Composition of permutation cycles

NanakiXIII
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I'm studying the S_n groups and I've been calculating a bunch of compositions of m-cycles. I haven't, however, been able to find a general pattern in how to do this for arbitrarily complicated cycles. Are there any general composition rules or is there no other way to find the compositions besides applying each cycle one by one and looking at the end result?
 
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You should state clearly what you mean by "calculating a bunch of compositions of m-cycles".

Are you asking how to multiply products of cyclic permutations?
 

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