TITLE: Galois group of a degree 6 polynomial
QUESTION [10 upvotes]: Problem 4-2 in https://www.jmilne.org/math/CourseNotes/FT.pdf asks

"It is a thought-provoking question that few graduate students would know how toapproach the question of determining the Galois group of, say, $$X^6+2X^5+3X^4+4X^3+5X^2+6X+7"$$

How would one actually solve this problem without using existing software that automatically calculates Galois groups?

I verified the polynomial is irreducible and I looked up the subgroup lattice of $S_6$ for groups transitive on 6 points. I calculated it's discriminant and checked that it was a nonsquare which eliminated some options. I made an attempt to check factorizations mod various primes and try to identify what group it is likely to be but I got the wrong answer from this and this approach would only give a lower bound on the group anyway.

For reference here is the factorizations mod 3,5 and 13:
? lift(factor(Mod(X^6+2*X^5+3*X^4+4*X^3+5*X^2+6*X+7,3)))
%2 = 
[X^6 + 2*X^5 + X^3 + 2*X^2 + 1 1]

? lift(factor(Mod(X^6+2*X^5+3*X^4+4*X^3+5*X^2+6*X+7,5)))
%3 = 
[          X^3 + X + 4 1]

[X^3 + 2*X^2 + 2*X + 3 1]

? lift(factor(Mod(X^6+2*X^5+3*X^4+4*X^3+5*X^2+6*X+7,13)))
%4 = 
[                        X + 11 1]

[X^5 + 4*X^4 + 11*X^3 + 5*X + 3 1]

Which gives cycle types (6), (3,3) and (1,5).

I found the polynomial is equal to $$\frac{x^8 - 8 x + 7}{(x-1)^2}$$

REPLY [2 votes]: Verification of the integer $P$:
? r = polroots(X^6+2*X^5+3*X^4+4*X^3+5*X^2+6*X+7)
%27 = [
-1.3078697439524358868574947207136826382 - 0.59329470741458755880701799527984032602*I, 
-1.3078697439524358868574947207136826382 + 0.59329470741458755880701799527984032602*I, 
0.71037886931271562450142054081955002103 - 1.1068452983838490198383021953838865547*I, 
0.71037886931271562450142054081955002103 + 1.1068452983838490198383021953838865547*I, 
-0.40250912536027973764392582010586738286 - 1.3416668277593834410394603953456211331*I, 
-0.40250912536027973764392582010586738286 + 1.3416668277593834410394603953456211331*I]~
? x = [r[1],r[2],r[3],r[6],r[4],r[5]]
%28 = [-1.3078697439524358868574947207136826382 - 0.59329470741458755880701799527984032602*I, 
-1.3078697439524358868574947207136826382 + 0.59329470741458755880701799527984032602*I, 
0.71037886931271562450142054081955002103 - 1.1068452983838490198383021953838865547*I, 
-0.40250912536027973764392582010586738286 + 1.3416668277593834410394603953456211331*I, 
0.71037886931271562450142054081955002103 + 1.1068452983838490198383021953838865547*I, 
-0.40250912536027973764392582010586738286 - 1.3416668277593834410394603953456211331*I]
? 4*x[6]^2*x[1]^2*x[2]*x[3] + 4*x[6]*x[1]*x[2]^2*x[3]^2 + 4*x[6]^2*x[1]*x[2]^2*x[4] + 4*x[1]^2*x[2]^2*x[3]*x[4] + 4*x[6]*x[1]^2*x[3]^2*x[4] + 4*x[6]^2*x[2]*x[3]^2*x[4] + 4*x[6]*x[1]^2*x[2]*x[4]^2 + 4*x[6]^2*x[1]*x[3]*x[4]^2 + 4*x[6]*x[2]^2*x[3]*x[4]^2 + 4*x[1]*x[2]*x[3]^2*x[4]^2 + 4*x[6]*x[1]^2*x[2]^2*x[5] + 4*x[6]^2*x[2]^2*x[3]*x[5] + 4*x[6]^2*x[1]*x[3]^2*x[5] + 4*x[1]^2*x[2]*x[3]^2*x[5] + 4*x[6]^2*x[1]^2*x[4]*x[5] + 4*x[2]^2*x[3]^2*x[4]*x[5] + 4*x[6]^2*x[2]*x[4]^2*x[5] + 4*x[1]*x[2]^2*x[4]^2*x[5] + 4*x[1]^2*x[3]*x[4]^2*x[5] + 4*x[6]*x[3]^2*x[4]^2*x[5] + 4*x[6]^2*x[1]*x[2]*x[5]^2 + 4*x[6]*x[1]^2*x[3]*x[5]^2 + 4*x[1]*x[2]^2*x[3]*x[5]^2 + 4*x[6]*x[2]*x[3]^2*x[5]^2 + 4*x[1]^2*x[2]*x[4]*x[5]^2 + 4*x[6]*x[2]^2*x[4]*x[5]^2 + 4*x[6]^2*x[3]*x[4]*x[5]^2 + 4*x[1]*x[3]^2*x[4]*x[5]^2 + 4*x[6]*x[1]*x[4]^2*x[5]^2 + 4*x[2]*x[3]*x[4]^2*x[5]^2
%29 = 264.00000000000000000000000000000000000 + 0.E-37*I