I have 2 questions but I am not sure if they are reasonable:
1) Given a finite set of characters $\chi$, is it always possible to find a finite group that has $\chi$ as its set of irreducible characters? Intuitively I feel this is not true, but I am not sure how to approach proving this.
2) Given a set of irreducible characters, and the knowledge that they are generated by a group, how could one go about finding the underlying group?
Thanks for your time!