 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.