On the Hall-Janko graph with 100 vertices and the near-octagon of order (2,4)

In this paper, we construct the Hall-Janko graph inside the split Cayley hexagon H(4). Using this construction, we then embed the near-octagon of order (2,4) as a subgeometry of the dual of H(4), with J_2:2 as its automorphism group. These constructions are based on a lemma determining the possibilities for the structure of the intersection of two subhexagons of order 2 in H(4).