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specific representation for groups inheriting from Sage's Group class

I am working on designing a class for the group of diagonal symmetries associated with a polynomial singularity. The representation of group elements matters - I want to store a tuple of rationals associated with each group element, and add a bunch of functions that use those values. I'm wondering if there is a way to do this while still inheriting from the Sage Group class (or something that inherits from the Group class).