I've seen a handful of cases of character tables for small groups like S3, S4 and the quaternion group.
I have seen these tables constructed by coming up with some representations and exploiting orthogonality relations and the rule that the sum of $\overline{x}x$ for each element $x$ in the row equals the order of the group (which falls out of the characters of irreducible representations being an orthonormal basis for the characters of all representations).
Computing the character table in this way basically proceeds row by row.
I'm wondering whether there's a way to efficiently compute a single column at once without computing the rest of the table, i.e. all the traces associated with a particular conjugacy class. For example, the column associated with the conjugacy class containing just the identity element gives the dimensions of all the irreducible representations.
I am mostly interested in small finite groups, but I have zero intuition for what kinds of algorithms exist or don't for computing columns of the character table, so I'll accept anything that works on an interesting subclass of groups.