Question on a permutation of the roots of a Galois number field, dihedral over the rationals The polynomial $f(x) = x^6 - 7x^5 + 21x^4 - 41x^3 + 63x^2 - 63x + 27$ defines a Galois extension $H$ of $\mathbb{Q}$. The Galois group of the extension $H/\mathbb{Q}$ is dihedral, and depending on your notation is $D_6$ or $D_3$.
(Aside: in fact I generated the polynomial by defining $g_1(\tau) = \eta(\tau/3)/\eta(\tau)$, where $\eta(\tau)$ is the Dedekind eta function, and letting $w = g_1(\tau)^2$ where $\tau = (-1 + \sqrt{-23})/2$ and taking the minimum polynomial of $w$ over $\mathbb{Q}$. In fact $w$ generates the Hilbert class field $H$ of $\mathbb{Q}(\sqrt{-23})$ over $\mathbb{Q}$.)
Suppose that the roots of $f(x)$ are written $r_1, r_2, \ldots, r_6$. Also suppose that they are ordered such that $r_1, r_2$ and $r_3, r_4$ and $r_5, r_6$ are complex conjugate pairs.
An automorphism in $G =$ Gal$(H/\mathbb{Q})$ defines a permutation on the roots.
My question is this: is there a $\sigma \in G$ which as a permutation on the roots is given by $(r_1 r_2)(r_3 r_4)(r_5 r_6)$.

The complex conjugate is such automorphism.
