so I've been studying the centrally extended Galilei group, by restricting to 1D translation, the group becomes abelian and that means that all the irreps of the groups are of dimension 1. A consequence of this statement is that this version of the group does not have any internal symmetries (well it has but they all act trivially on the Hilbert space).
So the way I understood it is that the internal structure of the group should come from the multiplicities (if the multiplicities of the regular representation are all of dimension 1, there's no internal structure and if they have higher dimensions then there's is).
This was the conclusion of a discussion I had with a professor, but I have tried to research these information and I couldn't find anything really clear. Even the definition of the concept of internal symmetries isn't clear at this point. Can someone help me with some references and some course notes, that'd be great.