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**universalsandbox** The symmetry group $\displaystyle S_{6}$ has a total number of elements 6!

How do you find the number of conjugacy classes and their sizes?

I have done the following:

6------------------(123456) ---- SIZE = 6!/6 = 120

5+1---------------(12345)(6) -- SIZE = 6!/5 = 144

4+2---------------(1234)(56) -- SIZE = $\displaystyle \color{red}{6\choose2}3! = 90$

4+1+1------------(1234)(5)(6) -- SIZE = $\displaystyle \color{red}{6\choose2}3! = 90$

3+3---------------(123)(456) -- SIZE = $\displaystyle \color{red}{6\choose3}\times2 = 40$

3+2+1------------(123)(45)(6) -- SIZE = $\displaystyle \color{red}{6\choose3}{3\choose1}\times2 = 120$

3+1+1+1---------(123)(4)(5)(6) -- SIZE = $\displaystyle \color{red}{6\choose3}\times2 = 40$

2+2+2------------(12)(34)(56) -- SIZE = $\displaystyle \color{red}{6\choose2}{4\choose2}\div3! = 15$

2+2+1+1---------(12)(34)(5)(6) -- SIZE = $\displaystyle \color{red}{6\choose2}{4\choose2}\div2 = 45$

2+1+1+1+1------(12)(3)(4)(5)(6) -- SIZE = $\displaystyle \color{red}{6\choose2} = 15$

1+1+1+1+1+1---(1)(2)(3)(4)(5)(6) = e ---- SIZE = 6!/6! = 1

It appears there is 11 conjugacy classes in total. The sizes must sum to 720.

Is this correct what I'm doing? How do you compute the size of the remaining classes? it's 6!/(something) I'm not quite sure.