I can see an api call to get a Group by name, but not one to get by id.
With categories, you can get the category by id by calling /c/{id}/show, but I haven’t been able to find something similar for groups. I tried /g/{id}/show and /groups/{id}/show, but neither worked.
I took a quick look and it seems you are right. We generally use the group name in URLs which is also a unique identifier for the group.
What you can do is retrieve a list of groups via /groups.json and then find the group by id in the JSON that is returned.
But I already have the id!