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? 



Character table does not determine group Vs Tannaka duality
If I'm not mistaken, the extra information is not contained in the representation ring, you have to look at the category of representations. In particular, you want to look at the representation category equipped with its forgetful functor to vector spaces. Then the group can be recovered as the automorphisms of this functor. Here's a blogpost I wrote which may be helpful: <http://concretenonsense.wordpress.com/2009/05/16/tannaka%E2%80%93krein-duality/>


