Cayley's theorem
The regular action used in the standard proof of Cayley's theorem does not produce the representation of "G" in a "minimal"-order permutation group. For example, formula_1, itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of formula_2 (a group of order 720). Indeed, the complex regular representation of finite groups is actually formed by the direct sum of all the irreducible representations with multiplicity equal to their dimension. The problem of finding an embedding of a group in a minimal-order symmetric group is rather difficult.