K = C₂ = ⟨(1 2)⟩ ≤ S₃ acts on X = {a, b} by the swap. The norm
N(X) = Map_K(S₃, X) is modelled on the three right cosets
C₂\S₃ (the triangle corners): a point of the norm is a choice of one
element of X at each corner. S₃ acts by the monomial rule — permute the three
fibers, then twist some by the C₂ swap (the cocycle). Click the two points of a
fiber to edit the section; pick an S₃ element to act.
Convention: right cosets C₂\S₃, action (h·f)(g)=f(gh), transversal
e, (1 3), (2 3). The twist at a corner is the cocycle
κ(i,h)∈C₂ from hᵢ·h = κ(i,h)·h_{i·h}: it swaps that fiber's a and b.
As an S₃-set, N(X) ≅ [S₃/1] ⊔ [S₃/A₃] (a free 6-point orbit and a 2-point orbit).