TITLE: Character table does not determine group Vs Tannaka duality
QUESTION [42 upvotes]: From the example $D_4$, $Q$, we see that the character table of a group doesn't determine the group up to isomorphism. On the other hand, Tannaka duality says that a group $G$ is determined by its representation ring $R(G)$.
What is the additional information contained in $R(G)$ as opposed to the character table?

REPLY [4 votes]: An  RWTH Aachen thesis by Helena Skrzipczyk [1992]  is referred to by Eick and J. Mu"ller
J. ALg 304 (2006) On Brauer pairs. She gives several examples of non-isomorphic groups with
bijections between the character tables and power maps. They are the smallest order Brauer
pairs assuming such to be p-groups.