Оn automorphisms of the generalized hexagon of order (3,27)

Possible orders and fixed-point subgraphs for automorphisms of the generalized hexagon $S$ of order (3,27) are found. It is proved that, if the automorphism group of $S$ acts transitively on points, then $S$ is isomorphic to the classical generalized hexagon corresponding to the building of the Steinberg group $^3D_4(3)$.