Why does $S_4$ act on $V$ as $S_3$ acts on $\{1,2,3\}$?

Question

Let $V=\{(12)(34),(13)(24),(14)(23)\}$ the conjugacy class of $(12)(34)$. I don't understand why $S_4$ acts on $V$ as $S_3$ acts on $X=\{1,2,3\}$. So, I agree that $X\underset{set}{\cong} V$, but that's all.

Answer 1

The action of $S_3$ on $\{1,2,3\}$ gives you all permutations of its elements. Similarly, the action of $S_4$ on $V$ gives you all permutations of its elements. In this sense they act the same.

I wouldn't say that $S_4$ acts on $V$ as $S_3$ acts on $\{1,2,3\}$, as $S_3$ acts faithfully on $\{1,2,3\}$ whereas $S_4$ doesn't do so on $V$. But perhaps I'm misunderstanding the use of the word 'as' here.