Am I computing the centralizer of $(1, 2, 3)$ in $A_4$ correctly?

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?

Sort of.

First of all, as I said in the comment above,

$$(123)=(231)=(312)$$

and

$$(132)=(213)=(321).$$

To find the centraliser of a permutation, there is a nice result: conjugation by permutations preserves cyclic structure.