I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $ K3$ -surface. It is known that this space is given by the bi-quotient $ SO(3,19;\mathbb{Z})\setminus SO(3,19)/SO(3)\times SO(19)$ and I know that this has the same rational cohomology as $ SO(3,19;\mathbb{Z})$ . Is there anything known about these groups? Unfortunately my knowledge on arithmetic groups is very limited.