The generating rank of the U4(3)U4(3) near hexagon

Yoshiara [Embeddings of flag-transitive classical locally polar geometries of rank 3, Geom. Dedicata 43 (1992) 121–165] and Bardoe [On the universal embedding of the near-hexagon for U4(3)U4(3), Geom. Dedicata 56 (1995) 7–17] showed that the embedding rank of the U4(3)U4(3) near hexagon E3E3 is equal to 21. The precise value of the generating rank of E3E3 has been unknown, see Cooperstein [On the generation of some embeddable GF(2)-geometries, J. Algebraic Combin. 13 (2001) 15–28]. In this note, we determine this generating rank. We give a short proof for the fact that the generating rank and the embedding rank of E3E3 are equal to 21.