Does this mean that anything after "danken" takes the dative case?
Yes, danken only takes an indirect (dative) object.
And here is a list of other verbs that go with dative: http://goo.gl/KlN6m
How would be "I thank the group"? Isn't it the same?
To me "Gruppen sounded" like Köchen!!! I replayed the phrase several times but just didn't heat the Grü. Is it my hearing?