I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit description.

Most interseting example for me is the case of complex Grassmannians. So to formalize a little bit...

For any two natural numbers $k, n$ with $k \leq n$ I want to have a finite subgroup $G_{n, k} \leq U(n)$ such that

- The generic orbit of $G_{n,k}$ on $U(n) / U(k)\times U(n-k)$ is
*big*. - There is a
*nice*model for $G_{n,k} \backslash U(n)/ U(k)\times U(n-k)$ where I would like to consider the double coset space as a metric space with distance function inherited from the symmetric space.

The much studied examples of discrete cocompact subgroups of noncompact symmetric spaces are messing up my Google / Mathscinet searches. I would be grateful for any pointers to relevant literature.

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