Assume that I am given the table of irreducible characters of a finite group $G$. I realize that we can see the order of the centralizer of any element $g \in G$ by summing the squares in the corresponding column: $$|C(g)| = \sum_{\chi} |\chi(g)|^2.$$

I need to know the centralizer more exactly: what conjugacy classes occur in the centralizer and how often. For example, from the second column in the character table of S4 we can see that the centralizer of $(1 2)$ has order $1^2 + 1^2 + 1^2 + 1^2.$ It is $$\{(), \, (1 2), \, (3 4), \, (1 2)(3 4)\}$$ so I would like to read off the tuple $(1,2,1,0,0)$ from this table somehow.

If that is impossible then it would still be useful to know that there are three conjugacy classes occuring in this centralizer. Can that information be read off the character table?


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