Right now I'm looking at a number field $E/\Bbb{Q}$ (that is Galois) with Galois group $S_3$ and defining polynomial $p(x)$. I've miraculously managed to find an integral basis in terms of a primitive element $\mathcal{A}=\{\alpha^5,\ldots, \alpha,1\}$ and I have explicit vectors in this $\mathcal{A}$-basis representing the roots of $p$. I've also got the generating transposition $\tau$ and $3$-cycle $\sigma$ for the Galois group $G=\operatorname{Gal}(E/\Bbb{Q}) = S_3$.
What I'm wanting to do from here is find a vector $v$ in this $\mathcal{A}$-basis and a representation $\rho: S_3 \rightarrow \operatorname{SL}^{\pm}(6,\Bbb{Z})$ (coming from the Galois action on the roots) so that the $\rho(S_3)$-orbit of $v$ is a linearly independent set. It seems that I'm butting up against the Normal Basis Theorem and a regular representation of $S_3$, but my representation theory knowledge is not at all strong enough to figure out how to piece this information together.
I had initially hoped that I could make some progress by sheer force, but it seems on the surface that I have way too many degrees of freedom in my choice of representation (maybe 8-10 for each of $\tau$ and $\sigma$) to make that worthwhile.
Is there anything I can say definitively that can simplify my search a bit? For example, I know that $\rho(\sigma)$ must have $3^{\text{rd}}$ roots of unity as eigenvalues; can I say anything about which ones should appear and their multiplicity (e.g. "$\rho(\sigma)$ should have eigenvalues $1$, $\zeta_3$, and $\zeta_3^2$ each with multiplicity $2$")?
Thanks in advance.
P.S. I'm happy to provide any of the specifics for my construction if it's useful.
EDIT: Because it was suggested I do so, the roots $\beta_i$ for $p(x)$ have the following vector forms in the ordered $\mathcal{A}$-basis above: $\beta_1=(0,0,0,0,1,0)$, $\beta_2=(0,0,0,0,-1,1)$, $\beta_3=(1,-2,-4,6,4,-2)$, $\beta_4 = (1,-3,-2,8,1,-2)$, $\beta_5 =(-1,3,2,-8,-1,3)$, $\beta_6 = (-1,2,4,-6,-4,3)$.
The Galois group is generated by the transposition $\tau = (\beta_1 \beta_2)(\beta_3 \beta_6)(\beta_4 \beta_5)$ and the $3$-cycle $\sigma= (\beta_1 \beta_3 \beta_4)(\beta_2 \beta_6 \beta_5)$