How to get a polynomial corresponding to a solvable Galois group of order 60?

Question

Of course, we know the $A_5$ of $60$ order is an unsolvable group. But as the wiki here, there are also $12$ solvable groups in the same $60$ order still:

Then I have generated many many irreducible polynomials whose Galois group is $60$ order with maple program to check, but none of them is solvable. What's wrong?

Is there a solvable polynomial whose Galois group is $60$ order? If it exists, can you please give any example?

Answer 1

The irreducible polynomial $$ x^{15}-30x^{10}-3708x^5-2 $$ has Galois group $$ C_3\rtimes F_5, $$ which is solvable of order $60$. Here $F_5$ is the Frobenius group of order $20$.

Reference: The tables here.