From my understanding, the centralizer of a permutation $p$ can be computed by including the identity permutation $()$ and then finding all the equivalent ways to represent $p$ (which can be done by rearranging the order of the disjoint cycles and rearranging the elements within the disjoint cycles).
So what I get that the centralizer of $(1, 2, 3)$ is the set:
$S = \{(), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 2, 1), (3, 1, 2)\}$
Is that correct?