[math-fun] cyclic groups; permutations

A cyclic group has a generator that generates all of the elements of the group; i.e., it produces a permutation of the elements of the group. Different generators may produce different permutations. Is there any algebraic structure that elegantly generates *all* permutations? One problem with generators is that there aren't nearly enough of them: there's only N elements, but N! permutations. Are there any other mechanisms -- perhaps high-degree polynomial evaluation or matrices -- that can generate all (& only) permutations? Obviously, matrices can permute the elements of rows & columns, but I'm interested in 1-1 functions of the elements themselves.