Every finite Coxeter group W has a standard action. For example, the standard action group of a Coxeter group of type An is the symmetric group of degree n + 1 acting on {1, ..., n}.
The standard action of the finite Coxeter group W.
The group G of the standard action of the finite Coxeter group W, together with an isomorphism W to G.
> W := CoxeterGroup("A3");
> G, h := StandardActionGroup(W);
> IsSymmetric(G);
true
> h(W.1); h(W.2); h(W.3);
(1, 2)
(2, 3)
(3, 4)