multiply transitive

Let $G$ be a group, $X$ a set on which it acts. Let $X^{(n)}$ be the set of order $n$-tuplesof distinct elements of $X$. This is a $G$-set by the diagonal action:

$$g\cdot (x_1,\mathrm{\dots },x_n)=(g\cdot x_1,\mathrm{\dots },g\cdot x_n)$$

The action of $G$ on $X$ is said to be $n$-transitive if it acts transitively on $X^{(n)}$.

For example, the standard action of $S^n$, the symmetric group, is $n$-transitive, and thestandard action of $A_n$, the alternating group, is $(n-2)$-transitive.