Small group number 860 of order 128

G is the group 128gp860

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, 3, 4.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 14 generators:

y₁ in degree 1, a nilpotent element
y₂ in degree 1
y₃ in degree 1
x₁ in degree 2
x₂ in degree 2
w₁ in degree 3
w₂ in degree 3
w₃ in degree 3
v in degree 4
u₁ in degree 5
u₂ in degree 5
t in degree 6
s in degree 7
r in degree 8, a regular element

There are 51 minimal relations:

y₁.y₃ = 0
y₁² = 0
y₃.x₂ = y₁.x₁
y₃.x₁ = y₁.x₁
y₁.y₂² = 0
x₁.x₂ = x₁² + y₁.w₁
y₃.w₃ = y₃.w₁ + y₁.w₂ + y₁.y₂.x₁
y₂².x₂ = y₁.w₁ + y₁.y₂.x₁
y₂².x₁ = y₁.w₂
y₁.w₃ = 0
x₂.w₃ = 0
x₂.w₂ = x₁.w₁ + y₂.x₁²
x₁.w₃ = 0
x₁.w₂ = x₁.w₁ + y₂.x₁² + y₁.y₂.w₂
y₂².w₃ = y₂².w₁ + y₁.y₂.w₂
y₁.v = y₁.y₂.w₁
w₃² = y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂².y₃⁴ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶
w₂.w₃ = w₁.w₂ + x₁.v + y₁.x₁.w₁ + y₁.y₂.x₁²
w₂² = x₁.v + y₃.u₁ + y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.x₁.w₁ + y₂².v + y₂².y₃.w₂ + y₂⁴.y₃² + y₁.x₁.w₁
w₁.w₃ = y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂².y₃⁴ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶
w₁² = x₂.v + y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.x₂.w₁ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂².y₃⁴ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶ + y₁.x₂.w₁ + y₁.y₂.x₁²
y₃.u₂ = y₃.u₁ + y₃³.w₂ + y₂.y₃.v + y₂.y₃².w₂ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂⁵.y₃ + y₁.x₁.w₁
y₁.u₂ = 0
y₁.u₁ = y₁.x₁.w₁
w₃.v = y₃.t + y₃.w₁.w₂ + y₃².u₁ + y₃³.v + y₂.y₃.u₁ + y₂.y₃³.w₂ + y₂².u₁ + y₂².y₃.v + y₂³.v + y₂³.y₃.w₂ + y₂⁴.w₂ + y₂⁴.w₁ + y₂⁷ + y₁.x₁³
x₂.u₂ = x₁.u₂
x₁.u₁ = x₁².w₁ + y₂.x₁.v + y₁.x₁³ + y₁.y₂.x₁.w₁
y₂².u₂ = y₂².u₁ + y₂².y₃².w₂ + y₂³.v + y₂³.y₃.w₂ + y₂³.y₃.w₁ + y₂⁴.w₁ + y₂⁷ + y₁.y₂.x₁.w₁
y₁.t = y₁.x₁³ + y₁.y₂.x₂.w₁
w₃.u₂ = y₃.s + y₃².w₁.w₂ + y₃³.u₁ + y₂.y₃.w₁.w₂ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².y₃³.w₁ + y₂².y₃⁶ + y₂³.y₃².w₂ + y₂³.y₃⁵ + y₂⁴.v + y₂⁴.y₃⁴ + y₂⁶.y₃² + y₁.x₁².w₁ + y₁.y₂.x₁³
w₃.u₁ = y₃.s + y₃³.u₁ + y₂.y₃.t + y₂.y₃.w₁.w₂ + y₂.y₃².u₁ + y₂.y₃⁴.w₂ + y₂².w₁.w₂ + y₂².y₃.u₁ + y₂².y₃².v + y₂².y₃³.w₁ + y₂².y₃⁶ + y₂³.u₁ + y₂³.y₃.v + y₂³.y₃².w₂ + y₂⁴.y₃.w₂ + y₂⁴.y₃.w₁ + y₂⁵.w₂ + y₂⁵.y₃³ + y₂⁶.y₃² + y₁.x₁².w₁
w₂.u₂ = w₂.u₁ + x₁.t + x₁⁴ + y₃³.u₁ + y₃⁴.v + y₃⁵.w₂ + y₃⁵.w₁ + y₂.w₂.v + y₂.x₁.u₂ + y₂.y₃.w₁.w₂ + y₂.y₃².u₁ + y₂.y₃³.v + y₂.y₃⁴.w₂ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².y₃².v + y₂².y₃³.w₂ + y₂³.y₃.v + y₂³.y₃².w₂ + y₂⁴.y₃⁴ + y₂⁵.w₂ + y₂⁵.y₃³ + y₁.x₁².w₁ + y₁.y₂.x₁³
w₁.u₂ = x₁.t + x₁².v + x₁⁴ + y₃.s + y₃².w₁.w₂ + y₃³.u₁ + y₂.y₃.w₁.w₂ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².y₃³.w₁ + y₂².y₃⁶ + y₂³.y₃².w₂ + y₂³.y₃⁵ + y₂⁴.v + y₂⁴.y₃⁴ + y₂⁶.y₃²
w₁.u₁ = x₂.t + x₂².v + x₁.t + y₃.s + y₃³.u₁ + y₂.w₁.v + y₂.x₁².w₁ + y₂.y₃³.v + y₂.y₃⁴.w₂ + y₂².w₁.w₂ + y₂².y₃².v + y₂².y₃³.w₂ + y₂².y₃³.w₁ + y₂².y₃⁶ + y₂³.y₃².w₂ + y₂⁴.v + y₂⁴.y₃.w₁ + y₂⁵.w₁ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁸ + y₁.x₂².w₁ + y₁.x₁².w₁
y₁.s = y₁.x₂².w₁ + y₁.y₂.x₁³
v.u₂ = v.u₁ + w₁.t + x₂.w₁.v + x₁.w₁.v + x₁³.w₁ + y₃.v² + y₃².s + y₃².w₂.v + y₃³.w₁.w₂ + y₃⁵.v + y₃⁶.w₂ + y₃⁶.w₁ + y₂.v² + y₂.x₂.t + y₂.x₂².v + y₂.x₁⁴ + y₂.y₃.s + y₂.y₃.w₂.v + y₂.y₃².t + y₂.y₃².w₁.w₂ + y₂.y₃⁵.w₂ + y₂².s + y₂².y₃⁴.w₂ + y₂².y₃⁴.w₁ + y₂².y₃⁷ + y₂³.y₃.u₁ + y₂³.y₃².v + y₂³.y₃³.w₂ + y₂³.y₃³.w₁ + y₂³.y₃⁶ + y₂⁴.y₃.v + y₂⁴.y₃².w₂ + y₂⁴.y₃².w₁ + y₂⁵.v + y₂⁵.y₃.w₂ + y₂⁵.y₃⁴ + y₂⁶.w₂ + y₂⁶.w₁ + y₂⁶.y₃³ + y₁.y₂.x₂².w₁
w₃.t = y₃.v² + y₃².s + y₃³.w₁.w₂ + y₃⁵.v + y₃⁶.w₂ + y₃⁶.w₁ + y₂.y₃.s + y₂.y₃³.u₁ + y₂.y₃⁴.v + y₂.y₃⁵.w₂ + y₂².s + y₂².y₃.t + y₂².y₃.w₁.w₂ + y₂².y₃³.v + y₂².y₃⁴.w₁ + y₂².y₃⁷ + y₂³.y₃.u₁ + y₂³.y₃³.w₁ + y₂³.y₃⁶ + y₂⁴.u₁ + y₂⁴.y₃.v + y₂⁴.y₃².w₁ + y₂⁵.v + y₂⁵.y₃.w₂ + y₂⁵.y₃.w₁ + y₂⁵.y₃⁴ + y₂⁶.y₃³ + y₂⁸.y₃ + y₂⁹ + y₁.y₂.x₁².w₁
x₂.s = x₂.w₁.v + x₂².u₁ + x₂³.w₁ + x₁.w₁.v + x₁³.w₁ + y₂.x₂².v + y₂.x₁⁴ + y₁.y₂.x₁².w₁
x₁.s = x₁³.w₁ + y₂.x₁⁴ + y₁.x₁⁴ + y₁.y₂.x₁².w₁
u₂² = x₁.v² + y₃.v.u₁ + y₃³.w₂.v + y₃⁴.w₁.w₂ + y₃⁵.u₁ + y₃⁶.v + y₃⁷.w₂ + y₂.y₃².w₂.v + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂.y₃⁹ + y₂².v² + y₂².y₃.w₂.v + y₂².y₃².w₁.w₂ + y₂².y₃⁴.v + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂³.y₃³.v + y₂³.y₃⁷ + y₂⁴.y₃².v + y₂⁵.y₃².w₂ + y₂⁵.y₃².w₁ + y₂⁵.y₃⁵ + y₂⁶.v + y₂⁶.y₃.w₁ + y₂⁶.y₃⁴ + y₂⁹.y₃ + y₃².r
u₁.u₂ = x₁².t + x₁³.v + x₁⁵ + y₃.v.u₁ + y₃².w₂.u₁ + y₃³.w₂.v + y₃⁴.w₁.w₂ + y₃⁷.w₁ + y₂.v.u₁ + y₂.w₁.t + y₂.x₂.w₁.v + y₂.x₁.w₁.v + y₂.x₁³.w₁ + y₂.y₃.v² + y₂.y₃.w₂.u₁ + y₂.y₃².w₂.v + y₂.y₃³.w₁.w₂ + y₂.y₃⁶.w₁ + y₂.y₃⁹ + y₂².y₃.w₂.v + y₂².y₃².t + y₂².y₃⁴.v + y₂².y₃⁵.w₁ + y₂³.s + y₂³.y₃.w₁.w₂ + y₂³.y₃³.v + y₂³.y₃⁴.w₂ + y₂³.y₃⁴.w₁ + y₂³.y₃⁷ + y₂⁴.w₁.w₂ + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁴.y₃³.w₂ + y₂⁵.u₁ + y₂⁵.y₃².w₂ + y₂⁵.y₃⁵ + y₂⁶.y₃.w₂ + y₂⁷.w₂ + y₂⁸.y₃² + y₂⁹.y₃ + y₂¹⁰ + y₁.x₁³.w₁ + y₁.y₂.x₁⁴ + y₃².r
u₁² = x₂.v² + x₁.v² + x₁³.v + y₃.v.u₁ + y₃³.w₂.v + y₃⁴.w₁.w₂ + y₃⁷.w₁ + y₂.x₁³.w₁ + y₂.y₃².w₂.v + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂.y₃⁹ + y₂².y₃.w₂.v + y₂².y₃².w₁.w₂ + y₂².y₃³.u₁ + y₂².y₃⁵.w₁ + y₂³.y₃³.v + y₂³.y₃⁴.w₁ + y₂³.y₃⁷ + y₂⁴.y₃².v + y₂⁵.y₃².w₂ + y₂⁵.y₃⁵ + y₂⁶.v + y₂⁷.y₃³ + y₁.x₁³.w₁ + y₁.y₂.x₁⁴ + y₃².r
w₃.s = y₃.v.u₁ + y₃².w₂.u₁ + y₂.y₃.v² + y₂.y₃².s + y₂.y₃².w₂.v + y₂.y₃³.w₁.w₂ + y₂.y₃⁴.u₁ + y₂².y₃.s + y₂².y₃.w₂.v + y₂².y₃².t + y₂².y₃².w₁.w₂ + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂³.y₃.w₁.w₂ + y₂³.y₃².u₁ + y₂³.y₃³.v + y₂³.y₃⁴.w₂ + y₂³.y₃⁷ + y₂⁴.t + y₂⁴.w₁.w₂ + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁴.y₃³.w₂ + y₂⁴.y₃³.w₁ + y₂⁶.v + y₂⁸.y₃² + y₂¹⁰
w₁.s = x₂.v² + x₂².t + x₁.v² + x₁².t + x₁³.v + y₃.v.u₁ + y₃².w₂.u₁ + y₂.x₂.w₁.v + y₂.x₂³.w₁ + y₂.x₁.w₁.v + y₂.x₁³.w₁ + y₂.y₃.v² + y₂.y₃².s + y₂.y₃².w₂.v + y₂.y₃³.w₁.w₂ + y₂.y₃⁴.u₁ + y₂².y₃.s + y₂².y₃.w₂.v + y₂².y₃².t + y₂².y₃².w₁.w₂ + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂³.y₃.w₁.w₂ + y₂³.y₃².u₁ + y₂³.y₃³.v + y₂³.y₃⁴.w₂ + y₂³.y₃⁷ + y₂⁴.t + y₂⁴.w₁.w₂ + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁴.y₃³.w₂ + y₂⁴.y₃³.w₁ + y₂⁶.v + y₂⁸.y₃² + y₂¹⁰ + y₁.y₂.x₁⁴
u₂.t = v.s + w₁.v² + x₂.w₁.t + x₁³.u₂ + x₁⁴.w₁ + y₃.v.t + y₃.w₂.s + y₃³.v² + y₃⁴.s + y₃⁵.t + y₃⁵.w₁.w₂ + y₃⁸.w₂ + y₂.x₂².t + y₂.x₂³.v + y₂.x₁³.v + y₂.x₁⁵ + y₂.y₃.v.u₁ + y₂.y₃.w₂.t + y₂.y₃².v² + y₂.y₃³.s + y₂.y₃³.w₂.v + y₂.y₃⁴.t + y₂.y₃⁴.w₁.w₂ + y₂.y₃⁵.u₁ + y₂.y₃⁷.w₁ + y₂.y₃¹⁰ + y₂².w₂.t + y₂².y₃.v² + y₂².y₃⁵.v + y₂².y₃⁶.w₂ + y₂².y₃⁹ + y₂³.y₃.w₂.v + y₂³.y₃².t + y₂³.y₃³.u₁ + y₂³.y₃⁴.v + y₂³.y₃⁵.w₂ + y₂³.y₃⁸ + y₂⁴.w₂.v + y₂⁴.y₃.t + y₂⁴.y₃³.v + y₂⁵.y₃⁶ + y₂⁶.u₁ + y₂⁶.y₃².w₂ + y₂⁷.v + y₂⁷.y₃.w₂ + y₂⁷.y₃.w₁ + y₂⁸.w₂ + y₂⁸.w₁ + y₂⁸.y₃³ + y₂¹¹ + y₁.y₂.x₂³.w₁ + y₃³.r + y₂.y₃².r + y₂².y₃.r
u₁.t = v.s + x₂².w₁.v + x₁.w₁.t + y₃.w₂.s + y₃².v.u₁ + y₃³.w₂.u₁ + y₃⁴.w₂.v + y₃⁵.w₁.w₂ + y₃⁸.w₁ + y₂.v.t + y₂.x₁³.v + y₂.y₃².w₂.u₁ + y₂.y₃⁴.w₁.w₂ + y₂.y₃¹⁰ + y₂².v.u₁ + y₂².w₂.t + y₂².y₃.v² + y₂².y₃.w₂.u₁ + y₂².y₃².w₂.v + y₂².y₃³.t + y₂².y₃⁶.w₂ + y₂².y₃⁶.w₁ + y₂².y₃⁹ + y₂³.v² + y₂³.y₃.w₂.v + y₂³.y₃².w₁.w₂ + y₂³.y₃³.u₁ + y₂³.y₃⁴.v + y₂³.y₃⁵.w₂ + y₂⁴.s + y₂⁴.y₃.t + y₂⁴.y₃.w₁.w₂ + y₂⁴.y₃³.v + y₂⁴.y₃⁴.w₂ + y₂⁴.y₃⁷ + y₂⁵.t + y₂⁵.y₃.u₁ + y₂⁵.y₃².v + y₂⁵.y₃⁶ + y₂⁶.u₁ + y₂⁶.y₃².w₁ + y₂⁶.y₃⁵ + y₂⁸.y₃³ + y₂¹¹ + y₁.x₁⁵ + y₁.y₂.x₁³.w₁ + y₃³.r + y₂.y₃².r + y₂².y₃.r
t² = v³ + x₂².v² + x₁⁶ + y₃.w₂.v² + y₃².v.t + y₃³.v.u₁ + y₃³.w₂.t + y₃⁴.v² + y₃⁶.t + y₃⁶.w₁.w₂ + y₃⁹.w₂ + y₂.w₁.v² + y₂.y₃².v.u₁ + y₂.y₃³.w₂.u₁ + y₂.y₃⁴.s + y₂.y₃⁵.t + y₂.y₃⁶.u₁ + y₂.y₃¹¹ + y₂².v.t + y₂².y₃.v.u₁ + y₂².y₃.w₂.t + y₂².y₃⁴.t + y₂².y₃⁴.w₁.w₂ + y₂².y₃⁵.u₁ + y₂².y₃⁶.v + y₂³.v.u₁ + y₂³.y₃.v² + y₂³.y₃.w₂.u₁ + y₂³.y₃³.t + y₂³.y₃³.w₁.w₂ + y₂³.y₃⁵.v + y₂³.y₃⁶.w₂ + y₂³.y₃⁶.w₁ + y₂⁴.v² + y₂⁴.w₂.u₁ + y₂⁴.y₃².t + y₂⁴.y₃⁵.w₁ + y₂⁴.y₃⁸ + y₂⁵.w₂.v + y₂⁵.y₃².u₁ + y₂⁵.y₃³.v + y₂⁵.y₃⁴.w₂ + y₂⁵.y₃⁷ + y₂⁶.t + y₂⁶.w₁.w₂ + y₂⁶.y₃³.w₂ + y₂⁶.y₃⁶ + y₂⁷.u₁ + y₂⁷.y₃.v + y₂⁷.y₃².w₂ + y₂⁸.y₃.w₁ + y₂⁸.y₃⁴ + y₂⁹.w₂ + y₂⁹.w₁ + y₂¹⁰.y₃² + y₃⁴.r + y₂².y₃².r + y₂⁴.r
u₂.s = x₁³.t + x₁⁴.v + x₁⁶ + y₃.v.s + y₃².w₂.s + y₃³.w₂.t + y₃⁴.w₂.u₁ + y₃⁵.s + y₃⁵.w₂.v + y₃⁶.t + y₃⁷.u₁ + y₃⁹.w₂ + y₂.x₁³.u₂ + y₂.y₃.v.t + y₂.y₃.w₂.s + y₂.y₃³.w₂.u₁ + y₂.y₃⁴.s + y₂.y₃⁷.v + y₂.y₃¹¹ + y₂².v.t + y₂².w₂.s + y₂².y₃.w₂.t + y₂².y₃².w₂.u₁ + y₂².y₃⁴.w₁.w₂ + y₂².y₃⁷.w₂ + y₂².y₃⁷.w₁ + y₂².y₃¹⁰ + y₂³.y₃.v² + y₂³.y₃².w₂.v + y₂³.y₃³.w₁.w₂ + y₂³.y₃⁴.u₁ + y₂³.y₃⁵.v + y₂⁴.v² + y₂⁴.w₂.u₁ + y₂⁴.y₃.s + y₂⁴.y₃.w₂.v + y₂⁴.y₃².t + y₂⁴.y₃⁵.w₂ + y₂⁴.y₃⁵.w₁ + y₂⁴.y₃⁸ + y₂⁵.w₂.v + y₂⁵.y₃³.v + y₂⁵.y₃⁷ + y₂⁶.w₁.w₂ + y₂⁶.y₃.u₁ + y₂⁶.y₃².v + y₂⁶.y₃³.w₂ + y₂⁷.y₃⁵ + y₂⁸.y₃.w₂ + y₂⁸.y₃.w₁ + y₂⁹.w₂ + y₁.x₁⁴.w₁ + y₁.y₂.x₁⁵ + y₃.w₁.r + y₃⁴.r + y₁.w₂.r + y₁.y₂.x₁.r
u₁.s = x₂.v.t + x₂³.t + x₂⁴.v + x₁.v.t + x₁³.t + y₃.v.s + y₃³.w₂.t + y₃⁴.w₂.u₁ + y₃⁵.s + y₃⁵.w₂.v + y₃⁶.t + y₃⁷.u₁ + y₃⁹.w₂ + y₂.v.s + y₂.y₃.v.t + y₂.y₃².v.u₁ + y₂.y₃⁴.s + y₂.y₃⁷.v + y₂.y₃¹¹ + y₂².v.t + y₂².w₂.s + y₂².y₃.v.u₁ + y₂².y₃.w₂.t + y₂².y₃².v² + y₂².y₃³.s + y₂².y₃³.w₂.v + y₂².y₃⁵.u₁ + y₂².y₃⁷.w₂ + y₂².y₃⁷.w₁ + y₂².y₃¹⁰ + y₂³.y₃².w₂.v + y₂³.y₃³.t + y₂³.y₃³.w₁.w₂ + y₂³.y₃⁵.v + y₂³.y₃⁶.w₂ + y₂³.y₃⁶.w₁ + y₂⁴.v² + y₂⁴.w₂.u₁ + y₂⁴.y₃³.u₁ + y₂⁴.y₃⁴.v + y₂⁴.y₃⁵.w₂ + y₂⁵.s + y₂⁵.w₂.v + y₂⁵.y₃.t + y₂⁵.y₃³.v + y₂⁵.y₃⁴.w₁ + y₂⁶.t + y₂⁶.y₃³.w₁ + y₂⁷.y₃.v + y₂⁷.y₃⁵ + y₂⁸.v + y₂⁸.y₃.w₂ + y₂⁸.y₃.w₁ + y₂⁹.w₂ + y₂⁹.y₃³ + y₂¹⁰.y₃² + y₂¹¹.y₃ + y₂¹² + y₁.x₂⁴.w₁ + y₁.x₁⁴.w₁ + y₃.w₁.r + y₃⁴.r + y₁.w₂.r + y₁.y₂.x₁.r
t.s = v².u₁ + x₂³.w₁.v + x₁.w₁.v² + x₁².w₁.t + x₁³.w₁.v + y₃⁴.w₂.t + y₃⁷.t + y₃⁷.w₁.w₂ + y₃¹⁰.w₂ + y₃¹⁰.w₁ + y₂.v³ + y₂.x₂.v.t + y₂.x₂².v² + y₂.x₂³.t + y₂.x₂⁴.v + y₂.x₁.v.t + y₂.x₁².v² + y₂.x₁⁴.v + y₂.y₃.v.s + y₂.y₃².v.t + y₂.y₃³.v.u₁ + y₂.y₃⁴.v² + y₂.y₃⁴.w₂.u₁ + y₂.y₃⁶.t + y₂.y₃⁸.v + y₂.y₃⁹.w₂ + y₂².w₂.v² + y₂².y₃.v.t + y₂².y₃⁴.s + y₂².y₃⁵.t + y₂².y₃⁵.w₁.w₂ + y₂².y₃⁶.u₁ + y₂².y₃⁸.w₂ + y₂².y₃⁸.w₁ + y₂².y₃¹¹ + y₂³.y₃.v.u₁ + y₂³.y₃².v² + y₂³.y₃³.w₂.v + y₂³.y₃⁴.t + y₂³.y₃⁴.w₁.w₂ + y₂³.y₃⁵.u₁ + y₂³.y₃⁷.w₁ + y₂³.y₃¹⁰ + y₂⁴.v.u₁ + y₂⁴.w₂.t + y₂⁴.y₃².s + y₂⁴.y₃³.w₁.w₂ + y₂⁴.y₃⁴.u₁ + y₂⁴.y₃⁹ + y₂⁵.w₂.u₁ + y₂⁵.y₃.w₂.v + y₂⁵.y₃².w₁.w₂ + y₂⁵.y₃⁴.v + y₂⁵.y₃⁵.w₂ + y₂⁶.w₂.v + y₂⁶.y₃.t + y₂⁶.y₃⁴.w₂ + y₂⁶.y₃⁴.w₁ + y₂⁶.y₃⁷ + y₂⁷.y₃.u₁ + y₂⁷.y₃².v + y₂⁷.y₃³.w₂ + y₂⁷.y₃⁶ + y₂⁸.y₃.v + y₂⁸.y₃².w₁ + y₂⁸.y₃⁵ + y₂⁹.y₃.w₂ + y₂⁹.y₃.w₁ + y₂⁹.y₃⁴ + y₂¹⁰.w₂ + y₂¹⁰.w₁ + y₂¹².y₃ + y₁.x₁⁶ + y₁.y₂.x₂⁴.w₁ + y₃².w₁.r + y₂.y₃.w₁.r + y₂.y₃⁴.r + y₂².w₁.r + y₂².y₃³.r
s² = x₂.v³ + x₂³.v² + x₂⁵.v + x₁.v³ + x₁³.v² + y₃.v².u₁ + y₃².w₂.v.u₁ + y₃³.v.s + y₃³.w₂.v² + y₃⁵.v.u₁ + y₃⁶.v² + y₃⁷.s + y₃⁷.w₂.v + y₃⁸.t + y₃⁸.w₁.w₂ + y₃⁹.u₁ + y₃¹¹.w₂ + y₂.x₂.w₁.v² + y₂.x₂⁵.w₁ + y₂.x₁.w₁.v² + y₂.y₃².v.s + y₂.y₃².w₂.v² + y₂.y₃³.v.t + y₂.y₃⁴.w₂.t + y₂.y₃⁶.w₂.v + y₂.y₃⁹.v + y₂.y₃¹⁰.w₁ + y₂.y₃¹³ + y₂².v³ + y₂².y₃.v.s + y₂².y₃³.v.u₁ + y₂².y₃³.w₂.t + y₂².y₃⁴.v² + y₂².y₃⁴.w₂.u₁ + y₂².y₃⁶.t + y₂².y₃⁶.w₁.w₂ + y₂².y₃⁷.u₁ + y₂².y₃⁸.v + y₂².y₃⁹.w₂ + y₂².y₃¹² + y₂³.y₃².v.u₁ + y₂³.y₃².w₂.t + y₂³.y₃³.v² + y₂³.y₃³.w₂.u₁ + y₂³.y₃⁴.s + y₂³.y₃⁴.w₂.v + y₂³.y₃⁶.u₁ + y₂³.y₃⁸.w₂ + y₂⁴.v.t + y₂⁴.y₃.v.u₁ + y₂⁴.y₃.w₂.t + y₂⁴.y₃².v² + y₂⁴.y₃³.s + y₂⁴.y₃³.w₂.v + y₂⁴.y₃⁴.w₁.w₂ + y₂⁴.y₃⁷.w₂ + y₂⁴.y₃⁷.w₁ + y₂⁵.y₃.v² + y₂⁵.y₃².s + y₂⁵.y₃⁶.w₂ + y₂⁵.y₃⁶.w₁ + y₂⁶.v² + y₂⁶.w₂.u₁ + y₂⁶.y₃.s + y₂⁶.y₃².w₁.w₂ + y₂⁶.y₃⁴.v + y₂⁶.y₃⁵.w₂ + y₂⁷.y₃.w₁.w₂ + y₂⁷.y₃⁷ + y₂⁸.t + y₂⁸.w₁.w₂ + y₂⁹.y₃.v + y₂⁹.y₃².w₂ + y₂⁹.y₃².w₁ + y₂¹⁰.v + y₂¹⁰.y₃.w₂ + y₂¹².y₃² + y₁.x₂⁵.w₁ + y₁.x₁⁵.w₁ + y₃².v.r + y₃³.w₂.r + y₃³.w₁.r + y₃⁶.r + y₂.y₃².w₁.r + y₂².y₃.w₁.r + y₂².y₃⁴.r + y₂⁴.y₃².r + y₂⁵.y₃.r + y₂⁶.r

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

y₁.w₁.w₂ = y₁.y₂.x₁.w₁
y₃.w₁.v = y₃².t + y₃².w₁.w₂ + y₃³.u₁ + y₃⁴.v + y₂.y₃².u₁ + y₂.y₃⁴.w₂ + y₂².y₃.u₁ + y₂².y₃².v + y₂³.y₃.v + y₂³.y₃².w₂ + y₂⁴.y₃.w₂ + y₂⁴.y₃.w₁ + y₂⁷.y₃
y₂².w₁.v = y₂².y₃.t + y₂².y₃.w₁.w₂ + y₂².y₃².u₁ + y₂².y₃³.v + y₂³.y₃.u₁ + y₂³.y₃³.w₂ + y₂⁴.u₁ + y₂⁴.y₃.v + y₂⁵.v + y₂⁵.y₃.w₂ + y₂⁶.w₂ + y₂⁶.w₁ + y₂⁹
w₁.w₂.v = x₁.v² + y₃.w₂.t + y₃².w₂.u₁ + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁷.w₂ + y₃⁷.w₁ + y₂.y₃.w₂.u₁ + y₂.y₃³.t + y₂.y₃³.w₁.w₂ + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂².w₂.u₁ + y₂².y₃.w₂.v + y₂².y₃².t + y₂².y₃².w₁.w₂ + y₂².y₃³.u₁ + y₂².y₃⁴.v + y₂³.w₂.v + y₂³.y₃².u₁ + y₂³.y₃⁴.w₂ + y₂³.y₃⁴.w₁ + y₂⁴.w₁.w₂ + y₂⁴.y₃³.w₂ + y₂⁴.y₃⁶ + y₂⁵.y₃².w₂ + y₂⁵.y₃⁵ + y₂⁶.v + y₂⁶.y₃.w₁ + y₂⁷.w₂ + y₂⁸.y₃² + y₂⁹.y₃ + y₁.x₁³.w₁ + y₁.y₂.x₁⁴
y₃.w₁.t = y₃².v² + y₃³.s + y₃⁴.w₁.w₂ + y₃⁶.v + y₃⁷.w₂ + y₃⁷.w₁ + y₂.y₃².s + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂².y₃.s + y₂².y₃².t + y₂².y₃².w₁.w₂ + y₂².y₃⁴.v + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.y₃².u₁ + y₂³.y₃⁴.w₁ + y₂³.y₃⁷ + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁴.y₃³.w₁ + y₂⁵.y₃.v + y₂⁵.y₃².w₂ + y₂⁵.y₃².w₁ + y₂⁵.y₃⁵ + y₂⁶.y₃⁴ + y₂⁸.y₃² + y₂⁹.y₃ + y₁.x₁³.w₁
x₂.v.u₁ = x₂.w₁.t + x₂².w₁.v + x₁.w₁.t + y₂.x₂.v² + y₂.x₂².t + y₂.x₂³.v + y₂.x₁².t + y₂.x₁³.v + y₁.y₂.x₂³.w₁
y₂².w₁.t = y₂².y₃.v² + y₂².y₃².s + y₂².y₃³.w₁.w₂ + y₂².y₃⁵.v + y₂².y₃⁶.w₂ + y₂².y₃⁶.w₁ + y₂³.y₃.s + y₂³.y₃³.u₁ + y₂³.y₃⁴.v + y₂³.y₃⁵.w₂ + y₂⁴.s + y₂⁴.y₃.t + y₂⁴.y₃.w₁.w₂ + y₂⁴.y₃³.v + y₂⁴.y₃⁴.w₁ + y₂⁴.y₃⁷ + y₂⁵.y₃.u₁ + y₂⁵.y₃³.w₁ + y₂⁵.y₃⁶ + y₂⁶.u₁ + y₂⁶.y₃.v + y₂⁶.y₃².w₁ + y₂⁷.v + y₂⁷.y₃.w₂ + y₂⁷.y₃.w₁ + y₂⁷.y₃⁴ + y₂⁸.y₃³ + y₂¹⁰.y₃ + y₂¹¹ + y₁.y₂.x₁³.w₁
w₁.w₂.t = x₁.v.t + y₃.w₂.v² + y₃².w₂.s + y₃⁵.s + y₃⁵.w₂.v + y₃⁶.t + y₃⁶.w₁.w₂ + y₃⁷.u₁ + y₃⁸.v + y₂.y₃.w₂.s + y₂.y₃³.w₂.u₁ + y₂.y₃⁴.w₂.v + y₂.y₃⁵.t + y₂.y₃⁵.w₁.w₂ + y₂.y₃⁶.u₁ + y₂.y₃⁷.v + y₂.y₃⁸.w₂ + y₂².w₂.s + y₂².y₃.w₂.t + y₂².y₃³.s + y₂².y₃³.w₂.v + y₂².y₃⁵.u₁ + y₂².y₃⁷.w₂ + y₂².y₃⁷.w₁ + y₂³.y₃.w₂.u₁ + y₂³.y₃³.t + y₂³.y₃⁵.v + y₂³.y₃⁶.w₂ + y₂³.y₃⁶.w₁ + y₂⁴.w₂.u₁ + y₂⁴.y₃.w₂.v + y₂⁴.y₃².t + y₂⁴.y₃⁵.w₁ + y₂⁵.w₂.v + y₂⁵.y₃.w₁.w₂ + y₂⁵.y₃².u₁ + y₂⁵.y₃⁴.w₂ + y₂⁵.y₃⁴.w₁ + y₂⁵.y₃⁷ + y₂⁶.y₃.u₁ + y₂⁶.y₃².v + y₂⁶.y₃⁶ + y₂⁷.y₃².w₂ + y₂⁷.y₃⁵ + y₂⁸.y₃.w₁ + y₂⁹.w₂ + y₂⁹.y₃³ + y₂¹¹.y₃ + y₁.x₁⁴.w₁ + y₁.y₂.x₁⁵

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 14 onwards, and Benson's tests detect stability from degree 14 onwards.

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

h₁ = r in degree 8
h₂ = x₂ + y₂² in degree 2
h₃ = v in degree 4
h₄ = y₃² in degree 2

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, 5, 10, 12.
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
y₂ in degree 1
y₁ in degree 1
x₁ in degree 2
y₂.y₃ in degree 2
y₂² in degree 2
y₁.y₂ in degree 2
w₃ in degree 3
w₂ in degree 3
w₁ in degree 3
y₂.x₁ in degree 3
y₂³ in degree 3
y₁.x₁ in degree 3
y₃.w₂ in degree 4
y₃.w₁ in degree 4
y₂.w₃ in degree 4
y₂.w₂ in degree 4
y₂.w₁ in degree 4
y₁.w₂ in degree 4
y₁.w₁ in degree 4
y₁.y₂.x₁ in degree 4
u₂ in degree 5
u₁ in degree 5
y₂.y₃.w₂ in degree 5
y₂.y₃.w₁ in degree 5
y₂².w₂ in degree 5
y₁.y₂.w₂ in degree 5
t in degree 6
w₁.w₂ in degree 6
y₃.u₁ in degree 6
y₂.u₂ in degree 6
y₂.u₁ in degree 6
y₂³.w₂ in degree 6
s in degree 7
y₃.w₁.w₂ in degree 7
y₂.t in degree 7
y₂.w₁.w₂ in degree 7
y₂.y₃.u₁ in degree 7
y₂².u₁ in degree 7
w₂.u₁ in degree 8
x₁.t in degree 8
y₃.s in degree 8
y₂.s in degree 8
y₂.y₃.w₁.w₂ in degree 8
y₂².t in degree 8
y₂³.u₁ in degree 8
w₂.t in degree 9
y₃.w₂.u₁ in degree 9
y₂.w₂.u₁ in degree 9
y₂.x₁.t in degree 9
y₂.y₃.s in degree 9
y₂³.t in degree 9
w₂.s in degree 10
y₂.w₂.t in degree 10
y₂.y₃.w₂.u₁ in degree 10
y₃.w₂.s in degree 11
y₂.w₂.s in degree 11
y₂.y₃.w₂.s in degree 12

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

y₁ in degree 1
x₁ in degree 2
y₂² in degree 2
y₁.y₂ in degree 2
w₃ + w₁ in degree 3
y₂.x₁ in degree 3
y₂³ in degree 3
y₁.x₁ in degree 3
y₂.w₃ + y₂.w₁ in degree 4
y₁.w₂ in degree 4
y₁.w₁ in degree 4
y₁.y₂.x₁ in degree 4
u₂ + u₁ + y₂.y₃.w₂ + y₂.y₃.w₁ + w₂.h in degree 5
y₂².w₂ in degree 5
y₁.y₂.w₂ in degree 5
t + w₁.w₂ + y₃.u₁ + y₂.u₁ + y₂.w₂.h in degree 6
y₂.u₂ + y₂.u₁ + y₂².y₃.w₂ + y₂².y₃.w₁ + y₂.w₂.h in degree 6
y₂³.w₂ in degree 6
y₂.t + y₂.w₁.w₂ + y₂.y₃.u₁ + y₂².u₁ + y₂².w₂.h in degree 7
y₂².u₁ in degree 7
x₁.t in degree 8
y₂².t in degree 8
y₂³.u₁ in degree 8
w₂.t + w₁.w₂² + y₃.w₂.u₁ + y₂.w₂.u₁ + y₂.w₂².h in degree 9
y₂.x₁.t in degree 9
y₂³.t in degree 9
y₂.w₂.t + y₂.w₁.w₂² + y₂.y₃.w₂.u₁ + y₂².w₂.u₁ + y₂².w₂².h in degree 10

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

y₁.y₂ in degree 2
y₁.x₁ in degree 3
y₁.w₂ in degree 4
y₁.w₁ in degree 4
y₁.y₂.x₁ in degree 4
y₁.y₂.w₂ in degree 5
y₁.h in degree 5

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
w₁ restricts to 0
w₂ restricts to 0
w₃ restricts to 0
v restricts to 0
u₁ restricts to 0
u₂ restricts to 0
t 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 0
x₁ restricts to 0
x₂ restricts to y₂²
w₁ restricts to y₂.y₃² + y₂².y₃
w₂ restricts to 0
w₃ restricts to 0
v restricts to y₃⁴ + y₂².y₃²
u₁ restricts to y₂.y₃⁴ + y₂³.y₃²
u₂ restricts to 0
t restricts to y₃⁶ + y₂.y₃⁵ + y₂³.y₃³ + y₂⁴.y₃²
s restricts to y₂.y₃⁶ + y₂².y₃⁵ + y₂⁴.y₃³ + y₂⁶.y₃
r restricts to y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁵.y₃³ + y₂⁶.y₃² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 3, which is 8gp5

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to 0
x₁ restricts to y₃²
x₂ restricts to y₃²
w₁ restricts to y₂.y₃² + y₂².y₃
w₂ restricts to y₂.y₃² + y₂².y₃
w₃ restricts to 0
v restricts to y₂².y₃² + y₂⁴
u₁ restricts to y₂.y₃⁴ + y₂².y₃³
u₂ restricts to y₂².y₃³ + y₂⁴.y₃
t restricts to y₃⁶ + y₂².y₃⁴ + y₂³.y₃³ + y₂⁵.y₃ + y₂⁶
s restricts to y₂.y₃⁶ + y₂².y₃⁵
r restricts to y₃⁸ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃³ + y₂⁶.y₃² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

y₁ restricts to 0
y₂ restricts to y₄
y₃ restricts to y₂
x₁ restricts to 0
x₂ restricts to 0
w₁ restricts to y₄³ + y₂.y₃² + y₂².y₄ + y₂².y₃
w₂ restricts to y₃.y₄² + y₃².y₄ + y₁.y₂² + y₁².y₂
w₃ restricts to y₄³ + y₂.y₃² + y₂².y₄ + y₂².y₃
v restricts to y₃².y₄² + y₃⁴ + y₂².y₄² + y₂².y₃.y₄ + y₂³.y₄ + y₂³.y₃ + y₁.y₂³ + y₁².y₂²
u₁ restricts to y₃.y₄⁴ + y₃².y₄³ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂.y₃⁴ + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₁².y₂³ + y₁⁴.y₂
u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃².y₄² + y₂.y₃⁴ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₁.y₂⁴ + y₁⁴.y₂
t restricts to y₃⁴.y₄² + y₃⁶ + y₂.y₄⁵ + y₂.y₃.y₄⁴ + y₂.y₃⁴.y₄ + y₂.y₃⁵ + y₂².y₃.y₄³ + y₂³.y₄³ + y₂³.y₃³ + y₂⁴.y₄² + y₂⁵.y₄ + y₂⁵.y₃ + y₁.y₂.y₄⁴ + y₁.y₂³.y₄² + y₁.y₂⁴.y₄ + y₁.y₂⁵ + y₁².y₄⁴ + y₁⁴.y₄² + y₁⁴.y₂.y₄ + y₁⁴.y₂²
s restricts to y₄⁷ + y₃.y₄⁶ + y₃⁶.y₄ + y₂.y₃².y₄⁴ + y₂.y₃⁴.y₄² + y₂.y₃⁵.y₄ + y₂.y₃⁶ + y₂².y₃.y₄⁴ + y₂².y₃².y₄³ + y₂².y₃⁴.y₄ + y₂².y₃⁵ + y₂³.y₃.y₄³ + y₂³.y₃².y₄² + y₂³.y₃³.y₄ + y₂⁴.y₄³ + y₂⁴.y₃.y₄² + y₂⁴.y₃³ + y₂⁵.y₄² + y₂⁵.y₃² + y₁.y₂.y₄⁵ + y₁.y₂².y₃².y₄² + y₁.y₂³.y₃.y₄² + y₁.y₂³.y₃².y₄ + y₁.y₂⁴.y₄² + y₁.y₂⁴.y₃.y₄ + y₁².y₄⁵ + y₁².y₂.y₃².y₄² + y₁².y₂².y₄³ + y₁².y₂².y₃.y₄² + y₁².y₂².y₃².y₄ + y₁².y₂³.y₄² + y₁².y₂³.y₃.y₄ + y₁².y₂³.y₃² + y₁².y₂⁴.y₄ + y₁².y₂⁴.y₃ + y₁².y₂⁵ + y₁⁴.y₄³ + y₁⁴.y₂.y₃² + y₁⁴.y₂².y₄ + y₁⁴.y₂².y₃ + y₁⁴.y₂³
r restricts to y₄⁸ + y₃.y₄⁷ + y₃⁶.y₄² + y₂.y₃⁵.y₄² + y₂².y₃.y₄⁵ + y₂².y₃⁴.y₄² + y₂².y₃⁶ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁵ + y₂⁴.y₄⁴ + y₂⁴.y₃⁴ + y₂⁵.y₃.y₄² + y₂⁵.y₃².y₄ + y₂⁵.y₃³ + y₂⁶.y₄² + y₂⁶.y₃.y₄ + y₂⁶.y₃² + y₂⁷.y₃ + y₁.y₂.y₄⁶ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃².y₄³ + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₃.y₄³ + y₁.y₂⁴.y₄³ + y₁.y₂⁵.y₄² + y₁.y₂⁵.y₃.y₄ + y₁².y₄⁶ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃².y₄³ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₃.y₄³ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₄³ + y₁².y₂⁵.y₄ + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁³.y₂⁵ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂³.y₄ + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁵.y₂³ + y₁⁶.y₂² + y₁⁸

Poincaré series

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

Back to the groups of order 128