\) over a finite field of characteristic 3 and defines a scalar product on the set of hyperbolic planes. This enables him to determine the order of the product of any two reflections. For \(\dim V=6\), then a graph \(\Gamma\) on a certain subset of hyperbolic planes is defined. It is shown that \(\Gamma\) is the regular near octagon of order \((2,4;0,3)\) (and on 315 vertices) associated with the finite simple Hall-Janko group \(HJ\) and that the corresponding 315 reflections generate the non-split extension \(2HJ\).