Exceptional Presentations of Three Generalized Hexagons of Order 2

摘要

Exceptional presentations of the generalized hexagon of order (2,1) on 21 points in the complex projective plane and of the dual of the classical generalized hexagon of order (2,2) on 63 points in the quaternionic projective plane are recalled. A third presentation of this kind is described, namely that of the (unique) generalized hexagon of order (2,8) on 819 points in the octonionic projective plane. The construction that is employed leads to an embedding of the finite group of Lie type (3)D sub 4 (2) in the Lie group of type F sub 4 (the set of all real numbers).