Embedded Thick Finite Generalized Hexagons in Projective Space

We show that every embedded finite thick generalized hexagon of order (s, t) in PG(n,q) which satisfies the conditions

1. s = q
2. the set of all points of generates PG(n, q)
3. for anypoint x of , the set of all points collinear in withx is containedin a plane of PG(n, q)
4. for any point x of , the set of allpoints of not oppositex in is contained in a hyperplane ofPG,(n, q)

is necessarily the standard representation of H(q) in PG(6,q) (on the quadric Q(6, q)), the standard representation ofH(q) for q even in PG(5, q) (inside a symplectic space), orthe standard representation of H(q, ) in PG(7, q) (where the lines of are the lines fixed by a trialityon the quadric Q⁺(7, q)). This generalizes a result by Cameronand Kantor 3], which is used in our proof.