So far, I've seen that the following permutations are in the centralizer: $(1 4), (2 5), (3 6)$, products of these transpositions(EDIT: not all of these are in the centralzier), $(1 2 3), (1 3 2), (4 5 6), (4 6 5)$, product of any two disjoint 3-cycles above and the identity permutation. Are these all the permutations in $C_{S_6}((1 2 3)(4 5 6))$?
EDIT: Okay, from using the fact that if $x$ commutes with $(1 2 3)(4 5 6)$, we have to have: $(1 2 3)(4 5 6) = (x(1) x(2) x(3))(x(4) x(5) x(6))$, is it right to think that $x$ either sends $(1 2 3)$ and $(4 5 6)$ to themselves or swap them?
For example, if we send $(1 2 3)$ and $(4 5 6)$ to themselves, we get the identity permutation. If we send $(1 2 3)$ to $(3 1 2)$ and $(4 5 6)$ to itself, we get $(1 3 2)$. If we send $(1 2 3)$ to $(2 3 1)$ and $(4 5 6)$ to itself, we get $(1 2 3)$.
We get more permutations: $(4 5 6)$, $(4 6 5)$ and products of two disjoint 3-cycles. These are all the odd permutations in the centralizer of $(1 2 3)(4 5 6)$.
The even permutations in the centralizer come from permutations that swap $(1 2 3)$ and $(4 5 6)$. We have: $(1 4)(2 5)(3 6)$, $(1 6)(2 4)(3 5)$ and $(1 5)(2 6)(3 4)$. Am I right to say that these, with the odd permutations above, are all the permutations in $C_{S_6}((1 2 3)(4 5 6))$?
HINT: Let $\sigma\in S_n$ and $(a_1,a_2,\ldots,a_n)$ be a cycle in $S_n$. Then $\sigma(a_1,a_2,\ldots,a_n)\sigma^{-1}=(\sigma(a_1),\sigma(a_2),\ldots,\sigma(a_n))$.