Suppose I have a polynomial:

poly := x^6 - 3*x + 3

And I know one of its roots and its galois group:

firstroot := [solve](poly, x)[1]:
G := GaloisGroup(poly, x):

Since G act transitively on the all roots. How do I get all the other roots of this polynomial just by the firstroot and G, but not by solving the equation.

This is my current code:

with~([NumberTheory, DocumentTools]):
tb := Tabulate([seq]([galois](CyclotomicPolynomial(i, x),x), i = 1 .. 10)):

I hope to export it into SVG file, but this code will report a error information:

Export("aatest.svg", tb)

As the code:

poly := x^4 + 8*x + 12:
galois(poly, x)

"4T4", {"A(4)"}, "+", 12, {"(1 2 4)", "(2 3 4)"}

 Then I know it's Galois group has to be (isomorphic to) A4. And I can draw its Subgroup Lattice:

DrawSubgroupLattice(GaloisGroup(poly, x), 'indices')

But according to Galois's theory, each subgroup represents an intermediate field. As far as I know, ⑤⑥⑦⑧ are Q(r₁),Q(r₂),Q(r₃) and Q(r₄), respectively, where r_i is the root of equation x^4+8x+12. But I have no idea what fields ②③④⑨ means. How do you calculate out those intermediate fields with maple?

This code can draw the subgroup lattice:

DrawSubgroupLattice(GaloisGroup(x^3 - 2, x), 'indices')

But I really want to know what the extension of field about each subgroup corresponds to, like this:

The root of the polynomial r₁=2^1/3, r₂=2^1/3(-1+sqrt(3) I)/2, r₂=2^1/3(-1-sqrt(3) I)/2 in the above graph. Further, how do we draw the extension relation of this polynomial:

x⁵+15x+44

The maple can draw this graph? If maple can't draw it, what software can?

As we know, choosing different generators can all represent the same group. But how can I get the combination of these different generators by maple? For example, how do I get the following different combinations of generators of C₆?