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!



A rank 3 geometry for the sporadic simple group of Suzuki
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](https://doi.org/10.1016/0097-3165%2879%2990041-4)). However, I have not studied yet how to build it geometrically.


