A rank 3 geometry for the sporadic simple group of Suzuki

Question

I am actually studying coset geometries (in the sense of Tits and Buekenhout) for the sporadic simple group of Suzuki. I came aware that Buekenhout found in 1979 a geometry over the following diagram

      c          6
O----------O----------O
1          4          4

However, I couldn't find any information about the maximal (or minimal) parabolic subgroups of this geometry.

Has anyone ever studied this geometry? Is there a paper where I could find the informations I am looking for?

As usual, thanks in advance!

Answer 1

I finally found the maximal parabolic subgroups of this geometry. Let us first denote the types of the elements with 0,1 and 2 when reading the diagram from left to right, and let us denote with $G_0$, $G_1$ and $G_2$ the stabilizer of an element of type 0, 1 and 2 respectively. Then we have:

$$ G_0 = G_2(4),\quad G_1 = 2^{2+8}:(A_5 \times S_3),\quad G_2 = 2^{4+6} : 3 A_6 $$ which are all maximal subgroups of $Sz$, and the Borel is $$B = 2^{12}.3^2$$ Historically, I read that this geometry was built using polar spaces (see Francis Buekenhout, Diagrams for geometries and groups, Journal of Combinatorial Theory A, 27, 121-151, 1979 doi:10.1016/0097-3165(79)90041-4). However, I have not studied yet how to build it geometrically.