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Is there a systematic way (other than trial-and-error) of finding the conjugacy classes, as well as the number of these and representatives of these classes, for a given finite group?

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For symmetric groups, you'll find this answer of mine helpful. For alternating groups, a similar argument thread can be woven. –  user21436 Mar 10 '12 at 19:03

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Many computer algebra packages do this in a way which must be considered reasonably systematic. If you want to do it by hand, then even for "well-known" groups, the answer is quiet difficult and complicated. The theory of the rational canonical form describes how to determine the conjugacy classes in ${\rm GL}(n,F)$ when $F$ is a finite field. For other finite "classical" groups, the description of the conjugacy classes becomes more difficult, and has received attention from some very strong mathematicians. As for exceptional groups of Lie type such as $E_{6}(q),E_{7}(q)$ and $E_{8}(q),$ the situation is yet more complex. In principle, (say when dealing with a reasonably small group), if you want to determine the representative for the conjgacy classes, the strategy is clear: pick an element $x \in G,$ determine $C_{G}(x),$ and exhibit the $[G:C_{G}(x)]$ conjugates of $x.$ Then choose an element $y \in G$ not already listed, determine $C_{G}(y),$ etc., until eventually all elements of $G$ are accounted for. A priori, there may not be any significant shortcut available for a general finite group, unless you have further information.

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