What are the necessary and sufficient conditions for two finite groups $G$ and $H$
to have same complex-valued character table?
Is there any criterion for which one could know about the character table similarity of two finite groups without direct computations of each table?


Obviously two isomorphic groups have same character table, but according to the case of $Q\_8$ and $D\_8$, I'm searching for a weaker criterion.



On finite groups with same complex-valued character table
Finite groups have the same complex character tables if and only if their group algebras 
are isomorphic as quasi-Hopf algebras (if and only if the group algebras
are twisted forms of each other as Drinfel'd quasi-bialgebras, if and only if there is non-associative bi-Galois algebra over these groups).
For details see [arXiv:math/0001119](http://arxiv.org/abs/math/0001119). 


