It seems that in the main Groups list the groups are sorted by slug (groupname) and not by the Full Name.
Since the slugs cannot be edited by group owners, but the full name can, this means that the order in the list can get very inconsistent if the full names are not the same as the slugs (groupname)
Could this either be an admin option to sort the groups list by Full Name rather than slug, or simply be changed to sorting by Full Name as the default.
From the user point of view the Full Name is what is what is probably expected everywhere, and the slug is just a techie convenience to make a better url.
Currently we have inconsistency between the use of Full Name and slug as a primary field. If slug has to be the sort field then Full Name becomes a bit pointless.