 n-transitive group action

Let $n\ge 1$ and let $X$ be a set with at least $n$ points. A group action on $X$ is n-transitive when it can send any ordered tuple of $n$ distinct points to any other such tuple. Here ordered means a sequence of entries, rather than increasing in a total order. For order-preserving group actions, the corresponding notion is order n-transitivity. The size condition excludes vacuous transitivity on an empty collection of distinct-point tuples.