The affine symmetric group is a subgroup of the extended affine symmetric group. The extended group is isomorphic to the wreath product Z ≀ ≀ S n . Its elements are extended affine permutations : bijections u : : Z → → Z such that u ( x + n ) = u ( x ) + n for all integers x. Unlike the affine symmetric group, the extended affine symmetric group is not a Coxeter group. But it has a natural generating set that extends the Coxeter generating set for S ~ ~ n : the shift operator τ τ whose window notation is τ τ = [ 2 , 3 , … … , n , n + 1 ] generates the extended group with the simple reflections, subject to the additional relations τ τ s i τ τ − − 1 = s i + 1 .