Consider a  finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. [MO 62088][1]). So we get ring with a basis and structure constants are natural numbers. Similar to what one has for product of irreps.
There many analogies between conjugacy classes and irreps in particular see [this article][2].

Tanaka-Krein duality states that group can be reconstructed from the  tensor category of its representations which is semisimple for finite groups, and hence carries the same information as ring + basis of irreps.

**Question:**  Can one reconstruct a group having  (ring + basis) made of  conjugacy classes ?

If not -  what partial information (e.g. character table) one can get ? 

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**Question:** Is there any relation between this ring and ring of irreps of the same group ? or may be some other group ?

(Remark. For abelian group they are isomorphic.)

**Question:** Are there any further analogies between ring of irreps and conjugacy classes except mentioned in the paper cited above ?

 


  [1]: http://mathoverflow.net/questions/62088/products-of-conjugacy-classes-in-s-n
  [2]: http://journals.cambridge.org/action/displayFulltext?type=1&fid=3077776&jid=PEM&volumeId=30&issueId=01&aid=3077768&bodyId=&membershipNumber=&societyETOCSession=