This presentation describes the construction of an outer automorphism of group $S_6$. To begin with, we construct a homomorphism from $S_5$ to $S_6$, using transitive action. It is further argued that $$|\ker f| \le\dfrac{|S_5|}{6} = 20$$
And the question is, where does this inequality come from?
Thanks for the help!
UPD: you can find detailed explanation here
Best Answer
If a group $G$ acts transitively on a set $X$ via $f\colon G\mapsto \operatorname{Sym}(X)$, then the image of $f$ has at least order $|X|$, hence the kernel is of index at least $|X|$.