# Order of roots in MAGMA and/or Pari for computing Galois groups

I am trying to understand the Galois group of splitting fields (over $\mathbb Q$) of some polynomials. Especially those for which the Galois group is not the full symmetric group. For instance the splitting field of $x^6-x^5+2x^4+x^3+2x^2+3x+1$ (Galois group isomorphic to $S_4$) or of $x^6 - x^5 + x^4 - x^3 - 4x^2 + 5$ (Galois group isomorphic to a transitive group of order $36$). In Pari/GP there is the command polgalois. But this just gives me what the Galois group is isomorphic to. I am also interested in the ordering of the root used for the polynomial input. Magma apparently gives me a bit more, at least this is my impression after reading a bit of its documentation here.

Magma gives an example from its documentation which I quote here

> Z:= Integers();
> P<x>:= PolynomialRing(Z);
> G, R, S := GaloisGroup(x^6-108);
> G;
Permutation group G acting on a set of cardinality 6
Order = 6 = 2 * 3
(1, 5, 3)(2, 6, 4)
(1, 2)(3, 6)(4, 5)
> R;
[ -58648*$.1 + 53139 + O(11^5), 58648*$.1 - 19478 +
O(11^5), -43755*$.1 - 72617 + O(11^5), 58648*$.1 -
53139 + O(11^5), -58648*$.1 + 19478 + O(11^5), 43755*$.1 + 72617 + O(11^5) ]
> S;
GaloisData over Z_11
> time G, _, S := GaloisGroup(x^32-x^16+2);
Time: 65.760
> #G;
2048


It seems the ordering of the root (for the above its the root of $x^6-108$) is given in R. But I cannot understand R here, it seems to be an encoding for an algebraic number. Is it correct for me to assume that the permutation group is given by the order given by R? If so, how do I read R as roots of $x^6-108$?