Enumerating all finite groups is a formidable task and as far as I know an unsolved problem in its full generality. I have thought about what if one approaches this by constructing the groups stepwise by constructing character tables by systematically combining sets of "valid" irreducible representations. That leads me to two basic questions:
Is every group uniquely characterised (up to isomorphism) by the full set of its irreducible representations?
Is every set of "characters" grouped into irreducible representations which fulfills the Schur theorem automatically a "character table" belonging to a group?
Edit: In case the answer to 1. is no: Which information is missing?