The geometric action of the affine symmetric group S ~ ~ n places it naturally in the family of affine Coxeter groups, each of which has a similar geometric action on an affine space. The combinatorial description of the S ~ ~ n may also be extended to many of these groups: in Eriksson & Eriksson (1998), an axiomatic description is given of certain permutation groups acting on Z (the "George groups", in honor of George Lusztig), and it is shown that they are exactly the "classical" Coxeter groups of finite and affine types A, B, C, and D. (In the classification of affine Coxeter groups, the affine symmetric group is type A.) Thus, the combinatorial interpretations of descents, inversions, etc., carry over in these cases. Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.