Display Category Group Name in Template
You're close! It's slightly counterintuitive but group is a property of a category. So you need to grab the group from an actual category. This would work:

{% set catName  = craft.categories.group('caseStudiesCategories').first.group %}


Basically you're getting the first category in the group and grabbing the group name (you could also use [0] instead of .first).

If your entry has a category field, you can access the group name in a similar fashion:

{% set catName = entry.yourCategoryField.first().group %}