 Sharp t-transitivity

An action is sharply $t$-transitive if each ordered $t$-tuple of distinct points is carried to any other by exactly one group element. For finite degree $n\ge t$, its order is $n(n-1)\cdots (n-t+1)$ and each ordered-tuple stabilizer is trivial. Sharp one-transitivity is a regular action. Finite sharp two-transitivity gives a regular kernel of a finite sharply two-transitive group.