I was studying representation theory when a question came to my mind.
It is known that there are non-isomorphic groups having the same character table, e.g. the 8-order dihedral group and the 8-order quaternion group. As a result, character theory is not enough descriptive to distinguish general non-isomorphic groups.
On the other hand, groups such as the alternating groups on 5 and 6 elements are known to be characterised by their character table. They are both known to be simple.
My question is the following. Which groups are characterised (up to isomorphism) by their character table? Certainly not all of them, but maybe some subclasses are known?
