Example of exotic $S_5$ as a Galois group

Question

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 $S_5\to S_6$.

Answer 1

The number field data base by Jürgen Klüners: http://galoisdb.math.upb.de gives polynomials for many extensions with prescribed Galois group. For example, $x^6+3x^4-2x^3+6x^2+1$ would be one whose Galois group is $S_5$ on 6 points (transitive group $T_{14}$ in degree 6.) One can verify this easily using software (e.g. GAP, Pari, Maple, Magma,...)

Answer 2

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 $S_6$.