Small group number 859 of order 128

G is the group 128gp859

G has 3 minimal generators, rank 4 and exponent 4. The centre has rank 1.

There are 3 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 4, 4.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 13 generators:

y₁ in degree 1, a nilpotent element
y₂ in degree 1
y₃ in degree 1
x₁ in degree 2
x₂ in degree 2
x₃ in degree 2
w in degree 3
u₁ in degree 5
u₂ in degree 5
t₁ in degree 6
t₂ in degree 6
s in degree 7
r in degree 8, a regular element

There are 44 minimal relations:

y₁.y₃ = 0
y₁² = 0
y₃.x₃ = y₁.x₁
y₃.x₁ = y₁.x₁
y₂².y₃ = y₁.x₂
x₁.x₃ = x₁² + y₁.y₂.x₁
y₃.w = y₂.y₃.x₂
y₁.w = y₁.y₂.x₂
y₁.x₂² = y₁.y₂².x₂
w² = x₂².x₃ + y₂².x₂² + y₂⁴.x₁ + y₁.y₂³.x₂
y₃.u₂ = y₃².x₂² + y₂.y₃.x₂² + y₂.y₃³.x₂ + y₁.y₂³.x₁
y₁.u₂ = y₁.y₂³.x₂ + y₁.y₂³.x₁
y₁.u₁ = 0
x₃.u₂ = x₃².w + x₂.x₃.w + y₂.x₂.x₃² + y₂.x₁².x₂ + y₂².x₃.w + y₂².x₁.w + y₂³.x₂.x₃ + y₂³.x₁.x₂ + y₁.t₂ + y₁.y₂².x₁² + y₁.y₂⁴.x₂
x₃.u₁ = x₁.x₂.w + y₂.x₁.x₂² + y₂.x₁².x₂ + y₂³.x₁² + y₁.t₁ + y₁.x₁³ + y₁.y₂².x₁² + y₁.y₂⁴.x₂ + y₁.y₂⁴.x₁
x₂.u₂ = x₂.x₃.w + x₂².w + y₃.x₂³ + y₂.x₂².x₃ + y₂.x₁.x₂² + y₂.y₃².x₂² + y₂².u₁ + y₂².x₂.w + y₂³.x₂² + y₂³.x₁.x₂ + y₂⁵.x₁ + y₁.y₂⁴.x₂ + y₁.y₂⁴.x₁
x₁.u₂ = x₁.x₂.w + x₁².w + y₁.t₁ + y₁.x₁³ + y₁.y₂⁴.x₂ + y₁.y₂⁴.x₁
x₁.u₁ = x₁.x₂.w + y₂.x₁.x₂² + y₂.x₁².x₂ + y₂³.x₁² + y₁.t₁ + y₁.x₁³ + y₁.y₂².x₁² + y₁.y₂⁴.x₂ + y₁.y₂⁴.x₁
y₃.t₂ = y₃.x₂³ + y₃².u₁ + y₁.t₁ + y₁.x₁³ + y₁.y₂².x₁² + y₁.y₂⁴.x₂
y₃.t₁ = y₃³.x₂² + y₂.y₃.u₁ + y₂.y₃².x₂² + y₂.y₃⁴.x₂ + y₁.t₁
w.u₂ = x₂².x₃² + x₂³.x₃ + y₂.x₂.x₃.w + y₂.x₁.x₂.w + y₂.y₃.x₂³ + y₂².x₂³ + y₂².x₁.x₂² + y₂³.u₁ + y₂³.x₂.w + y₂⁴.x₂² + y₂⁴.x₁² + y₂⁶.x₁ + y₁.y₂⁵.x₂ + y₁.y₂⁵.x₁
w.u₁ = x₁.x₂³ + y₂.x₂.u₁ + y₂.x₁.x₂.w + y₂².x₁.x₂² + y₂³.x₁.w
x₃.t₁ = x₁.t₁ + y₂.x₃².w + y₂.x₁².w + y₂².x₂².x₃ + y₂².x₁.x₂² + y₂³.x₃.w + y₂³.x₁.w + y₂⁴.x₂.x₃ + y₂⁴.x₁.x₂ + y₁.y₂.t₁ + y₁.y₂⁵.x₂
x₁.t₂ = x₁.t₁ + x₁.x₂³ + x₁².x₂² + x₁⁴ + y₂².x₁³ + y₂³.x₁.w + y₂⁴.x₁² + y₁.y₂.x₁³ + y₁.y₂³.x₁² + y₁.y₂⁵.x₁
y₃.s = y₃.x₂.u₁ + y₂.y₃.x₂³ + y₂.y₃³.x₂² + y₂.y₃⁵.x₂ + y₁.y₂³.x₁² + y₁.y₂⁵.x₂ + y₁.y₂⁵.x₁
y₁.s = y₁.y₂.t₂ + y₁.y₂.t₁ + y₁.y₂.x₁³ + y₁.y₂⁵.x₁
w.t₂ = x₃.s + x₃³.w + x₂².x₃.w + x₂³.w + x₁².x₂.w + y₂.x₃.t₂ + y₂.x₂.t₂ + y₂.x₂.x₃³ + y₂.x₂².x₃² + y₂.x₂⁴ + y₂.x₁.t₁ + y₂.x₁².x₂² + y₂.x₁³.x₂ + y₂.x₁⁴ + y₂².x₃².w + y₂².x₂².w + y₂³.x₂.x₃² + y₂³.x₂³ + y₂³.x₁.x₂² + y₂³.x₁³ + y₂⁴.x₃.w + y₂⁴.x₂.w + y₂⁵.x₂.x₃ + y₂⁵.x₂² + y₁.x₁.t₁ + y₁.y₂².t₂ + y₁.y₂⁶.x₂
w.t₁ = x₁.s + x₁².x₂.w + y₂.x₂.t₁ + y₂.x₂².x₃² + y₂.x₁³.x₂ + y₂².x₂.x₃.w + y₂².x₂².w + y₂².x₁.x₂.w + y₂³.x₂³ + y₂⁴.x₁.w + y₂⁵.x₁² + y₂⁷.x₁ + y₁.x₁.t₁ + y₁.y₂².x₁³ + y₁.y₂⁶.x₂
u₂² = x₂².x₃³ + x₂⁴.x₃ + y₃².x₂⁴ + y₂².x₂⁴ + y₂².x₁².x₂² + y₂⁴.x₂².x₃ + y₂⁴.x₁³ + y₁.y₂⁵.x₁²
u₁.u₂ = x₁.x₂⁴ + x₁².x₂³ + y₃.x₂².u₁ + y₂.x₂².u₁ + y₂.x₁².x₂.w + y₂.y₃².x₂.u₁ + y₂³.x₁.x₂.w + y₂³.x₁².w + y₂⁴.x₁².x₂ + y₁.y₂³.t₁ + y₁.y₂³.x₁³ + y₁.y₂⁵.x₁² + y₁.y₂⁷.x₂ + y₁.y₂⁷.x₁
u₁² = x₁.x₂⁴ + y₃.x₂².u₁ + y₃².x₂⁴ + y₃³.x₂.u₁ + y₃⁵.u₁ + y₃⁶.x₂² + y₂.y₃.x₂⁴ + y₂.y₃⁵.x₂² + y₂².x₁².x₂² + y₂⁴.x₁.x₂² + y₂⁶.x₁² + y₁.y₂⁷.x₂ + y₃².r
w.s = x₂².t₂ + x₂².x₃³ + x₂⁴.x₃ + x₂⁵ + x₁².x₂³ + y₃.x₂².u₁ + y₂.x₃.s + y₂.x₃³.w + y₂.x₂.s + y₂.x₁.s + y₂.x₁².x₂.w + y₂.x₁³.w + y₂².x₃.t₂ + y₂².x₂.x₃³ + y₂².x₂³.x₃ + y₂².x₂⁴ + y₂².x₁.t₁ + y₂².x₁².x₂² + y₂².x₁³.x₂ + y₂².x₁⁴ + y₂³.x₃².w + y₂³.x₂².w + y₂³.x₁².w + y₂⁴.t₁ + y₂⁴.x₂.x₃² + y₂⁴.x₂².x₃ + y₂⁴.x₁.x₂² + y₂⁴.x₁³ + y₂⁵.x₂.w + y₂⁶.x₂.x₃ + y₂⁶.x₁² + y₂⁷.w + y₂⁸.x₂ + y₂⁸.x₁ + y₁.y₂³.t₂ + y₁.y₂³.x₁³ + y₁.y₂⁷.x₂ + y₁.y₂⁷.x₁
u₂.t₂ = x₃².s + x₃⁴.w + x₂.x₃.s + x₂.x₃³.w + x₂².x₃².w + x₂⁴.w + x₁².x₂².w + x₁³.x₂.w + y₃.x₂⁵ + y₃².x₂².u₁ + y₂.x₃².t₂ + y₂.x₂.x₃.t₂ + y₂.x₂.x₃⁴ + y₂.x₂².t₂ + y₂.x₂³.x₃² + y₂.x₂⁴.x₃ + y₂.x₂⁵ + y₂.x₁.x₂⁴ + y₂.x₁².t₁ + y₂.x₁⁴.x₂ + y₂.x₁⁵ + y₂.y₃².x₂⁴ + y₂.y₃³.x₂.u₁ + y₂².x₃.s + y₂².x₂.x₃².w + y₂².x₂².u₁ + y₂².x₁.s + y₂².x₁.x₂².w + y₂².x₁³.w + y₂³.x₃.t₂ + y₂³.x₂³.x₃ + y₂³.x₁.t₁ + y₂³.x₁.x₂³ + y₂³.x₁².x₂² + y₂³.x₁³.x₂ + y₂⁴.x₂.u₁ + y₂⁴.x₁.x₂.w + y₂⁴.x₁².w + y₂⁵.x₁³ + y₂⁶.u₁ + y₂⁶.x₃.w + y₂⁶.x₂.w + y₂⁷.x₂.x₃ + y₂⁷.x₂² + y₂⁷.x₁.x₂ + y₂⁷.x₁² + y₂⁹.x₁ + y₁.x₃².t₂ + y₁.y₂².x₁.t₁ + y₁.y₂².x₁⁴ + y₁.y₂⁶.x₁² + y₁.y₂⁸.x₂ + y₁.y₂⁸.x₁ + y₁.x₃.r
u₂.t₁ = x₁.x₂.s + x₁².s + x₁².x₂².w + x₁³.x₂.w + y₃³.x₂⁴ + y₂.x₂².t₁ + y₂.x₂².x₃³ + y₂.x₂³.x₃² + y₂.x₁.x₂.t₁ + y₂.x₁³.x₂² + y₂.x₁⁴.x₂ + y₂.y₃.x₂².u₁ + y₂.y₃².x₂⁴ + y₂².x₂.x₃².w + y₂².x₂³.w + y₂².x₁.x₂².w + y₂².x₁².x₂.w + y₂³.x₂².x₃² + y₂³.x₂³.x₃ + y₂³.x₂⁴ + y₂³.x₁².x₂² + y₂⁴.x₂.u₁ + y₂⁴.x₂.x₃.w + y₂⁴.x₂².w + y₂⁴.x₁².w + y₂⁵.x₂³ + y₂⁵.x₁.x₂² + y₂⁵.x₁².x₂ + y₂⁵.x₁³ + y₂⁷.x₁² + y₁.y₂⁴.x₁³ + y₁.y₂⁶.x₁² + y₁.y₂⁸.x₁ + y₁.x₁.r
u₁.t₂ = x₂³.u₁ + x₁.x₂.s + x₁.x₂³.w + x₁².x₂².w + x₁³.x₂.w + y₃².x₂².u₁ + y₃³.x₂⁴ + y₃⁴.x₂.u₁ + y₃⁶.u₁ + y₃⁷.x₂² + y₂.x₁.x₂.t₁ + y₂.x₁.x₂⁴ + y₂.x₁⁴.x₂ + y₂.y₃².x₂⁴ + y₂.y₃⁶.x₂² + y₂².x₂².u₁ + y₂².x₁².x₂.w + y₂³.x₁.t₁ + y₂³.x₁³.x₂ + y₂³.x₁⁴ + y₂⁴.x₂.u₁ + y₂⁴.x₁.x₂.w + y₂⁵.x₁.x₂² + y₂⁵.x₁³ + y₂⁶.x₁.w + y₂⁷.x₁.x₂ + y₂⁷.x₁² + y₁.x₁⁵ + y₁.y₂².x₁⁴ + y₁.y₂⁴.x₁³ + y₃³.r + y₁.x₁.r
u₁.t₁ = x₁.x₂.s + x₁².x₂².w + y₃².x₂².u₁ + y₂.x₁.x₂.t₁ + y₂.x₁².x₂³ + y₂.x₁³.x₂² + y₂.y₃².x₂⁴ + y₂.y₃⁵.u₁ + y₂.y₃⁶.x₂² + y₂².x₂².u₁ + y₂³.x₁.t₁ + y₂³.x₁.x₂³ + y₂⁴.x₁.x₂.w + y₂⁵.x₁.x₂² + y₂⁵.x₁².x₂ + y₂⁷.x₁.x₂ + y₁.x₁².t₁ + y₁.y₂².x₁.t₁ + y₁.y₂².x₁⁴ + y₁.y₂⁴.x₁³ + y₁.y₂⁸.x₁ + y₂.y₃².r + y₁.x₁.r
t₂² = x₃³.t₂ + x₂.x₃².t₂ + x₂.x₃⁵ + x₂².x₃⁴ + x₂³.x₃³ + x₂⁴.x₃² + x₂⁶ + x₁².x₂.t₁ + x₁².x₂⁴ + x₁³.t₁ + x₁³.x₂³ + y₃³.x₂².u₁ + y₃⁴.x₂⁴ + y₃⁵.x₂.u₁ + y₃⁷.u₁ + y₃⁸.x₂² + y₂.x₃².s + y₂.x₃⁴.w + y₂.x₂.x₃³.w + y₂.x₁³.x₂.w + y₂.y₃³.x₂⁴ + y₂.y₃⁷.x₂² + y₂².x₂.x₃.t₂ + y₂².x₂².x₃³ + y₂².x₂³.x₃² + y₂².x₁².t₁ + y₂².x₁³.x₂² + y₂².x₁⁴.x₂ + y₂³.x₃³.w + y₂³.x₂.x₃².w + y₂³.x₂².x₃.w + y₂³.x₁².x₂.w + y₂⁴.x₃.t₂ + y₂⁴.x₂.x₃³ + y₂⁴.x₂⁴ + y₂⁴.x₁.t₁ + y₂⁴.x₁³.x₂ + y₂⁶.x₂².x₃ + y₂⁶.x₁².x₂ + y₂⁶.x₁³ + y₂⁷.x₃.w + y₂⁸.x₂.x₃ + y₂⁸.x₂² + y₂⁸.x₁.x₂ + y₁.y₂.x₃².t₂ + y₁.y₂.x₁².t₁ + y₁.y₂³.x₁.t₁ + y₁.y₂³.x₁⁴ + y₁.y₂⁵.t₁ + y₁.y₂⁵.x₁³ + y₁.y₂⁹.x₂ + x₃².r + y₃⁴.r
t₁.t₂ = x₂³.t₁ + x₁.x₂².t₁ + x₁³.t₁ + y₃³.x₂².u₁ + y₂.x₃².s + y₂.x₃⁴.w + y₂.x₂².x₃².w + y₂.x₁².x₂².w + y₂.y₃³.x₂⁴ + y₂.y₃⁶.u₁ + y₂.y₃⁷.x₂² + y₂².x₃².t₂ + y₂².x₂.x₃.t₂ + y₂².x₂.x₃⁴ + y₂².x₂².t₂ + y₂².x₂².t₁ + y₂².x₂².x₃³ + y₂².x₂⁴.x₃ + y₂².x₂⁵ + y₂².x₁².t₁ + y₂².x₁³.x₂² + y₂².x₁⁴.x₂ + y₂³.x₃.s + y₂³.x₂².x₃.w + y₂⁴.x₃.t₂ + y₂⁴.x₂.t₁ + y₂⁴.x₂².x₃² + y₂⁴.x₂³.x₃ + y₂⁴.x₂⁴ + y₂⁴.x₁.x₂³ + y₂⁴.x₁².x₂² + y₂⁴.x₁⁴ + y₂⁵.x₁².w + y₂⁶.x₂².x₃ + y₂⁶.x₂³ + y₂⁶.x₁².x₂ + y₂⁷.x₃.w + y₂⁷.x₁.w + y₂⁸.x₂.x₃ + y₂⁸.x₁.x₂ + y₁.y₂.x₁⁵ + y₁.y₂³.x₃.t₂ + y₁.y₂⁵.t₂ + y₁.y₂⁵.t₁ + y₁.y₂⁵.x₁³ + y₁.y₂⁷.x₁² + x₁².r + y₂.y₃³.r
t₁² = y₃⁴.x₂⁴ + y₂.x₁².s + y₂.x₁⁴.w + y₂².x₂².x₃³ + y₂².x₁.x₂.t₁ + y₂².x₁.x₂⁴ + y₂².x₁².t₁ + y₂².x₁³.x₂² + y₂².x₁⁴.x₂ + y₂².x₁⁵ + y₂³.x₁.x₂².w + y₂³.x₁².x₂.w + y₂⁴.x₂².x₃² + y₂⁴.x₂⁴ + y₂⁴.x₁.x₂³ + y₂⁴.x₁⁴ + y₂⁵.x₁.x₂.w + y₂⁶.x₂².x₃ + y₂⁶.x₁.x₂² + y₂¹⁰.x₁ + y₁.y₂.x₁².t₁ + y₁.y₂.x₁⁵ + y₁.y₂³.x₁.t₁ + y₁.y₂⁵.t₁ + y₁.y₂⁵.x₁³ + y₁.y₂⁷.x₁² + x₁².r
u₂.s = x₂².x₃.t₂ + x₂².x₃⁴ + x₂³.t₂ + x₂³.x₃³ + x₂⁴.x₃² + x₂⁶ + x₁².x₂⁴ + x₁³.x₂³ + y₂.x₃².s + y₂.x₃⁴.w + y₂.x₂.x₃.s + y₂.x₂.x₃³.w + y₂.x₂².s + y₂.x₁².s + y₂.x₁².x₂².w + y₂.x₁⁴.w + y₂.y₃.x₂⁵ + y₂.y₃².x₂².u₁ + y₂.y₃³.x₂⁴ + y₂.y₃⁵.x₂³ + y₂².x₃².t₂ + y₂².x₂.x₃.t₂ + y₂².x₂.x₃⁴ + y₂².x₂².t₂ + y₂².x₂².t₁ + y₂².x₂³.x₃² + y₂².x₂⁴.x₃ + y₂².x₁.x₂.t₁ + y₂².x₁².t₁ + y₂².x₁².x₂³ + y₂².x₁⁵ + y₂³.x₃.s + y₂³.x₂.x₃².w + y₂³.x₂³.w + y₂³.x₁.s + y₂³.x₁.x₂².w + y₂³.x₁².x₂.w + y₂⁴.x₃.t₂ + y₂⁴.x₂.t₁ + y₂⁴.x₁.x₂³ + y₂⁴.x₁².x₂² + y₂⁵.x₂.u₁ + y₂⁵.x₂.x₃.w + y₂⁵.x₂².w + y₂⁶.x₂².x₃ + y₂⁶.x₂³ + y₂⁷.u₁ + y₂⁷.x₃.w + y₂⁷.x₁.w + y₂⁸.x₂.x₃ + y₂⁸.x₁.x₂ + y₂¹⁰.x₁ + y₁.y₂.x₃².t₂ + y₁.y₂.x₁².t₁ + y₁.y₂.x₁⁵ + y₁.y₂³.x₁.t₁ + y₁.y₂³.x₁⁴ + y₁.y₂⁹.x₂ + y₁.y₂⁹.x₁ + y₁.y₂.x₃.r + y₁.y₂.x₁.r
u₁.s = x₂³.t₁ + x₁².x₂⁴ + y₃.x₂³.u₁ + y₃³.x₂².u₁ + y₃⁵.x₂.u₁ + y₃⁶.x₂³ + y₂.x₂³.x₃.w + y₂.x₁.x₂.s + y₂.x₁.x₂³.w + y₂.x₁².x₂².w + y₂.y₃².x₂².u₁ + y₂.y₃³.x₂⁴ + y₂.y₃⁴.x₂.u₁ + y₂.y₃⁵.x₂³ + y₂².x₂⁵ + y₂².x₁².x₂³ + y₂³.x₂².u₁ + y₂³.x₂³.w + y₂³.x₁.s + y₂⁴.x₂.t₁ + y₂⁴.x₂⁴ + y₂⁴.x₁².x₂² + y₂⁵.x₂.u₁ + y₂⁵.x₂.x₃.w + y₂⁶.x₂³ + y₂⁶.x₁.x₂² + y₂⁷.x₂.w + y₂⁸.x₂² + y₁.y₂³.x₁.t₁ + y₁.y₂³.x₁⁴ + y₁.y₂⁵.t₁ + y₁.y₂⁹.x₂ + y₃².x₂.r
t₂.s = x₂.x₃².s + x₂².x₃.s + x₂³.s + x₂³.x₃².w + x₂⁴.x₃.w + x₁.x₂⁴.w + x₁².x₂³.w + x₁³.s + x₁³.x₂².w + y₃².x₂³.u₁ + y₃³.x₂⁵ + y₃⁴.x₂².u₁ + y₃⁶.x₂.u₁ + y₃⁷.x₂³ + y₂.x₃³.t₂ + y₂.x₂.x₃⁵ + y₂.x₂².x₃.t₂ + y₂.x₂².x₃⁴ + y₂.x₂³.x₃³ + y₂.x₂⁴.x₃² + y₂.x₁.x₂⁵ + y₂.x₁².x₂.t₁ + y₂.x₁³.t₁ + y₂.x₁⁴.x₂² + y₂.x₁⁵.x₂ + y₂.x₁⁶ + y₂.y₃.x₂³.u₁ + y₂.y₃².x₂⁵ + y₂.y₃³.x₂².u₁ + y₂.y₃⁵.x₂.u₁ + y₂.y₃⁶.x₂³ + y₂².x₃².s + y₂².x₃⁴.w + y₂².x₂.x₃.s + y₂².x₂².s + y₂².x₂².x₃².w + y₂².x₂³.x₃.w + y₂².x₂⁴.w + y₂².x₁.x₂³.w + y₂².x₁².s + y₂³.x₂².t₂ + y₂³.x₂⁴.x₃ + y₂³.x₁³.x₂² + y₂³.x₁⁵ + y₂⁴.x₃³.w + y₂⁴.x₂.s + y₂⁴.x₂².x₃.w + y₂⁴.x₂³.w + y₂⁴.x₁.x₂².w + y₂⁴.x₁².x₂.w + y₂⁴.x₁³.w + y₂⁵.x₃.t₂ + y₂⁵.x₂.t₂ + y₂⁵.x₂.x₃³ + y₂⁵.x₂⁴ + y₂⁵.x₁.t₁ + y₂⁵.x₁⁴ + y₂⁶.x₂.u₁ + y₂⁶.x₂².w + y₂⁷.x₂³ + y₂⁷.x₁.x₂² + y₂⁸.u₁ + y₂⁸.x₃.w + y₂⁸.x₂.w + y₂⁸.x₁.w + y₂⁹.x₂.x₃ + y₂¹¹.x₁ + y₁.y₂².x₁².t₁ + y₁.y₂².x₁⁵ + y₁.y₂⁴.x₃.t₂ + y₁.y₂⁴.x₁⁴ + y₁.y₂⁶.t₂ + y₁.y₂⁸.x₁² + y₁.y₂¹⁰.x₂ + x₃.w.r + y₃³.x₂.r + y₂.x₃².r + y₂.x₂.x₃.r + y₂.x₁².r + y₁.x₁².r + y₁.y₂².x₃.r + y₁.y₂².x₁.r
t₁.s = x₁².x₂.s + x₁³.x₂².w + y₃².x₂³.u₁ + y₂.x₂².x₃.t₂ + y₂.x₂².x₃⁴ + y₂.x₂⁴.x₃² + y₂.x₂⁵.x₃ + y₂.x₁².x₂.t₁ + y₂.y₃⁴.x₂⁴ + y₂.y₃⁵.x₂.u₁ + y₂².x₃².s + y₂².x₃⁴.w + y₂².x₂.x₃.s + y₂².x₂².s + y₂².x₂³.u₁ + y₂².x₁².x₂².w + y₂³.x₃².t₂ + y₂³.x₂.x₃⁴ + y₂³.x₂².t₂ + y₂³.x₂².t₁ + y₂³.x₂².x₃³ + y₂³.x₂³.x₃² + y₂³.x₂⁵ + y₂³.x₁².t₁ + y₂³.x₁².x₂³ + y₂³.x₁³.x₂² + y₂³.x₁⁴.x₂ + y₂³.x₁⁵ + y₂⁴.x₃.s + y₂⁴.x₂².u₁ + y₂⁴.x₁³.w + y₂⁵.x₃.t₂ + y₂⁵.x₂.t₁ + y₂⁵.x₂².x₃² + y₂⁵.x₂³.x₃ + y₂⁵.x₁.t₁ + y₂⁵.x₁⁴ + y₂⁶.x₁².w + y₂⁷.t₁ + y₂⁷.x₁³ + y₂⁸.x₁.w + y₂⁹.x₂.x₃ + y₂⁹.x₂² + y₂¹⁰.w + y₂¹¹.x₂ + y₂¹¹.x₁ + y₁.y₂².x₁⁵ + y₁.y₂⁴.x₃.t₂ + y₁.y₂⁶.t₂ + y₁.y₂⁸.x₁² + y₁.y₂¹⁰.x₂ + x₁.w.r + y₂.x₁.x₂.r + y₂.y₃².x₂.r + y₁.x₁².r
s² = x₂².x₃².t₂ + x₂².x₃⁵ + x₂³.x₃.t₂ + x₂³.x₃⁴ + x₂⁴.x₃³ + x₂⁵.x₃² + x₁.x₂³.t₁ + x₁.x₂⁶ + x₁².x₂².t₁ + x₁².x₂⁵ + x₁³.x₂⁴ + y₃.x₂⁴.u₁ + y₃².x₂⁶ + y₃³.x₂³.u₁ + y₃⁵.x₂².u₁ + y₃⁶.x₂⁴ + y₂.x₂².x₃.s + y₂.x₂².x₃³.w + y₂.x₂³.x₃².w + y₂.x₁².x₂³.w + y₂.y₃.x₂⁶ + y₂.y₃⁵.x₂⁴ + y₂².x₃³.t₂ + y₂².x₂.x₃².t₂ + y₂².x₂.x₃⁵ + y₂².x₂².x₃⁴ + y₂².x₂³.t₂ + y₂².x₂³.x₃³ + y₂².x₂⁴.x₃² + y₂².x₂⁵.x₃ + y₂².x₁.x₂².t₁ + y₂².x₁².x₂.t₁ + y₂².x₁³.t₁ + y₂².x₁⁴.x₂² + y₂².x₁⁶ + y₂³.x₃².s + y₂³.x₃⁴.w + y₂³.x₂.x₃³.w + y₂³.x₂².x₃².w + y₂³.x₂³.x₃.w + y₂³.x₂⁴.w + y₂³.x₁.x₂³.w + y₂³.x₁².s + y₂³.x₁³.x₂.w + y₂³.x₁⁴.w + y₂⁴.x₂.x₃.t₂ + y₂⁴.x₂².t₂ + y₂⁴.x₂².t₁ + y₂⁴.x₁.x₂.t₁ + y₂⁴.x₁.x₂⁴ + y₂⁴.x₁².x₂³ + y₂⁴.x₁⁵ + y₂⁵.x₃³.w + y₂⁵.x₂.x₃².w + y₂⁵.x₁.s + y₂⁵.x₁³.w + y₂⁶.x₃.t₂ + y₂⁶.x₂.t₁ + y₂⁶.x₂.x₃³ + y₂⁶.x₂⁴ + y₂⁶.x₁⁴ + y₂⁷.x₂.x₃.w + y₂⁷.x₁.x₂.w + y₂⁸.x₂³ + y₂⁸.x₁.x₂² + y₂⁸.x₁².x₂ + y₂⁹.x₃.w + y₂⁹.x₂.w + y₂¹⁰.x₂.x₃ + y₂¹⁰.x₁.x₂ + y₂¹².x₁ + y₁.y₂³.x₃².t₂ + y₁.y₂³.x₁⁵ + y₁.y₂⁵.x₁⁴ + y₁.y₂⁷.t₁ + y₁.y₂⁷.x₁³ + y₁.y₂⁹.x₁² + x₂².x₃.r + y₃².x₂².r + y₂².x₃².r + y₂².x₁².r + y₂⁴.x₁.r + y₁.y₂³.x₂.r

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

y₁.x₂.x₃ = y₁.y₂².x₁
y₁.x₁.x₂ = y₁.y₂².x₁
y₁.x₂.t₂ = y₁.y₂².t₁ + y₁.y₂².x₁³ + y₁.y₂⁴.x₁²
y₁.x₂.t₁ = y₁.y₂².t₁

Completion information

This cohomology ring was obtained from a calculation out to degree 17. The cohomology ring approximation is stable from degree 14 onwards, and Benson's tests detect stability from degree 17 onwards.

This cohomology ring has dimension 4 and depth 2. Here is a homogeneous system of parameters:

h₁ = r in degree 8
h₂ = x₃² + x₂² + y₃⁴ + y₂.w + y₂².x₃ + y₂⁴ in degree 4
h₃ = x₂².x₃ + y₃².x₂² + y₂.u₂ + y₂².x₃² + y₂².x₁.x₂ + y₂³.w + y₂⁴.x₃ + y₂⁴.x₁ in degree 6
h₄ = y₂ in degree 1

The first 2 terms h₁, h₂ form a regular sequence of maximum length.

The first term h₁ forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.

Data for Benson's test:

Raw filter degree type: -1, -1, 6, 14, 15.
Filter degree type: -1, -2, -3, -4, -4.
α = 0
The system of parameters is very strongly quasi-regular.
The regularity conjecture is satisfied.

Koszul information

A basis for R/(h₁, h₂, h₃, h₄) is as follows.

1 in degree 0
y₃ in degree 1
y₁ in degree 1
x₃ in degree 2
x₂ in degree 2
x₁ in degree 2
y₃² in degree 2
w in degree 3
y₃.x₂ in degree 3
y₃³ in degree 3
y₁.x₃ in degree 3
y₁.x₁ in degree 3
x₂.x₃ in degree 4
x₂² in degree 4
x₁.x₂ in degree 4
x₁² in degree 4
y₃².x₂ in degree 4
y₃⁴ in degree 4
u₂ in degree 5
u₁ in degree 5
x₃.w in degree 5
x₂.w in degree 5
x₁.w in degree 5
y₃³.x₂ in degree 5
y₃⁵ in degree 5
y₁.x₁² in degree 5
t₂ in degree 6
t₁ in degree 6
x₂³ in degree 6
x₁².x₂ in degree 6
x₁³ in degree 6
y₃.u₁ in degree 6
y₃⁴.x₂ in degree 6
y₃⁶ in degree 6
s in degree 7
x₂.u₁ in degree 7
x₂.x₃.w in degree 7
x₂².w in degree 7
x₁.x₂.w in degree 7
x₁².w in degree 7
y₃².u₁ in degree 7
y₃⁵.x₂ in degree 7
y₁.t₂ in degree 7
y₁.t₁ in degree 7
x₃.t₂ in degree 8
x₂.t₂ in degree 8
x₂.t₁ in degree 8
x₁.t₁ in degree 8
x₁³.x₂ in degree 8
y₃.x₂.u₁ in degree 8
y₃³.u₁ in degree 8
y₃⁶.x₂ in degree 8
x₃.s in degree 9
x₂.s in degree 9
x₂³.w in degree 9
x₁.s in degree 9
x₁².x₂.w in degree 9
x₁³.w in degree 9
y₃².x₂.u₁ in degree 9
y₃⁴.u₁ in degree 9
y₁.x₁.t₁ in degree 9
x₂.x₃.t₂ in degree 10
x₂².t₂ in degree 10
x₁.x₂.t₁ in degree 10
x₁².t₁ in degree 10
y₃³.x₂.u₁ in degree 10
y₃⁵.u₁ in degree 10
x₂.x₃.s in degree 11
x₂².s in degree 11
x₁.x₂.s in degree 11
x₁².s in degree 11
y₃⁴.x₂.u₁ in degree 11
y₃⁶.u₁ in degree 11
x₂³.t₂ in degree 12
x₁².x₂.t₁ in degree 12
y₃⁵.x₂.u₁ in degree 12
x₂³.s in degree 13
x₁².x₂.s in degree 13
x₁³.s in degree 13
x₁³.x₂.s in degree 15

A basis for Ann_{R/(h₁, h₂, h₃)}(h₄) is as follows.

y₃².h in degree 3
y₃.x₂.h + y₃.h³ in degree 4
y₃³.h in degree 4
y₃².x₂.h in degree 5
y₃⁴.h in degree 5
y₃³.x₂.h in degree 6
y₃⁵.h in degree 6
y₁.x₁².h + y₁.x₁.h³ + y₃.h⁵ in degree 6
y₁.h⁵ in degree 6
y₃.u₁.h in degree 7
y₃⁴.x₂.h in degree 7
y₃⁶.h in degree 7
y₃².u₁.h in degree 8
y₃⁵.x₂.h in degree 8
y₁.x₃.h⁵ in degree 8
y₁.x₁.h⁵ in degree 8
y₃.h⁷ in degree 8
y₃.x₂.u₁.h + y₃.u₁.h³ in degree 9
y₃³.u₁.h in degree 9
y₃⁶.x₂.h in degree 9
x₁³.w.h + x₁².w.h³ + x₁³.h⁴ + u₂.h⁵ + x₃.w.h⁵ + x₂.w.h⁵ + x₁.w.h⁵ + x₂.x₃.h⁶ + x₁.x₂.h⁶ + x₁².h⁶ + w.h⁷ + x₂.h⁸ + x₁.h⁸ in degree 10
y₃².x₂.u₁.h in degree 10
y₃⁴.u₁.h in degree 10
y₃³.x₂.u₁.h in degree 11
y₃⁵.u₁.h in degree 11
y₃⁴.x₂.u₁.h in degree 12
y₃⁶.u₁.h in degree 12
y₁.t₂.h⁵ in degree 12
y₁.t₁.h⁵ in degree 12
y₃⁵.x₂.u₁.h in degree 13
y₁.x₁.t₁.h⁵ in degree 14
x₂.u₁.h⁷ + x₁².w.h⁷ + x₁².x₂.h⁸ + u₂.h⁹ + u₁.h⁹ + x₃.w.h⁹ + x₂.w.h⁹ + x₁.w.h⁹ + x₂.x₃.h¹⁰ + w.h¹¹ + x₂.h¹² in degree 14

A basis for Ann_{R/(h₁, h₂)}(h₃) is as follows.

y₁.x₁² + y₁.y₂².x₂ + y₁.y₂².x₁ in degree 5
y₁.y₂.x₁² + y₁.y₂³.x₂ + y₁.y₂³.x₁ in degree 6

Restriction information

Restriction to special subgroup number 1, which is 2gp1

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to 0
x₃ restricts to 0
w restricts to 0
u₁ restricts to 0
u₂ restricts to 0
t₁ restricts to 0
t₂ restricts to 0
s restricts to 0
r restricts to y⁸

Restriction to special subgroup number 2, which is 8gp5

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to y₂
x₁ restricts to 0
x₂ restricts to y₃² + y₂.y₃
x₃ restricts to 0
w restricts to 0
u₁ restricts to y₂.y₃⁴ + y₂³.y₃² + y₁².y₂³ + y₁⁴.y₂
u₂ restricts to y₂.y₃⁴ + y₂³.y₃²
t₁ restricts to y₂².y₃⁴ + y₂⁴.y₃²
t₂ restricts to y₃⁶ + y₂.y₃⁵ + y₂³.y₃³ + y₂⁴.y₃² + y₁².y₂⁴ + y₁⁴.y₂²
s restricts to y₂.y₃⁶ + y₂².y₃⁵ + y₂³.y₃⁴ + y₂⁴.y₃³ + y₁².y₂³.y₃² + y₁².y₂⁴.y₃ + y₁⁴.y₂.y₃² + y₁⁴.y₂².y₃
r restricts to y₃⁸ + y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁵.y₃³ + y₁².y₂².y₃⁴ + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁⁴.y₃⁴ + y₁⁴.y₂³.y₃ + y₁⁸

Restriction to special subgroup number 3, which is 16gp14

y₁ restricts to 0
y₂ restricts to y₂
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to y₄² + y₃.y₄
x₃ restricts to y₃²
w restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂.y₃.y₄
u₁ restricts to 0
u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃².y₄² + y₂².y₃.y₄² + y₂².y₃².y₄
t₁ restricts to y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃³.y₄ + y₂³.y₃.y₄² + y₂³.y₃².y₄
t₂ restricts to y₄⁶ + y₃.y₄⁵ + y₃³.y₄³ + y₃⁵.y₄ + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃.y₄³ + y₂².y₃².y₄² + y₂³.y₃.y₄² + y₂³.y₃².y₄ + y₂⁴.y₄² + y₁.y₃³.y₄² + y₁.y₃⁴.y₄ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₃⁴ + y₁⁴.y₃²
s restricts to y₃³.y₄⁴ + y₃⁶.y₄ + y₂.y₄⁶ + y₂.y₃⁵.y₄ + y₂².y₄⁵ + y₂².y₃².y₄³ + y₂³.y₃.y₄³ + y₂⁴.y₄³ + y₂⁴.y₃.y₄² + y₂⁵.y₄² + y₁.y₃².y₄⁴ + y₁.y₃⁴.y₄² + y₁.y₂.y₃³.y₄² + y₁.y₂.y₃⁴.y₄ + y₁².y₃.y₄⁴ + y₁².y₃⁴.y₄ + y₁².y₂.y₃².y₄² + y₁².y₂.y₃³.y₄ + y₁².y₂.y₃⁴ + y₁⁴.y₃.y₄² + y₁⁴.y₃².y₄ + y₁⁴.y₂.y₃²
r restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁵.y₄³ + y₂.y₃.y₄⁶ + y₂.y₃³.y₄⁴ + y₂.y₃⁴.y₄³ + y₂.y₃⁶.y₄ + y₂².y₄⁶ + y₂³.y₃.y₄⁴ + y₂³.y₃³.y₄² + y₂⁴.y₄⁴ + y₂⁴.y₃.y₄³ + y₂⁴.y₃³.y₄ + y₂⁵.y₃.y₄² + y₂⁵.y₃².y₄ + y₁.y₃³.y₄⁴ + y₁.y₃⁶.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₃⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃³.y₄ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

y₁ restricts to 0
y₂ restricts to y₂
y₃ restricts to 0
x₁ restricts to y₄²
x₂ restricts to y₃.y₄ + y₃² + y₂.y₄
x₃ restricts to y₄²
w restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂.y₃.y₄ + y₂.y₃²
u₁ restricts to y₃².y₄³ + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂².y₃.y₄² + y₂².y₃².y₄
u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃⁴ + y₂².y₄³ + y₂².y₃.y₄² + y₂².y₃².y₄
t₁ restricts to y₄⁶ + y₃².y₄⁴ + y₃⁴.y₄² + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃.y₄³ + y₂².y₃².y₄² + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂³.y₃.y₄² + y₂³.y₃².y₄ + y₂⁵.y₄ + y₁.y₃.y₄⁴ + y₁.y₃².y₄³ + y₁.y₂.y₄⁴ + y₁.y₂².y₄³ + y₁².y₄⁴ + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁².y₂.y₄³ + y₁².y₂².y₄² + y₁⁴.y₄²
t₂ restricts to y₃³.y₄³ + y₃⁴.y₄² + y₃⁵.y₄ + y₃⁶ + y₂.y₃.y₄⁴ + y₂.y₃³.y₄² + y₂².y₄⁴ + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂³.y₄³ + y₂⁴.y₃.y₄ + y₂⁴.y₃² + y₂⁵.y₄ + y₁.y₃.y₄⁴ + y₁.y₃².y₄³ + y₁.y₂.y₄⁴ + y₁.y₂².y₄³ + y₁².y₄⁴ + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁².y₂.y₄³ + y₁².y₂².y₄² + y₁⁴.y₄²
s restricts to y₃.y₄⁶ + y₃³.y₄⁴ + y₃⁵.y₄² + y₃⁶.y₄ + y₂.y₄⁶ + y₂.y₃.y₄⁵ + y₂.y₃².y₄⁴ + y₂.y₃⁴.y₄² + y₂.y₃⁶ + y₂².y₄⁵ + y₂².y₃.y₄⁴ + y₂².y₃².y₄³ + y₂².y₃⁴.y₄ + y₂².y₃⁵ + y₂⁴.y₄³ + y₂⁴.y₃².y₄ + y₂⁴.y₃³ + y₂⁵.y₄² + y₂⁵.y₃.y₄ + y₂⁵.y₃² + y₁.y₃².y₄⁴ + y₁.y₃⁴.y₄² + y₁.y₂².y₄⁴ + y₁.y₂⁴.y₄² + y₁².y₃.y₄⁴ + y₁².y₃⁴.y₄ + y₁².y₂.y₄⁴ + y₁².y₂⁴.y₄ + y₁⁴.y₃.y₄² + y₁⁴.y₃².y₄ + y₁⁴.y₂.y₄² + y₁⁴.y₂².y₄
r restricts to y₄⁸ + y₃⁴.y₄⁴ + y₃⁸ + y₂.y₃².y₄⁵ + y₂.y₃³.y₄⁴ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂².y₃³.y₄³ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂².y₃⁶ + y₂³.y₄⁵ + y₂³.y₃².y₄³ + y₂³.y₃⁴.y₄ + y₂⁴.y₄⁴ + y₂⁴.y₃.y₄³ + y₂⁴.y₃⁴ + y₂⁵.y₄³ + y₂⁷.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Poincaré series

(1 + 2t + 4t² + 5t³ + 5t⁴ + 6t⁵ + 6t⁶ + 6t⁷ + 5t⁸ + 4t⁹ + 3t¹⁰ + 2t¹¹ + t¹² - t¹⁴ - t¹⁵) / (1 - t) (1 - t⁴) (1 - t⁶) (1 - t⁸)

Back to the groups of order 128