In abstract terms, the 6- j symbols are precisely the information that is lost when passing from a semisimple monoidal category to its Grothendieck ring, since one can reconstruct a monoidal structure using the associator. For the case of representations of a finite group, it is well known that the character table alone (which determines the underlying abelian category and the Grothendieck ring structure) does not determine a group up to isomorphism, while the symmetric monoidal category structure does, by Tannaka-Krein duality. In particular, the two nonabelian groups of order 8 have equivalent abelian categories of representations and isomorphic Grothdendieck rings, but the 6- j symbols of their representation categories are distinct, meaning their representation categories are inequivalent as monoidal categories. Thus, the 6- j symbols give an intermediate level of information, that in fact uniquely determines the groups in many cases, such as when the group is odd order or simple.