Looking at finite groups $G$. Of course the character table is (up to permutation of rows and columns) determined by $G$ up to isomorphism. I thought about why the converse is not true (question 1)?
Given a complete set of characters of a finite group $G$, but not the group table (or the generators). What is exactly the minimum amount of information that is missing, necessary to determine the group $G$ (i.e. the group table) up to isomorphism, uniquely?
I have been looking at the famous example of the quaternion group $Q$ and the dihedral group $D_4$. They have up to permutation of rows and elements the same character table. However, they disagree in the order. I understand that a large part of the information necessary to determine the group table up to isomorphism must be contained in the character table, but I fail to pinpoint what is exactly the missing information in the general case.