A Realization Problem for Character Tables

Given an $n\times n$ table of complex numbers, are there known sufficient conditions for the table to be the character table of a finite group? Representation theory gives plenty of necessary conditions, but I can't imagine they'd be enough in general.

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certainly such conditions are not known in general, otherwise e.g. existence proofs for sporadic simple groups were much easier. For these groups, construction of the character table often was done before the construction of the group itself. –  Dima Pasechnik Mar 3 '12 at 10:16