Number of Automorphisms of $S_6$ Trivial question about counting the number of automorphisms of $S_6$:
I know that for all $n \geq 3$, $Z(S_n)=1$, so Inn$(S_n) \cong S_n$.
I also know that $S_6$ has nontrivial outer homomorphisms, Out$S_n \cong \mathbb{Z}_2$.
Does this mean there are $S_n + \mathbb{Z}_2$ automorphisms in total, since Inn$(S_n)$ and Out$(S_n)$ are disjoint sets? So, other symmetric groups (excluding $S_2$) have $n!$ elements, while $S_6$ has $6! + 2$.

We have $\operatorname{Out}(S_6)=\operatorname{Aut}(S_6)/\operatorname{Inn}(S_6)\cong C_2$ and hence
$$
|\operatorname{Aut}(S_6)|=|S_6|\cdot |C_2|=6!\cdot 2.
$$
Here we have used that $|G/N|=\frac{|G|}{|N|}$.
