Small group number 866 of order 128

G is the group 128gp866

G has 3 minimal generators, rank 4 and exponent 8. 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.x₂ = y₁.y₂²
x₂² = y₂².x₂ + y₁.w₁ + y₁.y₂.x₁
x₁.x₂ = y₂².x₁ + y₁.w₂ + y₁.y₂.x₁
x₁² = y₂².x₁ + y₁.w₂ + y₁.w₁ + y₁.y₂.x₁
y₃.w₃ = y₃.w₁ + y₁.w₂
y₁.w₃ = 0
x₂.w₃ = y₁.y₂².x₁
x₂.w₂ = x₁.w₁ + y₁.y₂².x₁
x₂.w₁ = y₂².w₃ + y₂².w₁ + y₁.y₂.w₂
x₁.w₃ = y₁.y₂².x₁
x₁.w₂ = x₁.w₁
y₁.v = y₁.y₂.w₂ + y₁.y₂.w₁ + y₁.y₂².x₁
w₃² = y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.y₃².w₂ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂⁴.x₂ + y₂⁵.y₃ + y₂⁶ + y₁.y₂².w₂ + y₁.y₂².w₁
w₂.w₃ = w₁.w₂ + x₁.v
w₂² = x₂.v + x₁.v + y₃.u₁ + y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.x₁.w₁ + y₂.y₃.v + y₂.y₃².w₂ + y₂².v + y₂².y₃.w₂ + y₂².y₃.w₁ + y₂³.w₂ + y₂³.y₃³ + y₂⁴.x₂ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶ + y₁.y₂².w₁ + y₁.y₂³.x₁
w₁.w₃ = y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.y₃².w₂ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂⁴.x₂ + y₂⁵.y₃ + y₂⁶ + y₁.y₂².w₁
w₁² = x₂.v + y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.x₁.w₁ + y₂.y₃².w₂ + y₂.y₃².w₁ + y₂².y₃.w₁ + y₂³.w₃ + y₂³.w₁ + y₂⁴.x₂ + y₂⁵.y₃ + y₂⁶ + y₁.y₂².w₁ + y₁.y₂³.x₁
y₃.u₂ = y₃.u₁ + y₃³.w₂ + y₃³.w₁ + y₂.y₃².w₂ + y₂².y₃.w₂ + y₂².y₃.w₁ + y₂⁴.y₃² + y₂⁵.y₃
y₁.u₂ = y₁.y₂².w₂
y₁.u₁ = y₁.y₂².w₂ + y₁.y₂³.x₁
w₃.v = x₁.u₁ + y₃.t + y₃.w₁.w₂ + y₃².u₁ + y₃⁴.w₂ + y₂.w₁.w₂ + y₂.x₁.v + y₂².u₁ + y₂².y₃.v + y₂².y₃².w₂ + y₂².y₃⁵ + y₂³.y₃⁴ + y₂⁴.w₃ + y₂⁵.x₂ + y₂⁵.y₃² + y₂⁶.y₃ + y₂⁷ + y₁.y₂³.w₁
x₂.u₂ = x₁.u₁ + y₂².u₂ + y₂².u₁ + y₂².x₁.w₁ + y₂².y₃².w₂ + y₂².y₃².w₁ + y₂³.y₃.w₂ + y₂⁴.w₃ + y₂⁴.w₂ + y₂⁵.x₂ + y₂⁶.y₃ + y₂⁷ + y₁.y₂³.w₁ + y₁.y₂⁴.x₁
x₂.u₁ = x₁.u₁ + y₁.y₂³.w₂ + y₁.y₂⁴.x₁
x₁.u₂ = y₂².x₁.w₁
y₁.t = y₁.y₂³.w₂ + y₁.y₂³.w₁ + y₁.y₂⁴.x₁
w₃.u₂ = y₃.s + y₃.w₂.v + y₃².w₁.w₂ + y₃³.u₁ + y₃⁴.v + y₃⁵.w₂ + y₂.y₃.t + y₂.y₃⁴.w₂ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².x₂.v + y₂².x₁.v + y₂².y₃.u₁ + y₂².y₃².v + y₂².y₃³.w₂ + y₂².y₃³.w₁ + y₂³.x₁.w₁ + y₂³.y₃.v + y₂³.y₃².w₂ + y₂⁴.v + y₂⁵.w₂ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁷.y₃ + y₁.y₂⁴.w₂
w₃.u₁ = y₃.s + y₃.w₂.v + y₃³.u₁ + y₃⁵.w₁ + y₂.y₃.t + y₂.y₃.w₁.w₂ + y₂².x₂.v + y₂².y₃.u₁ + y₂².y₃³.w₁ + y₂³.x₁.w₁ + y₂³.y₃.v + y₂³.y₃².w₁ + y₂⁴.v + y₂⁵.w₃ + y₂⁵.w₂ + y₂⁶.x₂ + y₂⁸ + y₁.y₂⁴.w₁ + y₁.y₂⁵.x₁
w₂.u₂ = w₂.u₁ + x₁.t + y₃².w₁.w₂ + y₃³.u₁ + y₃⁴.v + y₃⁵.w₂ + y₃⁵.w₁ + y₂.x₁.u₁ + y₂.y₃².u₁ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².x₂.v + y₂².y₃.u₁ + y₂².y₃².v + y₂².y₃³.w₂ + y₂³.y₃².w₂ + y₂³.y₃².w₁ + y₂³.y₃⁵ + y₂⁴.v + y₂⁴.y₃.w₂ + y₂⁴.y₃.w₁ + y₂⁵.y₃³ + y₂⁶.x₂ + y₂⁶.x₁ + y₂⁶.y₃² + y₂⁸
w₁.u₂ = x₂.t + x₁.t + y₃.s + y₃.w₂.v + y₃².w₁.w₂ + y₃³.u₁ + y₃⁴.v + y₃⁵.w₂ + y₂.y₃.t + y₂.y₃⁴.w₂ + y₂.y₃⁴.w₁ + y₂².w₁.w₂ + y₂².x₁.v + y₂².y₃.u₁ + y₂².y₃².v + y₂².y₃³.w₂ + y₂².y₃³.w₁ + y₂³.x₁.w₁ + y₂³.y₃.v + y₂³.y₃².w₂ + y₂⁴.v + y₂⁵.w₂ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁷.y₃ + y₁.y₂⁴.w₂
w₁.u₁ = x₁.t + y₃.s + y₃.w₂.v + y₃³.u₁ + y₃⁵.w₁ + y₂.x₁.u₁ + y₂.y₃.t + y₂.y₃.w₁.w₂ + y₂².x₂.v + y₂².y₃.u₁ + y₂².y₃³.w₁ + y₂³.x₁.w₁ + y₂³.y₃.v + y₂³.y₃².w₁ + y₂⁴.v + y₂⁵.w₃ + y₂⁵.w₂ + y₂⁶.x₂ + y₂⁶.x₁ + y₂⁸ + y₁.y₂⁴.w₂ + y₁.y₂⁵.x₁
y₁.s = y₁.y₂⁴.w₂ + y₁.y₂⁵.x₁
v.u₂ = v.u₁ + w₁.t + y₃.v² + y₃².s + y₃⁵.v + y₃⁶.w₂ + y₂.x₂.t + y₂.y₃.w₂.v + y₂.y₃³.u₁ + y₂.y₃⁴.v + y₂².s + y₂².w₁.v + y₂².y₃².u₁ + y₂².y₃⁴.w₁ + y₂³.t + y₂³.y₃.u₁ + y₂³.y₃².v + y₂³.y₃³.w₂ + y₂⁴.u₁ + y₂⁴.x₁.w₁ + y₂⁴.y₃².w₁ + y₂⁵.v + y₂⁵.y₃.w₁ + y₂⁶.w₃ + y₂⁶.w₁ + y₂⁶.y₃³ + y₂⁷.x₂ + y₂⁸.y₃ + y₂⁹ + y₁.y₂⁵.w₂ + y₁.y₂⁵.w₁ + y₁.y₂⁶.x₁
w₃.t = x₁.w₁.v + y₃.v² + y₃².s + y₃².w₂.v + y₃³.t + y₃³.w₁.w₂ + y₃⁴.u₁ + y₃⁵.v + y₂.y₃².w₁.w₂ + y₂.y₃³.u₁ + y₂.y₃⁴.v + y₂².s + y₂².w₂.v + y₂².x₁.u₁ + y₂².y₃³.v + y₂².y₃⁴.w₂ + y₂².y₃⁴.w₁ + y₂².y₃⁷ + y₂³.t + y₂³.x₁.v + y₂³.y₃.u₁ + y₂³.y₃².v + y₂³.y₃³.w₂ + y₂³.y₃⁶ + y₂⁴.u₁ + y₂⁴.y₃.v + y₂⁵.y₃.w₁ + y₂⁵.y₃⁴ + y₂⁶.w₃ + y₂⁶.w₁ + y₂⁷.x₂ + y₂⁷.x₁ + y₂⁷.y₃² + y₂⁸.y₃ + y₂⁹ + y₁.y₂⁵.w₂ + y₁.y₂⁶.x₁
x₂.s = y₂.x₂.t + y₂³.x₁.v + y₂⁶.w₃ + y₂⁶.w₁ + y₂⁷.x₁ + y₁.y₂⁵.w₁ + y₁.y₂⁶.x₁
x₁.s = y₂.x₁.t + y₂³.x₁.v + y₂⁴.x₁.w₁ + y₂⁷.x₁ + y₁.y₂⁵.w₂ + y₁.y₂⁵.w₁ + y₁.y₂⁶.x₁
u₂² = x₁.v² + y₃.v.u₁ + y₃².v² + y₃⁴.t + y₃⁶.v + y₂.y₃.v² + y₂.y₃.w₂.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂².v² + y₂².y₃⁸ + y₂³.y₃².u₁ + y₂³.y₃⁴.w₂ + y₂⁴.x₂.v + y₂⁴.y₃.u₁ + y₂⁴.y₃³.w₁ + y₂⁵.x₁.w₁ + y₂⁵.y₃².w₁ + y₂⁵.y₃⁵ + y₂⁶.y₃.w₁ + y₂⁷.w₃ + y₂⁷.w₁ + y₂⁷.y₃³ + y₂⁸.y₃² + y₁.y₂⁶.w₂ + y₃².r
u₁.u₂ = x₂.v² + y₃.v.u₁ + y₃².v² + y₃².w₂.u₁ + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁶.v + y₃⁷.w₁ + y₂.y₃.v² + y₂.y₃³.t + y₂.y₃³.w₁.w₂ + y₂.y₃⁶.w₂ + y₂.y₃⁶.w₁ + y₂².v² + y₂².w₂.u₁ + y₂².y₃.s + y₂².y₃.w₂.v + y₂².y₃³.u₁ + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.x₁.u₁ + y₂³.y₃.t + y₂³.y₃.w₁.w₂ + y₂³.y₃².u₁ + y₂³.y₃⁴.w₂ + y₂³.y₃⁴.w₁ + y₂³.y₃⁷ + y₂⁴.y₃³.w₂ + y₂⁴.y₃³.w₁ + y₂⁴.y₃⁶ + y₂⁵.u₁ + y₂⁶.y₃.w₂ + y₂⁶.y₃.w₁ + y₂⁶.y₃⁴ + y₂⁷.w₃ + y₂⁷.y₃³ + y₂⁸.y₃² + y₁.y₂⁶.w₂ + y₃².r
u₁² = x₂.v² + x₁.v² + y₃.v.u₁ + y₃².v² + y₃⁴.t + y₃⁵.u₁ + y₃⁶.v + y₂.y₃.v² + y₂.y₃.w₂.u₁ + y₂.y₃⁶.w₂ + y₂.y₃⁶.w₁ + y₂².v² + y₂².y₃³.u₁ + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.y₃².u₁ + y₂³.y₃³.v + y₂³.y₃⁴.w₂ + y₂³.y₃⁷ + y₂⁴.x₂.v + y₂⁴.x₁.v + y₂⁴.y₃².v + y₂⁴.y₃³.w₂ + y₂⁴.y₃⁶ + y₂⁵.x₁.w₁ + y₂⁵.y₃.v + y₂⁵.y₃².w₂ + y₂⁶.v + y₂⁶.y₃.w₂ + y₂⁶.y₃.w₁ + y₂⁶.y₃⁴ + y₂⁷.w₂ + y₂⁷.y₃³ + y₂⁸.x₂ + y₂¹⁰ + y₁.y₂⁶.w₁ + y₃².r
w₃.s = y₃.v.u₁ + y₃.w₂.t + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁵.u₁ + y₃⁶.v + y₂.x₁.w₁.v + y₂.y₃.v² + y₂.y₃.w₂.u₁ + y₂.y₃².s + y₂.y₃⁶.w₂ + y₂².w₂.u₁ + y₂².x₂.t + y₂².x₁.t + y₂².y₃.w₂.v + y₂².y₃³.u₁ + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.s + y₂³.w₂.v + y₂³.x₁.u₁ + y₂³.y₃.w₁.w₂ + y₂³.y₃³.v + y₂³.y₃⁴.w₁ + y₂³.y₃⁷ + y₂⁴.w₁.w₂ + y₂⁴.x₂.v + y₂⁴.y₃³.w₂ + y₂⁴.y₃³.w₁ + y₂⁴.y₃⁶ + y₂⁵.y₃².w₂ + y₂⁵.y₃⁵ + y₂⁶.v + y₂⁷.w₃ + y₂⁷.w₁ + y₂⁸.x₂ + y₂¹⁰ + y₁.y₂⁶.w₁ + y₁.y₂⁷.x₁
w₁.s = y₃.v.u₁ + y₃.w₂.t + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁵.u₁ + y₃⁶.v + y₂.w₁.t + y₂.x₁.w₁.v + y₂.y₃.w₂.u₁ + y₂.y₃².w₂.v + y₂.y₃³.t + y₂.y₃³.w₁.w₂ + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂.y₃⁶.w₂ + y₂².w₂.u₁ + y₂².x₂.t + y₂².x₁.t + y₂².y₃.w₂.v + y₂².y₃².w₁.w₂ + y₂².y₃⁴.v + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.y₃.w₁.w₂ + y₂³.y₃⁴.w₂ + y₂⁴.t + y₂⁴.w₁.w₂ + y₂⁴.x₁.v + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁴.y₃³.w₁ + y₂⁵.u₁ + y₂⁵.y₃.v + y₂⁵.y₃².w₂ + y₂⁵.y₃⁵ + y₂⁶.v + y₂⁶.y₃.w₁ + y₂⁶.y₃⁴ + y₂⁷.w₃ + y₂⁷.w₁ + y₂⁸.x₁ + y₂⁸.y₃² + y₂⁹.y₃ + y₁.y₂⁶.w₁ + y₁.y₂⁷.x₁
u₂.t = v.s + w₂.v² + w₁.v² + x₁.w₁.t + y₃.v.t + y₃.w₂.s + y₃³.v² + y₃³.w₂.u₁ + y₃⁵.t + y₃⁶.u₁ + y₃⁸.w₂ + y₃⁸.w₁ + y₂.v.t + y₂.w₂.s + y₂.x₂.v² + y₂.x₁.v² + y₂.y₃.v.u₁ + y₂.y₃².v² + y₂.y₃⁴.t + y₂.y₃⁴.w₁.w₂ + y₂.y₃⁵.u₁ + y₂.y₃⁷.w₂ + y₂².w₁.t + y₂².x₁.w₁.v + y₂².y₃.v² + y₂².y₃.w₂.u₁ + y₂².y₃³.t + y₂².y₃³.w₁.w₂ + y₂².y₃⁴.u₁ + y₂².y₃⁶.w₂ + y₂².y₃⁶.w₁ + y₂².y₃⁹ + y₂³.v² + y₂³.x₂.t + y₂³.x₁.t + y₂³.y₃.w₂.v + y₂³.y₃².t + y₂³.y₃⁵.w₂ + y₂³.y₃⁵.w₁ + y₂⁴.s + y₂⁴.w₁.v + y₂⁴.y₃.w₁.w₂ + y₂⁴.y₃².u₁ + y₂⁴.y₃³.v + y₂⁴.y₃⁴.w₁ + y₂⁴.y₃⁷ + y₂⁵.t + y₂⁵.w₁.w₂ + y₂⁵.x₂.v + y₂⁵.y₃.u₁ + y₂⁵.y₃³.w₂ + y₂⁵.y₃³.w₁ + y₂⁵.y₃⁶ + y₂⁶.y₃.v + y₂⁷.v + y₂⁷.y₃.w₂ + y₂⁷.y₃.w₁ + y₂⁸.y₃³ + y₂⁹.x₁ + y₂⁹.y₃² + y₁.y₂⁷.w₁ + y₃³.r + y₂².y₃.r + y₁.x₁.r
u₁.t = v.s + w₂.v² + x₁.w₁.t + y₃.w₂.s + y₃².v.u₁ + y₃⁴.w₂.v + y₃⁵.t + y₃⁵.w₁.w₂ + y₃⁶.u₁ + y₃⁷.v + y₂.v.t + y₂.w₂.s + y₂.x₂.v² + y₂.y₃.v.u₁ + y₂.y₃².v² + y₂.y₃².w₂.u₁ + y₂.y₃⁴.t + y₂.y₃⁶.v + y₂.y₃⁷.w₂ + y₂².v.u₁ + y₂².w₂.t + y₂².y₃³.t + y₂².y₃⁴.u₁ + y₂².y₃⁶.w₂ + y₂².y₃⁶.w₁ + y₂².y₃⁹ + y₂³.v² + y₂³.w₂.u₁ + y₂³.x₂.t + y₂³.y₃⁴.v + y₂³.y₃⁵.w₁ + y₂⁴.s + y₂⁴.w₁.v + y₂⁴.x₁.u₁ + y₂⁴.y₃.t + y₂⁴.y₃².u₁ + y₂⁴.y₃⁴.w₂ + y₂⁵.w₁.w₂ + y₂⁵.x₁.v + y₂⁵.y₃³.w₂ + y₂⁵.y₃³.w₁ + y₂⁶.x₁.w₁ + y₂⁶.y₃.v + y₂⁶.y₃².w₂ + y₂⁶.y₃².w₁ + y₂⁸.w₂ + y₂⁸.w₁ + y₂⁸.y₃³ + y₂⁹.x₁ + y₂⁹.y₃² + y₂¹⁰.y₃ + y₁.y₂⁷.w₂ + y₁.y₂⁷.w₁ + y₃³.r + y₂².y₃.r + y₁.x₁.r
t² = v³ + x₂.v.t + y₃.w₂.v² + y₃².v.t + y₃³.v.u₁ + y₃³.w₂.t + y₃⁵.w₂.v + y₃⁷.u₁ + y₃⁹.w₁ + y₂.w₂.v² + y₂.w₁.v² + y₂.x₁.w₁.t + y₂.y₃².w₂.t + y₂.y₃³.w₂.u₁ + y₂.y₃⁴.s + y₂.y₃⁵.t + y₂.y₃⁶.u₁ + y₂.y₃⁷.v + y₂.y₃⁸.w₂ + y₂².v.t + y₂².y₃.v.u₁ + y₂².y₃.w₂.t + y₂².y₃².w₂.u₁ + y₂².y₃⁶.v + y₂².y₃⁷.w₂ + y₂³.w₂.t + y₂³.x₁.w₁.v + y₂³.y₃.v² + y₂³.y₃.w₂.u₁ + y₂³.y₃².s + y₂³.y₃².w₂.v + y₂³.y₃³.w₁.w₂ + y₂⁴.w₂.u₁ + y₂⁴.x₂.t + y₂⁴.x₁.t + y₂⁴.y₃.w₂.v + y₂⁴.y₃².t + y₂⁴.y₃⁵.w₁ + y₂⁵.w₂.v + y₂⁵.y₃.t + y₂⁵.y₃.w₁.w₂ + y₂⁵.y₃⁴.w₁ + y₂⁵.y₃⁷ + y₂⁶.t + y₂⁶.y₃.u₁ + y₂⁶.y₃².v + y₂⁶.y₃³.w₂ + y₂⁶.y₃³.w₁ + y₂⁷.u₁ + y₂⁷.y₃.v + y₂⁷.y₃².w₂ + y₂⁷.y₃⁵ + y₂⁸.v + y₂⁸.y₃.w₂ + y₂⁸.y₃.w₁ + y₂⁸.y₃⁴ + y₂⁹.w₃ + y₂⁹.w₂ + y₂⁹.w₁ + y₂⁹.y₃³ + y₂¹⁰.x₂ + y₂¹² + y₁.y₂⁸.w₁ + y₃⁴.r + y₂².x₂.r + y₂⁴.r + y₁.w₁.r + y₁.y₂.x₁.r
u₂.s = w₂.v.u₁ + x₂.v.t + x₁.v.t + y₃.v.s + y₃.w₂.v² + y₃².v.t + y₃².w₂.s + y₃⁴.w₂.u₁ + y₃⁵.s + y₃⁶.w₁.w₂ + y₃⁸.v + y₃⁹.w₂ + y₃⁹.w₁ + y₂.v.s + y₂.w₂.v² + y₂.w₁.v² + y₂.x₁.w₁.t + y₂.y₃.v.t + y₂.y₃².v.u₁ + y₂.y₃⁵.w₁.w₂ + y₂.y₃⁶.u₁ + y₂.y₃⁷.v + y₂.y₃⁸.w₁ + y₂².x₁.v² + y₂².y₃³.s + y₂².y₃³.w₂.v + y₂².y₃⁵.u₁ + y₂².y₃⁷.w₂ + y₂².y₃⁷.w₁ + y₂³.w₁.t + y₂³.y₃.w₂.u₁ + y₂³.y₃².s + y₂³.y₃³.w₁.w₂ + y₂⁴.w₂.u₁ + y₂⁴.y₃².w₁.w₂ + y₂⁴.y₃³.u₁ + y₂⁴.y₃⁴.v + y₂⁴.y₃⁵.w₂ + y₂⁴.y₃⁸ + y₂⁵.s + y₂⁵.w₂.v + y₂⁵.w₁.v + y₂⁵.x₁.u₁ + y₂⁵.y₃.t + y₂⁵.y₃².u₁ + y₂⁵.y₃³.v + y₂⁵.y₃⁴.w₂ + y₂⁶.t + y₂⁶.x₁.v + y₂⁶.y₃.u₁ + y₂⁷.u₁ + y₂⁷.y₃².w₁ + y₂⁸.v + y₂⁸.y₃.w₂ + y₂⁸.y₃.w₁ + y₂⁸.y₃⁴ + y₂⁹.w₂ + y₂⁹.y₃³ + y₂¹⁰.x₁ + y₂¹¹.y₃ + y₁.y₂⁸.w₁ + y₁.y₂⁹.x₁ + y₃.w₁.r + y₃⁴.r + y₂.y₃³.r + y₂².y₃².r + y₂³.y₃.r + y₁.w₂.r + y₁.y₂.x₁.r
u₁.s = w₂.v.u₁ + x₂.v.t + x₁.v.t + y₃.v.s + y₃.w₂.v² + y₃².v.t + y₃³.v.u₁ + y₃³.w₂.t + y₃⁴.w₂.u₁ + y₃⁵.w₂.v + y₃⁶.t + y₃⁶.w₁.w₂ + y₃⁷.u₁ + y₃⁹.w₂ + y₃⁹.w₁ + y₂.v.s + y₂.w₂.v² + y₂.y₃.w₂.s + y₂.y₃².w₂.t + y₂.y₃³.v² + y₂.y₃⁵.t + y₂.y₃⁵.w₁.w₂ + y₂.y₃⁷.v + y₂².w₂.s + y₂².y₃.v.u₁ + y₂².y₃³.w₂.v + y₂².y₃⁴.t + y₂².y₃⁵.u₁ + y₂².y₃⁶.v + y₂².y₃¹⁰ + y₂³.v.u₁ + y₂³.x₁.w₁.v + y₂³.y₃.v² + y₂³.y₃.w₂.u₁ + y₂³.y₃².s + y₂³.y₃².w₂.v + y₂³.y₃³.t + y₂³.y₃⁴.u₁ + y₂³.y₃⁵.v + y₂⁴.w₂.u₁ + y₂⁴.x₁.t + y₂⁴.y₃.s + y₂⁴.y₃².w₁.w₂ + y₂⁴.y₃⁵.w₂ + y₂⁵.w₂.v + y₂⁵.w₁.v + y₂⁵.x₁.u₁ + y₂⁵.y₃.t + y₂⁵.y₃³.v + y₂⁵.y₃⁴.w₁ + y₂⁶.w₁.w₂ + y₂⁶.x₂.v + y₂⁶.x₁.v + y₂⁶.y₃.u₁ + y₂⁷.y₃.v + y₂⁷.y₃⁵ + y₂⁸.v + y₂⁸.y₃.w₁ + y₂⁸.y₃⁴ + y₂⁹.w₃ + y₂¹⁰.x₁ + y₂¹⁰.y₃² + y₂¹¹.y₃ + y₁.y₂⁸.w₂ + y₁.y₂⁸.w₁ + y₃.w₁.r + y₃⁴.r + y₂.y₃³.r + y₂².y₃².r + y₂³.y₃.r + y₁.w₂.r + y₁.y₂.x₁.r
t.s = v².u₁ + w₂.v.t + y₃².v.s + y₃².w₂.v² + y₃³.v.t + y₃³.w₂.s + y₃⁴.w₂.t + y₃⁵.v² + y₃⁶.s + y₃⁶.w₂.v + y₃¹⁰.w₁ + y₂.v³ + y₂.x₂.v.t + y₂.y₃.v.s + y₂.y₃.w₂.v² + y₂.y₃².v.t + y₂.y₃².w₂.s + y₂.y₃³.v.u₁ + y₂.y₃⁴.w₂.u₁ + y₂.y₃⁵.w₂.v + y₂.y₃⁷.u₁ + y₂.y₃⁸.v + y₂.y₃⁹.w₂ + y₂².w₂.v² + y₂².w₁.v² + y₂².x₁.w₁.t + y₂².y₃.v.t + y₂².y₃².w₂.t + y₂².y₃⁵.w₁.w₂ + y₂².y₃⁸.w₁ + y₂³.v.t + y₂³.y₃.v.u₁ + y₂³.y₃².v² + y₂³.y₃².w₂.u₁ + y₂³.y₃³.w₂.v + y₂³.y₃⁴.t + y₂³.y₃⁵.u₁ + y₂³.y₃⁷.w₂ + y₂⁴.v.u₁ + y₂⁴.w₂.t + y₂⁴.w₁.t + y₂⁴.y₃.w₂.u₁ + y₂⁴.y₃³.t + y₂⁴.y₃⁵.v + y₂⁴.y₃⁶.w₂ + y₂⁵.w₂.u₁ + y₂⁵.x₂.t + y₂⁵.y₃.w₂.v + y₂⁵.y₃².w₁.w₂ + y₂⁵.y₃⁴.v + y₂⁵.y₃⁸ + y₂⁶.s + y₂⁶.w₂.v + y₂⁶.x₁.u₁ + y₂⁶.y₃.t + y₂⁶.y₃².u₁ + y₂⁶.y₃⁴.w₂ + y₂⁶.y₃⁷ + y₂⁷.x₂.v + y₂⁷.y₃.u₁ + y₂⁷.y₃³.w₂ + y₂⁷.y₃⁶ + y₂⁸.x₁.w₁ + y₂⁸.y₃.v + y₂⁸.y₃⁵ + y₂⁹.y₃.w₂ + y₂⁹.y₃.w₁ + y₂⁹.y₃⁴ + y₂¹⁰.y₃³ + y₂¹¹.x₁ + y₂¹¹.y₃² + y₁.y₂⁹.w₂ + y₁.y₂⁹.w₁ + y₃².w₁.r + y₂.y₃⁴.r + y₂².w₃.r + y₂².y₃³.r + y₂³.x₂.r + y₂⁴.y₃.r + y₂⁵.r + y₁.y₂.w₁.r + y₁.y₂².x₁.r
s² = y₃².w₂.v.u₁ + y₃³.v.s + y₃³.w₂.v² + y₃⁴.v.t + y₃⁵.v.u₁ + y₃⁵.w₂.t + y₃⁶.v² + y₃⁶.w₂.u₁ + y₃⁷.w₂.v + y₃⁸.w₁.w₂ + y₃⁹.u₁ + y₃¹⁰.v + y₃¹¹.w₂ + y₂.y₃².v.s + y₂.y₃³.v.t + y₂.y₃³.w₂.s + y₂.y₃⁵.w₂.u₁ + y₂.y₃⁶.w₂.v + y₂.y₃⁸.u₁ + y₂.y₃⁹.v + y₂.y₃¹⁰.w₂ + y₂.y₃¹⁰.w₁ + y₂².v³ + y₂².y₃.v.s + y₂².y₃.w₂.v² + y₂².y₃².w₂.s + y₂².y₃³.v.u₁ + y₂².y₃³.w₂.t + y₂².y₃⁴.v² + y₂².y₃⁴.w₂.u₁ + y₂².y₃⁵.s + y₂².y₃⁶.w₁.w₂ + y₂².y₃⁷.u₁ + y₂².y₃⁹.w₂ + y₂².y₃⁹.w₁ + y₂³.w₂.v² + y₂³.w₁.v² + y₂³.x₁.w₁.t + y₂³.y₃.v.t + y₂³.y₃.w₂.s + y₂³.y₃².v.u₁ + y₂³.y₃².w₂.t + y₂³.y₃³.v² + y₂³.y₃³.w₂.u₁ + y₂³.y₃⁴.s + y₂³.y₃⁴.w₂.v + y₂³.y₃⁵.t + y₂³.y₃⁶.u₁ + y₂³.y₃⁷.v + y₂³.y₃¹¹ + y₂⁴.x₁.v² + y₂⁴.y₃.v.u₁ + y₂⁴.y₃.w₂.t + y₂⁴.y₃².v² + y₂⁴.y₃².w₂.u₁ + y₂⁴.y₃⁴.t + y₂⁴.y₃⁴.w₁.w₂ + y₂⁴.y₃⁵.u₁ + y₂⁴.y₃⁷.w₂ + y₂⁴.y₃⁷.w₁ + y₂⁴.y₃¹⁰ + y₂⁵.v.u₁ + y₂⁵.y₃².w₂.v + y₂⁵.y₃³.t + y₂⁵.y₃³.w₁.w₂ + y₂⁵.y₃⁹ + y₂⁶.w₂.u₁ + y₂⁶.y₃.s + y₂⁶.y₃.w₂.v + y₂⁶.y₃².t + y₂⁶.y₃².w₁.w₂ + y₂⁶.y₃⁴.v + y₂⁶.y₃⁵.w₁ + y₂⁷.y₃.w₁.w₂ + y₂⁷.y₃⁷ + y₂⁸.y₃.u₁ + y₂⁸.y₃².v + y₂⁸.y₃⁶ + y₂⁹.x₁.w₁ + y₂⁹.y₃.v + y₂¹⁰.y₃.w₂ + y₂¹¹.w₃ + y₂¹¹.y₃³ + y₂¹².y₃² + y₂¹³.y₃ + y₃².v.r + y₃³.w₂.r + y₃³.w₁.r + y₃⁶.r + y₂.y₃².w₂.r + y₂.y₃².w₁.r + y₂.y₃⁵.r + y₂².y₃.w₁.r + y₂⁴.y₃².r + y₂⁵.y₃.r + y₁.y₂².w₂.r + y₁.y₂³.x₁.r

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

y₁.x₁.w₁ = y₁.y₂².w₂
y₁.w₁.w₂ = y₁.y₂⁴.x₁
y₃.w₁.v = y₃².t + y₃².w₁.w₂ + y₃³.u₁ + y₃⁵.w₂ + y₂.y₃.w₁.w₂ + y₂².y₃.u₁ + y₂².y₃².v + y₂².y₃³.w₂ + y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁴.y₃.w₁ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁷.y₃ + y₁.y₂⁴.w₂ + y₁.y₂⁵.x₁
w₁.w₂.v = x₁.v² + y₃.w₂.t + y₃².w₂.u₁ + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁵.u₁ + y₃⁷.w₂ + y₃⁷.w₁ + y₂.y₃².s + y₂.y₃².w₂.v + y₂.y₃³.t + y₂.y₃⁴.u₁ + y₂.y₃⁶.w₂ + y₂.y₃⁶.w₁ + y₂².w₂.u₁ + y₂².x₁.t + y₂².y₃.w₂.v + y₂².y₃².t + y₂².y₃².w₁.w₂ + y₂².y₃³.u₁ + y₂².y₃⁸ + y₂³.y₃.t + y₂³.y₃.w₁.w₂ + y₂³.y₃³.v + y₂⁴.w₁.w₂ + y₂⁴.x₁.v + y₂⁴.y₃².v + y₂⁴.y₃³.w₁ + y₂⁴.y₃⁶ + y₂⁵.u₁ + y₂⁵.x₁.w₁ + y₂⁵.y₃⁵ + y₂⁶.y₃.w₁ + y₂⁷.w₂ + y₂⁸.x₂ + y₂⁸.x₁ + y₂⁸.y₃² + y₂⁹.y₃ + y₂¹⁰ + y₁.y₂⁶.w₂
y₃.w₁.t = y₃².v² + y₃³.s + y₃³.w₂.v + y₃⁴.t + y₃⁴.w₁.w₂ + y₃⁵.u₁ + y₃⁶.v + y₂.y₃³.w₁.w₂ + y₂.y₃⁴.u₁ + y₂.y₃⁵.v + y₂².y₃.s + y₂².y₃.w₂.v + y₂².y₃⁴.v + y₂².y₃⁵.w₂ + y₂².y₃⁵.w₁ + y₂².y₃⁸ + y₂³.y₃.t + y₂³.y₃².u₁ + y₂³.y₃³.v + y₂³.y₃⁴.w₂ + y₂³.y₃⁷ + y₂⁴.y₃.u₁ + y₂⁴.y₃².v + y₂⁵.y₃².w₁ + y₂⁵.y₃⁵ + y₂⁷.y₃³ + y₂⁸.y₃² + y₂⁹.y₃
x₁.v.u₁ = x₁.w₁.t + y₂³.x₁.t + y₂⁴.x₁.u₁ + y₂⁶.x₁.w₁ + y₂⁹.x₁ + y₁.y₂⁷.w₂ + y₁.y₂⁷.w₁
w₁.w₂.t = x₁.v.t + y₃.w₂.v² + y₃².w₂.s + y₃³.v.u₁ + y₃³.w₂.t + y₃⁴.v² + y₃⁴.w₂.u₁ + y₃⁵.s + y₃⁵.w₂.v + y₃⁶.w₁.w₂ + y₃⁷.u₁ + y₃⁸.v + y₃⁹.w₂ + y₂.y₃³.v² + y₂.y₃³.w₂.u₁ + y₂.y₃⁴.s + y₂.y₃⁴.w₂.v + y₂.y₃⁵.t + y₂.y₃⁵.w₁.w₂ + y₂.y₃⁶.u₁ + y₂.y₃⁷.v + y₂.y₃⁸.w₁ + y₂².w₂.s + y₂².x₂.v² + y₂².y₃.v.u₁ + y₂².y₃³.w₂.v + y₂².y₃⁴.t + y₂².y₃⁷.w₁ + y₂³.w₂.t + y₂³.y₃.v² + y₂³.y₃.w₂.u₁ + y₂³.y₃².w₂.v + y₂³.y₃³.t + y₂³.y₃³.w₁.w₂ + y₂³.y₃⁴.u₁ + y₂³.y₃⁵.v + y₂³.y₃⁶.w₁ + y₂⁴.v² + y₂⁴.w₂.u₁ + y₂⁴.x₁.t + y₂⁴.y₃².t + y₂⁴.y₃².w₁.w₂ + y₂⁴.y₃³.u₁ + y₂⁴.y₃⁴.v + y₂⁵.w₂.v + y₂⁵.x₁.u₁ + y₂⁵.y₃².u₁ + y₂⁵.y₃³.v + y₂⁵.y₃⁴.w₂ + y₂⁵.y₃⁴.w₁ + y₂⁵.y₃⁷ + y₂⁶.x₂.v + y₂⁶.x₁.v + y₂⁶.y₃.u₁ + y₂⁶.y₃².v + y₂⁶.y₃³.w₂ + y₂⁶.y₃³.w₁ + y₂⁷.y₃.v + y₂⁷.y₃².w₂ + y₂⁷.y₃⁵ + y₂⁸.v + y₂⁸.y₃.w₂ + y₂⁸.y₃.w₁ + y₂⁹.w₂ + y₂⁹.y₃³ + y₂¹⁰.x₁ + y₂¹¹.y₃ + y₁.y₂⁸.w₂ + y₁.y₂⁸.w₁ + y₁.y₂⁹.x₁

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 17 onwards.

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

h₁ = r in degree 8
h₂ = v + y₃.w₂ + y₃.w₁ + y₃⁴ + y₂.w₃ + y₂.w₂ + y₂.w₁ + y₂².x₂ + y₂².x₁ + y₂².y₃² + y₂³.y₃ in degree 4
h₃ = x₂.v + y₃².v + y₃³.w₂ + y₃³.w₁ + y₂.u₂ + y₂.u₁ + y₂.y₃.v + y₂.y₃².w₂ + y₂².v + y₂².y₃.w₂ + y₂².y₃.w₁ + y₂².y₃⁴ + y₂³.w₃ + y₂³.y₃³ + y₂⁴.x₁ + y₂⁴.y₃² + y₂⁵.y₃ in degree 6
h₄ = y₃ 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, 7, 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
y₂ in degree 1
y₁ in degree 1
x₂ in degree 2
x₁ 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.x₂ in degree 3
y₂.x₁ in degree 3
y₂³ in degree 3
y₁.y₂² in degree 3
y₂.w₃ in degree 4
y₂.w₂ in degree 4
y₂.w₁ in degree 4
y₂².x₂ in degree 4
y₂².x₁ in degree 4
y₂⁴ in degree 4
y₁.w₁ in degree 4
y₁.y₂³ in degree 4
u₂ in degree 5
u₁ in degree 5
x₁.w₁ in degree 5
y₂².w₃ in degree 5
y₂².w₂ in degree 5
y₂².w₁ in degree 5
y₂³.x₂ in degree 5
y₂³.x₁ in degree 5
y₂⁵ in degree 5
y₁.y₂.w₁ in degree 5
t in degree 6
w₁.w₂ in degree 6
y₂.u₁ in degree 6
y₂.x₁.w₁ in degree 6
y₂³.w₃ in degree 6
y₂³.w₂ in degree 6
y₂³.w₁ in degree 6
y₂⁴.x₂ in degree 6
y₂⁴.x₁ in degree 6
y₂⁶ in degree 6
y₁.y₂².w₁ in degree 6
s in degree 7
y₂.t in degree 7
y₂.w₁.w₂ in degree 7
y₂².u₁ in degree 7
y₂².x₁.w₁ in degree 7
y₂⁴.w₃ in degree 7
y₂⁴.w₂ in degree 7
y₂⁴.w₁ in degree 7
y₂⁵.x₂ in degree 7
y₂⁵.x₁ in degree 7
y₂⁷ in degree 7
y₁.y₂³.w₁ in degree 7
w₂.u₁ in degree 8
x₂.t in degree 8
x₁.t in degree 8
y₂.s in degree 8
y₂².t in degree 8
y₂².w₁.w₂ in degree 8
y₂³.x₁.w₁ in degree 8
y₂⁵.w₃ in degree 8
y₂⁵.w₂ in degree 8
y₂⁵.w₁ in degree 8
y₂⁶.x₁ in degree 8
y₂⁸ in degree 8
w₂.t in degree 9
y₂.w₂.u₁ in degree 9
y₂².s in degree 9
y₂³.t in degree 9
y₂³.w₁.w₂ in degree 9
y₂⁴.x₁.w₁ in degree 9
y₂⁶.w₂ in degree 9
y₂⁶.w₁ in degree 9
y₂⁷.x₁ in degree 9
y₂⁹ in degree 9
w₂.s in degree 10
y₂.w₂.t in degree 10
y₂³.s in degree 10
y₂⁴.t in degree 10
y₂⁴.w₁.w₂ in degree 10
y₂⁵.x₁.w₁ in degree 10
y₂⁷.w₂ in degree 10
y₂⁷.w₁ in degree 10
y₂⁸.x₁ in degree 10
y₂¹⁰ in degree 10
y₂.w₂.s in degree 11
y₂².w₂.t in degree 11
y₂⁴.s in degree 11
y₂⁵.t in degree 11
y₂⁵.w₁.w₂ in degree 11
y₂⁸.w₂ in degree 11
y₂⁸.w₁ in degree 11
y₂².w₂.s in degree 12
y₂³.w₂.t in degree 12
y₂⁵.s in degree 12
y₂⁹.w₂ in degree 12
y₂⁹.w₁ in degree 12
y₂³.w₂.s in degree 13
y₂⁴.w₂.t in degree 13
y₂¹⁰.w₂ in degree 13
y₂¹⁰.w₁ in degree 13
y₂⁴.w₂.s in degree 14
y₂⁵.w₂.t in degree 14
y₂⁵.w₂.s in degree 15

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

y₁ in degree 1
x₂ + x₁ in degree 2
y₁.y₂ in degree 2
y₂.x₂ + y₂.x₁ in degree 3
y₁.y₂² in degree 3
x₁.h in degree 3
y₂².x₂ + y₂².x₁ in degree 4
y₁.w₁ in degree 4
y₁.y₂³ in degree 4
w₃.h + w₁.h in degree 4
y₂.x₁.h in degree 4
u₂ + u₁ + y₂².w₂ + y₂².w₁ + y₂⁵ + y₂.w₂.h + y₂⁴.h + w₂.h² + w₁.h² in degree 5
x₁.w₁ + y₂².w₃ + y₂².w₁ in degree 5
y₂³.x₂ + y₂³.x₁ in degree 5
y₁.y₂.w₁ in degree 5
y₂.w₃.h + y₂.w₁.h in degree 5
y₂².x₁.h in degree 5
y₂.x₁.w₁ + y₂³.w₃ + y₂³.w₁ in degree 6
y₂⁴.x₂ + y₂⁴.x₁ in degree 6
y₂⁴.x₁ in degree 6
y₂⁶ + y₂⁵.h + y₂⁴.h² + y₂³.h³ + y₂².h⁴ + y₂.h⁵ + h⁶ in degree 6
y₁.y₂².w₁ in degree 6
y₂².w₃.h + y₂².w₁.h in degree 6
y₂³.x₁.h in degree 6
y₂².u₁ + y₂⁴.w₁ + t.h + y₂³.w₂.h + y₂³.w₁.h + u₁.h² + y₂².w₁.h² + y₂⁵.h² + y₂.w₁.h³ + y₂⁴.h³ + w₂.h⁴ + w₁.h⁴ + y₂².h⁵ in degree 7
y₂².x₁.w₁ + y₂⁴.w₃ + y₂⁴.w₁ in degree 7
y₂⁴.w₃ + y₂⁴.w₁ in degree 7
y₂⁵.x₂ + y₂⁵.x₁ in degree 7
y₂⁵.x₁ in degree 7
y₂⁷ + y₂⁶.h + y₂⁵.h² + y₂⁴.h³ + y₂³.h⁴ + y₂².h⁵ + y₂.h⁶ in degree 7
y₁.y₂³.w₁ in degree 7
y₂³.w₃.h + y₂³.w₁.h in degree 7
x₂.t + x₁.t in degree 8
x₁.t in degree 8
y₂³.x₁.w₁ + y₂⁵.w₃ + y₂⁵.w₁ in degree 8
y₂⁵.w₃ + y₂⁵.w₁ in degree 8
y₂⁶.x₁ in degree 8
y₂⁸ + y₂⁷.h + y₂⁶.h² + y₂⁵.h³ + y₂⁴.h⁴ + y₂³.h⁵ + y₂².h⁶ in degree 8
y₂⁴.x₁.w₁ + y₂⁶.w₃ + y₂⁶.w₁ in degree 9
y₂⁶.w₂ + y₂⁵.w₂.h + y₂⁴.w₂.h² + y₂³.w₂.h³ + y₂².w₂.h⁴ + y₂.w₂.h⁵ + w₂.h⁶ in degree 9
y₂⁶.w₁ + y₂⁵.w₁.h + y₂⁴.w₁.h² + y₂³.w₁.h³ + y₂².w₁.h⁴ + y₂.w₁.h⁵ + w₁.h⁶ in degree 9
y₂⁷.x₁ in degree 9
y₂⁹ + y₂⁸.h + y₂⁷.h² + y₂⁶.h³ + y₂⁵.h⁴ + y₂⁴.h⁵ + y₂³.h⁶ in degree 9
y₂⁵.x₁.w₁ + y₂⁷.w₃ + y₂⁷.w₁ in degree 10
y₂⁷.w₂ + y₂⁶.w₂.h + y₂⁵.w₂.h² + y₂⁴.w₂.h³ + y₂³.w₂.h⁴ + y₂².w₂.h⁵ + y₂.w₂.h⁶ in degree 10
y₂⁷.w₁ + y₂⁶.w₁.h + y₂⁵.w₁.h² + y₂⁴.w₁.h³ + y₂³.w₁.h⁴ + y₂².w₁.h⁵ + y₂.w₁.h⁶ in degree 10
y₂⁸.x₁ in degree 10
y₂¹⁰ + y₂⁹.h + y₂⁸.h² + y₂⁷.h³ + y₂⁶.h⁴ + y₂⁵.h⁵ + y₂⁴.h⁶ in degree 10
y₂⁸.w₂ + y₂⁷.w₂.h + y₂⁶.w₂.h² + y₂⁵.w₂.h³ + y₂⁴.w₂.h⁴ + y₂³.w₂.h⁵ + y₂².w₂.h⁶ in degree 11
y₂⁸.w₁ + y₂⁷.w₁.h + y₂⁶.w₁.h² + y₂⁵.w₁.h³ + y₂⁴.w₁.h⁴ + y₂³.w₁.h⁵ + y₂².w₁.h⁶ in degree 11
y₂⁴.t.h + y₂³.t.h² + y₂⁵.w₂.h³ + y₂⁴.w₁.h⁴ + t.h⁵ + y₂.u₁.h⁵ + y₂³.w₂.h⁵ + y₂².w₂.h⁶ + y₂².w₁.h⁶ + y₂⁵.h⁶ + y₂.w₂.h⁷ + w₂.h⁸ + w₁.h⁸ + y₂³.h⁸ + y₂².h⁹ + y₂.h¹⁰ in degree 11
y₂⁹.w₂ + y₂⁸.w₂.h + y₂⁷.w₂.h² + y₂⁶.w₂.h³ + y₂⁵.w₂.h⁴ + y₂⁴.w₂.h⁵ + y₂³.w₂.h⁶ in degree 12
y₂⁹.w₁ + y₂⁸.w₁.h + y₂⁷.w₁.h² + y₂⁶.w₁.h³ + y₂⁵.w₁.h⁴ + y₂⁴.w₁.h⁵ + y₂³.w₁.h⁶ in degree 12
y₂⁵.t.h + y₂⁴.t.h² + y₂⁶.w₂.h³ + y₂⁵.w₁.h⁴ + y₂.t.h⁵ + y₂².u₁.h⁵ + y₂⁴.w₂.h⁵ + y₂³.w₂.h⁶ + y₂³.w₁.h⁶ + y₂⁶.h⁶ + y₂².w₂.h⁷ + y₂.w₂.h⁸ + y₂.w₁.h⁸ + y₂⁴.h⁸ + y₂³.h⁹ + y₂².h¹⁰ in degree 12
y₂⁴.w₂.t + y₂³.w₂.t.h + y₂⁴.w₁.w₂.h³ + w₂.t.h⁴ + y₂.w₂.u₁.h⁴ + y₂³.t.h⁴ + y₂².w₁.w₂.h⁵ + y₂⁵.w₁.h⁵ + y₂⁴.w₁.h⁶ + t.h⁷ + w₁.w₂.h⁷ + y₂.u₁.h⁷ + y₂³.w₂.h⁷ + y₂³.w₁.h⁷ + y₂².w₂.h⁸ + y₂⁵.h⁸ + y₂⁴.h⁹ + y₂³.h¹⁰ + y₂².h¹¹ + y₂.h¹² in degree 13
y₂¹⁰.w₂ + y₂⁹.w₂.h + y₂⁸.w₂.h² + y₂⁷.w₂.h³ + y₂⁶.w₂.h⁴ + y₂⁵.w₂.h⁵ + y₂⁴.w₂.h⁶ in degree 13
y₂¹⁰.w₁ + y₂⁹.w₁.h + y₂⁸.w₁.h² + y₂⁷.w₁.h³ + y₂⁶.w₁.h⁴ + y₂⁵.w₁.h⁵ + y₂⁴.w₁.h⁶ in degree 13
y₂⁵.w₂.t + y₂⁴.w₂.t.h + y₂⁵.w₁.w₂.h³ + y₂.w₂.t.h⁴ + y₂².w₂.u₁.h⁴ + y₂⁴.t.h⁴ + y₂³.w₁.w₂.h⁵ + y₂⁶.w₁.h⁵ + y₂⁵.w₁.h⁶ + y₂.t.h⁷ + y₂.w₁.w₂.h⁷ + y₂².u₁.h⁷ + y₂⁴.w₂.h⁷ + y₂⁴.w₁.h⁷ + y₂³.w₂.h⁸ + y₂⁶.h⁸ + y₂⁵.h⁹ + y₂⁴.h¹⁰ + y₂³.h¹¹ + y₂².h¹² in degree 14

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

y₁ in degree 1
y₁.y₂ in degree 2
y₁.x₁ in degree 3
y₁.y₂² in degree 3
y₁.w₂ in degree 4
y₁.w₁ in degree 4
y₁.y₂.x₁ in degree 4
y₁.y₂³ in degree 4
y₁.y₂.w₂ in degree 5
y₁.y₂.w₁ in degree 5
y₁.y₂².x₁ in degree 5
y₁.y₂².w₂ in degree 6
y₁.y₂².w₁ in degree 6
y₁.y₂³.x₁ in degree 6
y₁.y₂³.w₂ in degree 7
y₁.y₂³.w₁ in degree 7

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 y₃
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to y₃²
w₁ restricts to y₂.y₃² + y₂².y₃
w₂ restricts to 0
w₃ restricts to 0
v restricts to y₂.y₃³ + y₂⁴
u₁ restricts to 0
u₂ restricts to y₂².y₃³ + y₂⁴.y₃
t restricts to y₂.y₃⁵ + y₂³.y₃³ + y₂⁵.y₃ + y₂⁶
s restricts to y₂².y₃⁵ + y₂³.y₃⁴ + y₂⁵.y₃² + y₂⁶.y₃
r restricts to y₂³.y₃⁵ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁸ + 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 y₃
y₃ restricts to 0
x₁ restricts to y₃²
x₂ restricts to y₃²
w₁ restricts to y₃³ + y₂.y₃² + y₂².y₃
w₂ restricts to y₃³ + y₂.y₃² + y₂².y₃
w₃ restricts to 0
v restricts to y₃⁴ + y₂².y₃² + y₂⁴
u₁ restricts to y₂.y₃⁴ + y₂⁴.y₃
u₂ restricts to y₃⁵ + y₂.y₃⁴ + y₂².y₃³
t restricts to y₃⁶ + y₂².y₃⁴ + y₂³.y₃³ + y₂⁵.y₃ + y₂⁶
s restricts to y₂.y₃⁶ + y₂².y₃⁵ + y₂³.y₃⁴ + y₂⁴.y₃³ + y₂⁵.y₃² + y₂⁶.y₃
r restricts to y₃⁸ + y₂.y₃⁷ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁸ + 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 + y₂.y₃² + y₂².y₃ + y₁.y₂² + y₁².y₂
w₃ restricts to y₄³ + y₂.y₃² + y₂².y₄ + y₂².y₃
v restricts to y₄⁴ + y₃.y₄³ + y₃⁴ + y₂.y₃².y₄ + y₂².y₃² + y₂³.y₄ + y₁.y₂².y₄ + y₁.y₂³ + y₁².y₂.y₄ + y₁².y₂²
u₁ restricts to y₄⁵ + y₃².y₄³ + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂².y₃.y₄² + y₂³.y₄² + y₂³.y₃.y₄ + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
u₂ restricts to y₄⁵ + y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₄⁴ + y₂³.y₄² + y₂⁴.y₄ + y₁.y₂⁴ + y₁⁴.y₂
t restricts to y₃.y₄⁵ + y₃⁶ + y₂.y₃.y₄⁴ + y₂.y₃⁴.y₄ + y₂.y₃⁵ + y₂².y₃².y₄² + y₂³.y₃².y₄ + y₂³.y₃³ + y₂⁴.y₄² + y₂⁴.y₃.y₄ + y₂⁵.y₃ + y₁.y₂.y₄⁴ + y₁.y₂³.y₄² + y₁.y₂⁴.y₄ + y₁.y₂⁵ + y₁².y₄⁴ + y₁².y₂³.y₄ + y₁⁴.y₄² + y₁⁴.y₂²
s restricts to y₃.y₄⁶ + y₃².y₄⁵ + y₃³.y₄⁴ + y₃⁴.y₄³ + y₃⁵.y₄² + y₃⁶.y₄ + y₂.y₄⁶ + y₂.y₃³.y₄³ + y₂.y₃⁴.y₄² + y₂.y₃⁶ + y₂².y₃².y₄³ + y₂².y₃⁵ + y₂³.y₄⁴ + y₂³.y₃.y₄³ + y₂³.y₃².y₄² + y₂³.y₃³.y₄ + y₂³.y₃⁴ + y₂⁴.y₃².y₄ + y₂⁴.y₃³ + y₂⁵.y₄² + y₂⁵.y₃.y₄ + y₂⁵.y₃² + y₂⁶.y₄ + y₂⁶.y₃ + y₁.y₂².y₃⁴ + y₁.y₂³.y₃².y₄ + y₁.y₂⁴.y₃.y₄ + y₁.y₂⁵.y₄ + y₁.y₂⁵.y₃ + y₁².y₂.y₃⁴ + y₁².y₂².y₃².y₄ + y₁².y₂³.y₄² + y₁².y₂³.y₃.y₄ + y₁².y₂³.y₃² + y₁⁴.y₂.y₄² + y₁⁴.y₂.y₃² + y₁⁴.y₂².y₄ + y₁⁴.y₂².y₃
r restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁶.y₄² + y₃⁸ + y₂.y₄⁷ + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁶.y₄ + y₂².y₃².y₄⁴ + y₂².y₃³.y₄³ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂².y₃⁶ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂³.y₃⁵ + y₂⁴.y₄⁴ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + y₂⁶.y₄² + y₂⁶.y₃² + y₂⁷.y₄ + y₂⁷.y₃ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₃².y₄ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₄ + y₁².y₂⁶ + y₁³.y₂⁵ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₄ + y₁⁴.y₂⁴ + y₁⁵.y₂³ + y₁⁶.y₂² + y₁⁸

Poincaré series

(1 + 2t + 3t² + 5t³ + 5t⁴ + 5t⁵ + 5t⁶ + 4t⁷ + 4t⁸ + 3t⁹ + 3t¹⁰ + 3t¹¹ + 2t¹² + 2t¹³ + t¹⁴) / (1 - t) (1 - t⁴) (1 - t⁶) (1 - t⁸)

Back to the groups of order 128