TITLE: Problems which use S₄ → S₃
QUESTION [10 upvotes]: I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).
Obvious candidates:

Lagrange resolvent (the reduction of quartic to cubic equations). 
Tait's theorem on equivalence of 4-coloring of normal map and 3-coloring of its edges.  


Do you have more interesting examples?

REPLY [13 votes]: I use this or something equivalent in teaching projective geometry to show that the cross-ratio has at most 6 distinct values (when the points are permuted) as opposed to the 24 naively expected. This involves checking that the elements in the Klein 4-group act trivially on the cross-ratio $R(A,B,C,D)$.