# On finite groups with same complex-valued character table

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.

-
s/to/two/ # That is what he said... –  jmc May 18 '13 at 18:13

This is a complicated question. A pair of non-isomorphic groups with the same character table is sometimes called a "Brauer Pair". There are many such pairs, especially among $p$-groups.