Automorphisms acting on Symmetric Groups

Question

I am unsure of how the Automorphism group of a symmetric group acts on the symmetric group. I know that $$Aut(n)=S_n$$ for all $n\neq 2,6$. However, how does then for example the automorphism $(12)$ act on $S_4$, or in particular the subgroup: $$\{e,(12)(34),(14)(34),(13)(24)\}$$

Thank you.

Answer 1

Just calculating these conjugates with GAP:

gap> ((1,2)*(3,4))^(1,2);
(1,2)(3,4)
gap> (1,4)*(3,4);
(1,3,4)
gap> ((1,4)*(3,4))^(1,2);
(2,3,4)
gap> ((1,3)*(2,4))^(1,2);
(1,4)(2,3)

Answer 2

There is a very quick way to calculate conjugation by a permutation: if we are conjugating $\sigma$ by $\tau$, and $\sigma$ is given in cycle notation, just write $\sigma$ down again, but with all entries of $\sigma$ being acted upon by $\tau$. For example $(1,2,3)^{(1,2)}$ is simply $(2,1,3)=(1,3,2)$.