Example of exotic $S_5$ as a Galois group Is there an example of a sextic irreducible polynomial over $\mathbb{Q}$ with Galois group isomorphic to $S_5$?
The transitive action of the Galois group of this polynomial on the 6 roots of $p(x)$ would give rise to the exotic embedding $S_5\to S_6$.

The Galois groups of sextic polynomials have been determined here. In table $2$ on page $5$, the group T14 is $S_5$, generated by $(15364), (16)(24), (3465)$ in $S_6$.
