Suppose that you are given two finite groups, for example, via their Cayley tables.
One can efficiently compute their character tables (efficiently = polynomial time in the order of the group), this is a result of Babai and Rònyai from 1990.
It is well-known that the character table does not determine the group (e.g. $Q_8$ and $D_4$ are non-isomorphic groups with the same character table), nevertheless, I am interested in the following:
Question: Is there an efficient way to check whether two character tables are the same?