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Question: How to generate a polynomial corresponding to a solvable Galois group of order 60?

Question:How to generate a polynomial corresponding to a solvable Galois group of order 60?

jud 155 Maple 2021
polynomial group-theory
May 19 2022
1

Of course, we know the A5 of 60 order is an unsolvable group, but as the wiki here, There are also some solvable groups in the same 60 order. Similarly, although map(IsSolubleNumber, [60, 120, 168, 180]) will give false, there are some solvable groups in orders 60, 120, order 168, and order 180. But how to find these corresponding solvable polynomials by maple? I tried to generate them using random polynomials like this:

with(GroupTheory);
do
    do poly := randpoly(x, degree = rand(6 .. 8)()); until irreduc(poly);
    G := GaloisGroup(poly, x);
until IsSoluble(G) and is(GroupOrder(G) in {60, 120, 168, 180});
poly;
galois(poly, x);

But I didn't get any result even after one night..
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