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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](http://math.stackexchange.com/a/102198/21436) of mine helpful. For alternating groups, a similar argument thread can be woven.2012-03-10

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