The affine symmetric group S ~ ~ n has Λ as a normal subgroup, and is isomorphic to the semidirect product S ~ ~ n ≅ ≅ S n ⋉ ⋉ Λ Λ of this subgroup with the finite symmetric group S n , where the action of S n on Λ is by permutation of coordinates. Consequently, every element u of S ~ ~ n has a unique realization as a product u = r ⋅ ⋅ t where r is a permutation in the standard copy of S n in S ~ ~ n and t is a translation in Λ.