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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?

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constructing character table of $S_4$? – user91132 Jan 17 at 14:50

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 $R(A,B,C,D)$.

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