List all of the partitions of 6. For each partition π , give a permutation σπ is in S6 whose cycle structure is given by that partition. For each σπ , list all of the powers of σπ and indicate the order of σπ.
First, try to find a partition of 6 using the method shown in the link. You can also find some values of partition function in here. Each partition of 6 corresponds to a conjugacy class in S_6.
For instance, the partitions of 6 can be described as,
1. 6 : example cycle type (1, 2, 3, 4 ,5 ,6)
2. 5+1 : example cycle type (1, 2, 3, ,4 ,5)(6) = (1, 2, 3, 4 ,5)
3. 4+2 : example cycle type (1, 2, 3, 4)(5,6)
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Since P(6)=11, there are 11 conjugacy classes in S_6. Note that each cycle type in S_6 represents a conjugacy class(partition) in S_6.