Find all elements of the set $ \{ \sigma \in S_6 : \sigma(12)(34)\sigma^{-1}=(12)(34)\}$

Question

Find all elements of the set $ \{ \sigma \in S_6 : \sigma(12)(34)\sigma^{-1}=(12)(34)\}$

I have read that two permutations conjugate if they have the same cycle type. So we have a cycle type of $(1,1,2,2)$ Now is their a particular algorithm to find all these quickly? I belive their should be $16$ of them given $6\choose 2$$=15$.

Answer 1

Hint: $\sigma(12)(34)\sigma^{-1} = (\sigma(1),\sigma(2))(\sigma(3),\sigma(4))$.

And the size of this set is \begin{equation*} \frac{|S_6|}{\text{number of elements of this cycle type}} = \frac{6!}{{6\choose 2}{4\choose 2}/2} = 16. \end{equation*}