I’ll work this out from first principles: exact discriminant via a fraction-free (Bareiss) determinant, then Frobenius cycle types mod several unramified primes via distinct-degree factorization that I implement myself, and finally cross-check everything with a numerically-computed resolvent built directly from the roots. Let me set up the computation.
Discriminant is exact: disc(f) = 418195493, positive but not a perfect square ⇒ √disc ∉ ℚ ⇒ Galois group is not contained in A₈.
Now let’s factor the discriminant (to know which primes are ramified and must be avoided) and then compute Frobenius cycle types at several good primes.
Excellent — disc(f) = 53⁵, positive, not a square. So the only ramified prime is 53, and we already know the group is not inside A₈ (it contains an odd permutation).
Now let me implement, from scratch, modular polynomial arithmetic over 𝔽ₚ and a distinct-degree factorization routine, and run it over many good primes to get Frobenius cycle types (Dedekind’s theorem).
This is very informative:
p=2 gives a single irreducible factor of degree 8 ⇒ f is irreducible mod 2 ⇒ f is irreducible over ℤ (hence over ℚ, Gauss), confirmed independently of the rational-root check. So Gal(f) is a transitive subgroup of S₈.
We see a cycle type (7,1) (e.g. at p=11,37,…). A permutation with a 7-cycle and a fixed point cannot preserve any nontrivial block system on 8 points (7 doesn’t divide 2 or 4), so G is primitive.
Cycle types (8) [odd], (6,1,1) [odd], (2,2,2,1,1) [odd] confirm G ⊄ A₈ (matches the discriminant test).
Primitive transitive groups of degree 8 with order divisible by 7 are very restricted: PSL(2,7), PGL(2,7), AGL(3,2), AΓL(1,8), A₈, S₈. Let me build these explicitly from their definitions (matrix/affine/semilinear actions — not looked up, constructed) and compare their exact cycle-type spectra against much more extensive Frobenius data.
Now let’s explicitly construct the candidate primitive groups of degree 8 (from their algebraic definitions — matrix/affine/semilinear actions on 8 points) and compute their exact cycle-type spectra to compare against the empirical Frobenius distribution.
This is a striking match. Let me also rule out the other candidates (AGL(3,2), AΓL(1,8), A₈, S₈) explicitly, and run a statistical goodness-of-fit check against PGL(2,7).
The match with PGL(2,7) is exact on the set of cycle types (all 7 nontrivial classes present, nothing extra), while AGL(3,2) and AΓL(1,8) each contain classes (like (6,2), (4,2,1,1), (2,2,1,1,1,1)) with substantial predicted frequency (12%–33%) that never once appeared in 429 primes — that alone rules them out. Let’s quantify the PGL(2,7) fit with a chi-square test, then do an independent symbolic resolvent (the degree-28 “pair-sum” resolvent) as a further cross-check.
Chi-square = 0.49 with 6 degrees of freedom — an essentially perfect statistical fit to PGL(2,7) (p-value ≈ 1). Now the symbolic/numeric resolvent cross-check: build the degree-28 “pairwise-sum” resolvent ∏(y − (rᵢ+rⱼ)) from the actual roots at high precision, recover its exact integer coefficients, and test its factorization pattern mod primes.
Summary of the computation
Polynomial: f(x) = x⁸ − x⁷ + 3x⁶ − 3x⁵ + 2x⁴ − 2x³ + 5x² + 5x + 1
Step 1 — Irreducibility. No rational roots (f(±1) ≠ 0), and f mod 2 is irreducible (a single degree‑8 factor found by our own distinct‑degree‑factorization routine). A mod‑p factorization pattern is always a coarsening of the factorization over ℤ, so an irreducible reduction forces f irreducible over ℤ (Gauss). ⇒ Gal(f) is transitive on the 8 roots.
Step 2 — Discriminant. Built the 15×15 Sylvester matrix of (f, f′) and computed its determinant exactly by the Bareiss fraction‑free algorithm (no black‑box resultant/discriminant call):
Step 3 — Frobenius/Dedekind data. For each unramified prime p (p ≠ 53), Dedekind’s theorem says the factorization‑degree pattern of f mod p is the cycle type of an actual element of Gal(f) ⊆ S₈. I implemented 𝔽ₚ arithmetic and distinct‑degree factorization from scratch and ran it over 429 primes. Only 7 cycle types ever occurred:
cycle type
freq. (observed, 429 primes)
(8)
.2517
(6,1,1)
.1678
(3,3,1,1)
.1655
(7,1)
.1469
(4,4)
.1166
(2,2,2,1,1)
.0839
(2,2,2,2)
.0676
A (7,1)‑element (7 coprime to 2,4) rules out any block system ⇒ Gal(f) is primitive.
Step 4 — Candidate construction. The primitive transitive groups of degree 8 with order divisible by 7 are (built explicitly from definitions — matrix/affine/semilinear actions, not looked up): PSL(2,7) [168], PGL(2,7) [336] (Möbius maps on P¹(𝔽₇)), AGL(3,2) [1344] (affine maps on 𝔽₂³), AΓL(1,8) [168] (semilinear maps on 𝔽₈), plus A₈, S₈. Computing each group’s exact cycle‑type spectrum:
PGL(2,7) gives exactly the 7 observed cycle types, with theoretical frequencies 84:56:56:48:42:28:21 (÷336) — matching the data to χ² = 0.49 on 6 d.o.f. (essentially a perfect fit).
AGL(3,2) and AΓL(1,8) each contain cycle types (e.g. (6,2), (4,2,1,1), (2,2,1,1,1,1)) predicted with 12–33% frequency that never once appeared in 429 primes — decisively excluded. PSL(2,7) is all‑even, contradicting the discriminant test. A₈/S₈ would show far more than 7 distinct cycle types.
Step 5 — Independent resolvent check. Computed the 8 roots to 80‑digit precision and built the degree‑28 resolvent R(y) = ∏_{i<j}(y − (rᵢ+rⱼ)) directly from the roots; its coefficients round to integers to within 10⁻⁷⁶ (i.e. they are the exact integers). Its mod‑p factorization pattern at many primes matches — factor‑for‑factor — the combinatorial orbit structure predicted for PGL(2,7) acting on the 28 unordered pairs of points (e.g. a (7,1)-Frobenius element predicts pattern [7,7,7,7] on pairs; observed exactly [7,7,7,7], and similarly for every other type tested). This confirms genericity (no hidden coincidences among the roots) and independently reconfirms the group.
Conclusion
a group of order 336, acting 3‑transitively on the 8 roots as the Möbius group on the projective line P¹(𝔽₇). (In the standard classification of the 50 transitive groups of degree 8 this is the group usually catalogued as 8T35; that label is just nomenclature — the derivation above pins down the group intrinsically by order, primitivity, parity, and its full conjugacy‑class/cycle‑type profile, independently cross‑checked by a symbolic resolvent.)