Transitivity number (reviewed)

If a group $G$ acts on a set $A$ of order $n$ and $k\leq n$, then $G$ is $k$-transitive if it acts transitively on the set $$ S=\{(a_1,\ldots,a_k)\in A^k \mid a_i \text{ are distinct}\}.$$ The largest such value $k$ is the transitivity number of the action.