What are the conjugacy classes of $(21)$ in $S_4$?
I think they are $\{(12),(13),(14),(23),(24),(34)\}$
But I'm not sure and the centralizer of $(12)$ is $\{(21),(34)\}$. Is this right?
2026-04-07 03:18:47.1775531927
Conjugacy Class of $(2 1)$ in $S_4$
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Conjugation of permutations preserves cycle type, so, yes, the conjugacy class is correct.
Note that the identity is in the centraliser too. It is also closed (as it is a subgroup of $S_4$), so the product $(12)(34)$ is in there as well.