A Distance-2-Spread of the Generalized Hexagon H(3)

Abstract

In this paper, we construct a distance-2-spread of the known generalized hexagon of order 3 (the split Cayley hexagon H(3)). Furthermore we prove the uniqueness of this distance-2-spread in H(3) and show that its automorphism group is the linear group L2(13). We remark that a distance-2-spread in any split Cayley hexagon H(q) is a line spread of the underlying polar space Q(6, q) and we construct a line spread of Q(6, 2) that is not a distance-2-spread in any H(2) defined on Q(6, 2).