The conjugacy classes are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$.  What happens when you take products of two such elements?  I saw in a paper,
$$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$
Which I take to mean if you multiply a 6-cycle (**abcdef**) and a product of disjoint 3-cycles (**pq**)(**rs**)(**tv**), you can get 

 - a three-cycle (**abc**),
 - two two-cycles (**ab**)(**cd**),
 - a five-cycles (**abcde**),
 - a four-cycles and a two-cycle (**abcd**)(**ef**),
 - two three cycles (**abc**)(**def**)

With certain multiplicities. Is it predictable what kinds of conjugacy classes you get?  And there an interpretation of this as the intersection cohomology of some moduli space?