How can you get the route name/path of groups?
There is probably a *.routing.yml in the group module where you can find the route name, but routes can also be defined dynamically. So better use a tool, for the command line Drupal Console or for UI the devel module:

Install the devel module, visit the page you want to find out the route name for and click on "Current route info". The route name is in the variable _route. You find also the route parameters you might need to further narrow down the route.