Small group number 1751 of order 128

G is the group 128gp1751

G has 4 minimal generators, rank 4 and exponent 8. The centre has rank 1.

There are 5 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 4, 4, 4, 4.

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 11 generators:

• y₁ in degree 1
• y₂ in degree 1
• y₃ in degree 1
• y₄ in degree 1
• x₁ in degree 2
• x₂ in degree 2
• u₁ in degree 5
• u₂ in degree 5
• u₃ in degree 5
• t in degree 6
• r in degree 8, a regular element

There are 27 minimal relations:

• y₂.y₄ = 0
• y₁.y₄ = 0
• y₄.x₂ = 0
• y₃².y₄ = y₁.x₁
• x₂² = y₂².x₂ + y₁.y₂.x₂
• y₁.x₁² = y₁.y₃².x₁
• x₁².x₂ = y₃⁴.x₂ + y₂.u₃ + y₂.u₂ + y₂.y₃.x₁.x₂ + y₂.y₃.x₁² + y₂.y₃³.x₁ + y₂².x₁.x₂ + y₂².y₃².x₂ + y₂².y₃².x₁ + y₁.u₃ + y₁.u₁
• y₄.u₃ = 0
• y₄.u₁ = 0
• y₃².x₁.x₂ = y₃⁴.x₂ + y₂.u₃ + y₁.u₃ + y₁.u₁
• y₁.u₂ = y₁.y₂.y₃².x₂
• x₂.u₃ = y₂².u₃ + y₁.t + y₁.y₃⁴.x₁ + y₁.y₂.u₁ + y₁.y₂².y₃².x₂ + y₁².u₃ + y₁².y₃³.x₂ + y₁².y₂.y₃².x₂ + y₁³.y₃².x₂
• x₂.u₂ = y₂.y₃.u₂ + y₂.y₃².x₁² + y₂.y₃⁴.x₁ + y₂².u₂ + y₂².y₃.x₁.x₂ + y₂².y₃.x₁² + y₂².y₃³.x₂ + y₂³.y₃².x₁ + y₁.y₂².y₃².x₂
• x₂.u₁ = y₂.t + y₂.y₃.u₂ + y₂.y₃⁴.x₂ + y₂.y₃⁴.x₁ + y₂².u₂ + y₂².y₃.x₁² + y₂².y₃³.x₂ + y₂².y₃³.x₁ + y₂³.x₁.x₂ + y₁.t + y₁.y₃⁴.x₁ + y₁.y₂.u₃ + y₁.y₂.u₁ + y₁.y₂.y₃³.x₂ + y₁².u₃ + y₁².y₃³.x₂ + y₁³.y₃².x₂
• x₁.u₃ = y₃².u₂ + y₃³.x₁² + y₃⁵.x₂ + y₃⁵.x₁ + y₂.y₃.u₃ + y₂.y₃⁴.x₁ + y₂².u₃ + y₁.y₃.u₃ + y₁.y₃.u₁ + y₁.y₂.u₃ + y₁.y₂.u₁
• y₄.t = y₃.y₄.u₂ + y₁.y₃⁴.x₁
• x₂.t = y₃⁶.x₂ + y₂.y₃².u₃ + y₂.y₃².u₂ + y₂.y₃³.x₁² + y₂.y₃⁵.x₁ + y₂².t + y₂².y₃².x₁² + y₂².y₃⁴.x₂ + y₂².y₃⁴.x₁ + y₂³.y₃³.x₁ + y₂⁴.y₃².x₁ + y₁.y₃².u₃ + y₁.y₃².u₁ + y₁.y₂².u₃ + y₁.y₂².u₁ + y₁.y₂³.y₃².x₂ + y₁².t + y₁².y₂.u₁ + y₁².y₂.y₃³.x₂ + y₁³.u₃ + y₁³.y₃³.x₂ + y₁⁴.y₃².x₂
• u₃² = y₂.y₃⁴.u₃ + y₂.y₃⁴.u₂ + y₂.y₃⁵.x₁² + y₂.y₃⁷.x₂ + y₂.y₃⁷.x₁ + y₂².y₃³.u₃ + y₂².y₃⁶.x₁ + y₂³.y₃².u₃ + y₁.y₃⁴.u₁ + y₁.y₂.y₃³.u₃ + y₁.y₂.y₃³.u₁ + y₁.y₂².y₃².u₃ + y₁.y₂².y₃².u₁ + y₁².y₃².t + y₁².y₂.y₃.t + y₁².y₂.y₃².u₁ + y₁².y₂².y₃.u₃ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁².y₂⁴.y₃².x₂ + y₁³.y₂.y₃.u₃ + y₁³.y₂.y₃.u₁ + y₁⁴.y₃.u₃ + y₁⁴.y₂.y₃³.x₂ + y₁⁵.y₂.y₃².x₂ + y₁⁵.y₂².y₃.x₂ + y₁⁶.y₂.y₃.x₂ + y₁².r
• u₂.u₃ = y₃³.x₁.u₂ + y₃⁴.x₁³ + y₃⁵.u₂ + y₃⁸.x₁ + y₂.y₃².x₁.u₂ + y₂.y₃³.x₁³ + y₂.y₃⁴.u₂ + y₂.y₃⁵.x₁² + y₂².y₃³.u₃ + y₂².y₃³.u₂ + y₂².y₃⁶.x₂ + y₂³.y₃².u₃ + y₂³.y₃⁵.x₁ + y₂⁴.y₃.u₃ + y₁.y₂.y₃².t + y₁.y₂.y₃³.u₃ + y₁.y₂.y₃³.u₁ + y₁.y₂².y₃².u₁ + y₁.y₂³.y₃.u₃ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₃ + y₁².y₂.y₃².u₃ + y₁².y₂².y₃.u₃ + y₁².y₂³.u₁ + y₁³.y₂.y₃.u₃ + y₁³.y₂.y₃.u₁ + y₁³.y₂².u₁ + y₁⁴.y₂.u₃ + y₁⁴.y₂.u₁
• u₂² = y₄.x₁².u₂ + y₄³.x₁.u₂ + y₄⁵.u₂ + y₃.y₄⁴.u₂ + y₃.y₄⁷.x₁ + y₃².x₁⁴ + y₃⁶.x₁² + y₂.x₁².u₂ + y₂.y₃.x₁⁴ + y₂.y₃².x₁.u₂ + y₂.y₃⁵.x₁² + y₂².y₃.x₁.u₂ + y₂².y₃⁴.x₁² + y₂³.x₁.u₂ + y₂³.y₃.x₁³ + y₂³.y₃².u₂ + y₂³.y₃³.x₁² + y₂³.y₃⁵.x₁ + y₂⁴.y₃².x₁² + y₂⁴.y₃⁴.x₁ + y₁.y₂⁴.u₃ + y₁².y₂³.u₃ + y₁².y₂³.u₁ + y₄².r
• u₁.u₃ = y₃².x₁.t + y₃³.x₁.u₂ + y₃⁴.t + y₃⁵.u₂ + y₃⁶.x₁² + y₃⁸.x₁ + y₂.y₃².x₁.u₂ + y₂.y₃³.x₁³ + y₂.y₃⁴.u₃ + y₂.y₃⁴.u₂ + y₂.y₃⁷.x₁ + y₂².y₃³.u₃ + y₂³.y₃².u₂ + y₂³.y₃³.x₁² + y₂³.y₃⁵.x₂ + y₂³.y₃⁵.x₁ + y₂⁴.y₃.u₃ + y₂⁴.y₃⁴.x₁ + y₂⁵.u₃ + y₁.y₃⁴.u₃ + y₁.y₃⁴.u₁ + y₁.y₂.y₃².t + y₁.y₂.y₃³.u₁ + y₁.y₂².y₃.t + y₁.y₂².y₃².u₃ + y₁.y₂².y₃².u₁ + y₁.y₂⁴.u₃ + y₁.y₂⁵.y₃².x₂ + y₁².y₃².t + y₁².y₃³.u₃ + y₁².y₃³.u₁ + y₁².y₂.y₃.t + y₁².y₂³.u₁ + y₁².y₂⁴.y₃².x₂ + y₁³.y₃².u₃ + y₁³.y₃².u₁ + y₁³.y₂.y₃.u₁ + y₁³.y₂².y₃³.x₂ + y₁⁴.y₃.u₃ + y₁⁴.y₂.y₃³.x₂ + y₁⁴.y₂².y₃².x₂ + y₁⁴.y₂³.y₃.x₂ + y₁⁵.y₂.y₃².x₂ + y₁⁶.y₂.y₃.x₂ + y₁.y₂.r + y₁².r
• u₁.u₂ = x₁².t + y₃.x₁².u₂ + y₃.x₁².u₁ + y₃².x₁.t + y₃³.x₁.u₂ + y₃³.x₁.u₁ + y₃⁴.x₁³ + y₃⁶.x₁² + y₂.x₁².u₂ + y₂.y₃.x₁.t + y₂.y₃.x₁⁴ + y₂.y₃².x₁.u₁ + y₂.y₃⁴.u₂ + y₂.y₃⁵.x₁² + y₂.y₃⁷.x₁ + y₂².x₁.t + y₂².y₃².t + y₂².y₃².x₁³ + y₂².y₃⁴.x₁² + y₂².y₃⁶.x₁ + y₂³.y₃².u₂ + y₂³.y₃³.x₁² + y₂⁴.y₃².x₁² + y₁.y₂.y₃².t + y₁.y₂².y₃².u₃ + y₁.y₂².y₃².u₁ + y₁.y₂³.y₃.u₃ + y₁².y₂.y₃².u₃ + y₁².y₂².y₃.u₁ + y₁³.y₂.y₃.u₃ + y₁³.y₂.y₃.u₁ + y₁³.y₂².u₃ + y₁⁴.y₂.u₃ + y₁⁴.y₂.u₁
• u₁² = y₃⁴.x₁³ + y₃⁸.x₁ + y₂.y₃².x₁.u₁ + y₂.y₃⁴.u₁ + y₂².y₃.x₁.u₁ + y₂².y₃².t + y₂².y₃⁴.x₁² + y₂².y₃⁶.x₂ + y₂².y₃⁶.x₁ + y₂³.x₁.u₂ + y₂³.x₁.u₁ + y₂³.y₃.t + y₂³.y₃².u₃ + y₂³.y₃².u₂ + y₂³.y₃².u₁ + y₂⁴.y₃.u₃ + y₂⁴.y₃.u₂ + y₂⁴.y₃.u₁ + y₂⁴.y₃⁴.x₁ + y₂⁵.u₃ + y₂⁵.u₂ + y₂⁵.u₁ + y₂⁵.y₃.x₁.x₂ + y₂⁵.y₃³.x₂ + y₂⁵.y₃³.x₁ + y₂⁶.x₁.x₂ + y₂⁶.y₃².x₁ + y₁.y₃⁴.u₁ + y₁.y₂².y₃².u₃ + y₁.y₂².y₃².u₁ + y₁.y₂³.y₃.u₃ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₃ + y₁.y₂⁴.u₁ + y₁².y₃².t + y₁².y₂.y₃.t + y₁².y₂.y₃².u₁ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁².y₂³.y₃³.x₂ + y₁².y₂⁴.y₃².x₂ + y₁³.y₂.y₃.u₃ + y₁³.y₂.y₃.u₁ + y₁³.y₂³.y₃².x₂ + y₁³.y₂⁴.y₃.x₂ + y₁⁴.y₃.u₃ + y₁⁴.y₂.y₃³.x₂ + y₁⁴.y₂³.y₃.x₂ + y₁⁵.y₂.y₃².x₂ + y₁⁵.y₂².y₃.x₂ + y₁⁶.y₂.y₃.x₂ + y₂².r + y₁².r
• u₃.t = y₃⁴.x₁.u₂ + y₃⁵.x₁³ + y₃⁷.x₁² + y₃⁹.x₂ + y₂.y₃².x₁.t + y₂.y₃⁴.t + y₂.y₃⁴.x₁³ + y₂.y₃⁵.u₃ + y₂².y₃⁵.x₁² + y₂².y₃⁷.x₁ + y₂³.y₃⁴.x₁² + y₂³.y₃⁶.x₁ + y₁.y₃⁴.t + y₁.y₃⁵.u₃ + y₁.y₃⁵.u₁ + y₁.y₃⁸.x₁ + y₁.y₂.y₃⁴.u₃ + y₁.y₂.y₃⁴.u₁ + y₁.y₂².y₃².t + y₁.y₂³.y₃².u₃ + y₁.y₂⁴.t + y₁.y₂⁵.u₃ + y₁.y₂⁵.y₃³.x₂ + y₁².y₃³.t + y₁².y₃⁴.u₃ + y₁².y₃⁴.u₁ + y₁².y₂.y₃².t + y₁².y₂².y₃².u₃ + y₁².y₂².y₃².u₁ + y₁².y₂³.t + y₁².y₂³.y₃.u₃ + y₁².y₂³.y₃.u₁ + y₁².y₂⁴.u₁ + y₁³.y₃².t + y₁³.y₃³.u₁ + y₁³.y₂.y₃².u₁ + y₁³.y₂².y₃.u₁ + y₁³.y₂³.u₁ + y₁³.y₂⁴.y₃².x₂ + y₁⁴.y₃.t + y₁⁴.y₂².u₃ + y₁⁴.y₂².u₁ + y₁⁴.y₂⁴.y₃.x₂ + y₁⁵.y₃.u₃ + y₁⁵.y₃.u₁ + y₁⁵.y₂.u₃ + y₁⁵.y₂.y₃³.x₂ + y₁⁵.y₂³.y₃.x₂ + y₁⁶.u₃ + y₁⁶.u₁ + y₁⁶.y₃³.x₂ + y₁⁶.y₂.y₃².x₂ + y₁⁶.y₂².y₃.x₂ + y₁⁷.y₂.y₃.x₂ + y₁.x₂.r + y₁².y₂.r + y₁³.r
• u₂.t = y₃.x₁².t + y₃.y₄.x₁².u₂ + y₃.y₄³.x₁.u₂ + y₃.y₄⁵.u₂ + y₃².x₁².u₂ + y₃³.x₁.t + y₃³.x₁⁴ + y₃⁵.x₁³ + y₂.x₁².t + y₂.y₃².x₁⁴ + y₂.y₃³.x₁.u₂ + y₂.y₃⁴.x₁³ + y₂.y₃⁶.x₁² + y₂².y₃.x₁.t + y₂².y₃².x₁.u₂ + y₂².y₃³.x₁³ + y₂².y₃⁵.x₁² + y₂³.x₁.t + y₂³.y₃².t + y₂⁴.y₃⁵.x₁ + y₂⁵.y₃².x₁² + y₂⁵.y₃⁴.x₁ + y₁.y₂³.y₃².u₃ + y₁.y₂³.y₃².u₁ + y₁.y₂⁵.u₃ + y₁².y₂.y₃².t + y₁².y₂².y₃².u₁ + y₁².y₂³.y₃.u₃ + y₁².y₂⁴.u₃ + y₁².y₂⁴.u₁ + y₁³.y₂.y₃².u₃ + y₁³.y₂².y₃.u₁ + y₁⁴.y₂.y₃.u₃ + y₁⁴.y₂.y₃.u₁ + y₁⁴.y₂².u₃ + y₁⁵.y₂.u₃ + y₁⁵.y₂.u₁ + y₃.y₄².r
• u₁.t = y₃.x₁².t + y₃².x₁².u₂ + y₃².x₁².u₁ + y₃³.x₁.t + y₃⁶.u₂ + y₃⁷.x₁² + y₃⁹.x₁ + y₂.x₁².t + y₂.y₃².x₁.t + y₂.y₃².x₁⁴ + y₂.y₃³.x₁.u₁ + y₂.y₃⁴.x₁³ + y₂.y₃⁶.x₁² + y₂.y₃⁸.x₂ + y₂.y₃⁸.x₁ + y₂².x₁².u₂ + y₂².y₃.x₁.t + y₂².y₃.x₁⁴ + y₂².y₃².x₁.u₁ + y₂².y₃⁴.u₂ + y₂².y₃⁵.x₁² + y₂².y₃⁷.x₁ + y₂³.x₁.t + y₂³.y₃².t + y₂³.y₃².x₁³ + y₂³.y₃³.u₂ + y₂³.y₃⁶.x₂ + y₂⁴.x₁.u₂ + y₂⁴.y₃.x₁³ + y₂⁴.y₃³.x₁² + y₂⁴.y₃⁵.x₂ + y₂⁵.t + y₂⁵.y₃².x₁² + y₂⁵.y₃⁴.x₁ + y₂⁶.u₃ + y₂⁶.y₃.x₁.x₂ + y₂⁶.y₃³.x₂ + y₂⁶.y₃³.x₁ + y₂⁷.y₃².x₁ + y₁.y₂.y₃³.t + y₁.y₂.y₃⁴.u₁ + y₁.y₂².y₃³.u₃ + y₁.y₂³.y₃².u₃ + y₁.y₂⁴.y₃.u₃ + y₁.y₂⁴.y₃.u₁ + y₁.y₂⁵.u₃ + y₁.y₂⁵.u₁ + y₁.y₂⁵.y₃³.x₂ + y₁².y₃³.t + y₁².y₃⁴.u₁ + y₁².y₂².y₃².u₃ + y₁².y₂³.t + y₁².y₂³.y₃.u₃ + y₁².y₂⁵.y₃².x₂ + y₁³.y₃².t + y₁³.y₃³.u₃ + y₁³.y₂.y₃.t + y₁³.y₂.y₃².u₃ + y₁³.y₂.y₃².u₁ + y₁³.y₂².y₃.u₁ + y₁³.y₂³.u₃ + y₁³.y₂⁴.y₃².x₂ + y₁³.y₂⁵.y₃.x₂ + y₁⁴.y₃.t + y₁⁴.y₃².u₃ + y₁⁴.y₃².u₁ + y₁⁴.y₂.y₃.u₃ + y₁⁴.y₂.y₃.u₁ + y₁⁴.y₂².u₁ + y₁⁴.y₂².y₃³.x₂ + y₁⁵.y₃.u₃ + y₁⁵.y₃.u₁ + y₁⁵.y₂.u₁ + y₁⁵.y₂².y₃².x₂ + y₁⁶.u₃ + y₁⁶.u₁ + y₁⁶.y₃³.x₂ + y₁⁶.y₂.y₃².x₂ + y₁⁷.y₂.y₃.x₂ + y₂.x₂.r + y₁.x₂.r + y₁.y₂².r + y₁³.r
• t² = y₃⁴.x₁⁴ + y₂.y₃².x₁².u₂ + y₂.y₃³.x₁⁴ + y₂.y₃⁵.x₁³ + y₂.y₃⁶.u₂ + y₂.y₃⁷.x₁² + y₂.y₃⁹.x₁ + y₂².y₃².x₁.t + y₂².y₃⁴.t + y₂².y₃⁸.x₂ + y₂².y₃⁸.x₁ + y₂³.x₁².u₂ + y₂³.y₃.x₁.t + y₂³.y₃.x₁⁴ + y₂³.y₃⁴.u₂ + y₂⁴.x₁.t + y₂⁴.y₃³.u₂ + y₂⁴.y₃⁶.x₂ + y₂⁴.y₃⁶.x₁ + y₂⁵.x₁.u₂ + y₂⁵.y₃.x₁³ + y₂⁵.y₃³.x₁² + y₂⁵.y₃⁵.x₂ + y₂⁵.y₃⁵.x₁ + y₂⁶.t + y₂⁶.y₃².x₁² + y₂⁷.u₃ + y₂⁷.y₃.x₁.x₂ + y₂⁷.y₃³.x₂ + y₂⁷.y₃³.x₁ + y₂⁸.y₃².x₁ + y₁.y₂.y₃⁴.t + y₁.y₂².y₃⁴.u₁ + y₁.y₂³.y₃².t + y₁.y₂³.y₃³.u₃ + y₁.y₂⁴.y₃.t + y₁.y₂⁴.y₃².u₃ + y₁.y₂⁶.u₃ + y₁.y₂⁶.y₃³.x₂ + y₁.y₂⁷.y₃².x₂ + y₁².y₂².y₃².t + y₁².y₂².y₃³.u₁ + y₁².y₂³.y₃.t + y₁².y₂³.y₃².u₃ + y₁².y₂³.y₃².u₁ + y₁².y₂⁴.t + y₁².y₂⁴.y₃.u₃ + y₁².y₂⁵.u₁ + y₁³.y₃⁴.u₁ + y₁³.y₂.y₃².t + y₁³.y₂.y₃³.u₃ + y₁³.y₂.y₃³.u₁ + y₁³.y₂².y₃².u₃ + y₁³.y₂³.y₃.u₃ + y₁³.y₂³.y₃.u₁ + y₁³.y₂⁴.u₃ + y₁³.y₂⁴.u₁ + y₁³.y₂⁴.y₃³.x₂ + y₁³.y₂⁶.y₃.x₂ + y₁⁴.y₃².t + y₁⁴.y₂.y₃².u₃ + y₁⁴.y₂.y₃².u₁ + y₁⁴.y₂².y₃.u₁ + y₁⁴.y₂⁵.y₃.x₂ + y₁⁵.y₂.y₃.u₃ + y₁⁵.y₂².u₁ + y₁⁵.y₂².y₃³.x₂ + y₁⁶.y₃.u₃ + y₁⁶.y₂.u₃ + y₁⁶.y₂.u₁ + y₁⁷.y₂.y₃².x₂ + y₁⁷.y₂².y₃.x₂ + y₁⁸.y₂.y₃.x₂ + y₂².x₂.r + y₁.y₂.x₂.r + y₁².y₂².r + y₁⁴.r

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

• y₁.y₂.x₁ = 0
• y₁².x₁ = 0
• y₁.x₁.x₂ = 0
• y₁.y₃⁴.x₂ = y₁.y₂.u₃ + y₁².u₃ + y₁².u₁
• y₁.x₁.u₁ = 0
• y₁.x₁.t = y₁.y₃⁶.x₁

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 12 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 = x₁² + y₄⁴ + y₃².x₁ + y₃⁴ + y₂.y₃.x₁ + y₂².y₃² + y₂⁴ + y₁.y₂.y₃² + y₁.y₂².y₃ + y₁².y₃² + y₁².y₂.y₃ + y₁².y₂² + y₁⁴ in degree 4
• h₃ = y₄².x₁² + y₃².x₁² + y₃⁴.x₁ + y₂.y₃.x₁² + y₂².x₁² + y₂².y₃⁴ + y₂³.y₃.x₁ + y₂⁴.y₃² + y₁.y₂.y₃⁴ + y₁.y₂⁴.y₃ + y₁².y₃⁴ + y₁².y₂².y₃² + y₁².y₂⁴ + y₁⁴.y₃² + y₁⁴.y₂.y₃ + y₁⁴.y₂² in degree 6
• h₄ = y₃ + y₁ 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
• y₄ in degree 1
• y₂ in degree 1
• y₁ in degree 1
• x₂ in degree 2
• x₁ in degree 2
• y₄² in degree 2
• y₂² in degree 2
• y₁.y₂ in degree 2
• y₁² in degree 2
• y₄.x₁ in degree 3
• y₄³ in degree 3
• y₂.x₂ in degree 3
• y₂.x₁ in degree 3
• y₂³ in degree 3
• y₁.x₂ in degree 3
• y₁.y₂² in degree 3
• y₁².y₂ in degree 3
• y₁³ in degree 3
• x₁.x₂ in degree 4
• y₄².x₁ in degree 4
• y₄⁴ in degree 4
• y₂².x₂ in degree 4
• y₂².x₁ in degree 4
• y₂⁴ in degree 4
• y₁.y₂.x₂ in degree 4
• y₁.y₂³ in degree 4
• y₁².x₂ in degree 4
• y₁².y₂² in degree 4
• y₁³.y₂ in degree 4
• y₁⁴ in degree 4
• u₃ in degree 5
• u₂ in degree 5
• u₁ in degree 5
• y₄³.x₁ in degree 5
• y₄⁵ in degree 5
• y₂.x₁.x₂ in degree 5
• y₂³.x₂ in degree 5
• y₂³.x₁ in degree 5
• y₂⁵ in degree 5
• y₁.y₂².x₂ in degree 5
• y₁².y₂.x₂ in degree 5
• y₁².y₂³ in degree 5
• y₁³.x₂ in degree 5
• y₁³.y₂² in degree 5
• y₁⁴.y₂ in degree 5
• y₁⁵ in degree 5
• t in degree 6
• y₄.u₂ in degree 6
• y₄⁴.x₁ in degree 6
• y₂.u₁ in degree 6
• y₂².x₁.x₂ in degree 6
• y₂⁴.x₂ in degree 6
• y₂⁴.x₁ in degree 6
• y₂⁶ in degree 6
• y₁.u₃ in degree 6
• y₁.u₁ in degree 6
• y₁.y₂³.x₂ in degree 6
• y₁².y₂².x₂ in degree 6
• y₁³.y₂.x₂ in degree 6
• y₁³.y₂³ in degree 6
• y₁⁴.x₂ in degree 6
• y₁⁴.y₂² in degree 6
• y₁⁵.y₂ in degree 6
• y₁⁶ in degree 6
• x₁.u₂ in degree 7
• x₁.u₁ in degree 7
• y₄².u₂ in degree 7
• y₄⁵.x₁ in degree 7
• y₂.t in degree 7
• y₂².u₁ in degree 7
• y₂³.x₁.x₂ in degree 7
• y₂⁵.x₂ in degree 7
• y₂⁵.x₁ in degree 7
• y₁.t in degree 7
• y₁.y₂.u₁ in degree 7
• y₁².u₃ in degree 7
• y₁².u₁ in degree 7
• y₁².y₂³.x₂ in degree 7
• y₁³.y₂².x₂ in degree 7
• y₁⁴.y₂.x₂ in degree 7
• y₁⁴.y₂³ in degree 7
• y₁⁵.x₂ in degree 7
• y₁⁵.y₂² in degree 7
• y₁⁶.y₂ in degree 7
• x₁.t in degree 8
• y₄.x₁.u₂ in degree 8
• y₄³.u₂ in degree 8
• y₂.x₁.u₁ in degree 8
• y₂².t in degree 8
• y₂³.u₁ in degree 8
• y₂⁴.x₁.x₂ in degree 8
• y₂⁶.x₁ in degree 8
• y₁.y₂.t in degree 8
• y₁.y₂².u₁ in degree 8
• y₁².t in degree 8
• y₁².y₂.u₁ in degree 8
• y₁³.u₃ in degree 8
• y₁³.u₁ in degree 8
• y₁³.y₂³.x₂ in degree 8
• y₁⁴.y₂².x₂ in degree 8
• y₁⁵.y₂.x₂ in degree 8
• y₁⁵.y₂³ in degree 8
• y₁⁶.x₂ in degree 8
• y₁⁶.y₂² in degree 8
• y₄².x₁.u₂ in degree 9
• y₄⁴.u₂ in degree 9
• y₂.x₁.t in degree 9
• y₂².x₁.u₁ in degree 9
• y₂³.t in degree 9
• y₂⁴.u₁ in degree 9
• y₂⁵.x₁.x₂ in degree 9
• y₁.y₂².t in degree 9
• y₁².y₂.t in degree 9
• y₁².y₂².u₁ in degree 9
• y₁³.t in degree 9
• y₁³.y₂.u₁ in degree 9
• y₁⁴.u₃ in degree 9
• y₁⁴.u₁ in degree 9
• y₁⁴.y₂³.x₂ in degree 9
• y₁⁵.y₂².x₂ in degree 9
• y₁⁶.y₂.x₂ in degree 9
• y₁⁶.y₂³ in degree 9
• y₄³.x₁.u₂ in degree 10
• y₄⁵.u₂ in degree 10
• y₂².x₁.t in degree 10
• y₂³.x₁.u₁ in degree 10
• y₂⁴.t in degree 10
• y₂⁵.u₁ in degree 10
• y₁.y₂³.t in degree 10
• y₁².y₂².t in degree 10
• y₁³.y₂.t in degree 10
• y₁³.y₂².u₁ in degree 10
• y₁⁴.t in degree 10
• y₁⁴.y₂.u₁ in degree 10
• y₁⁵.u₃ in degree 10
• y₁⁵.u₁ in degree 10
• y₁⁵.y₂³.x₂ in degree 10
• y₁⁶.y₂².x₂ in degree 10
• y₄⁴.x₁.u₂ in degree 11
• y₂³.x₁.t in degree 11
• y₂⁴.x₁.u₁ in degree 11
• y₂⁵.t in degree 11
• y₁².y₂³.t in degree 11
• y₁³.y₂².t in degree 11
• y₁⁴.y₂.t in degree 11
• y₁⁴.y₂².u₁ in degree 11
• y₁⁵.t in degree 11
• y₁⁵.y₂.u₁ in degree 11
• y₁⁶.u₁ in degree 11
• y₁⁶.y₂³.x₂ in degree 11
• y₄⁵.x₁.u₂ in degree 12
• y₂⁴.x₁.t in degree 12
• y₂⁵.x₁.u₁ in degree 12
• y₁³.y₂³.t in degree 12
• y₁⁴.y₂².t in degree 12
• y₁⁵.y₂.t in degree 12
• y₁⁵.y₂².u₁ in degree 12
• y₁⁶.t in degree 12
• y₁⁶.y₂.u₁ in degree 12
• y₂⁵.x₁.t in degree 13
• y₁⁴.y₂³.t in degree 13
• y₁⁵.y₂².t in degree 13
• y₁⁶.y₂.t in degree 13
• y₁⁶.y₂².u₁ in degree 13
• y₁⁵.y₂³.t in degree 14
• y₁⁶.y₂².t in degree 14
• y₁⁶.y₂³.t in degree 15

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

• y₄².h in degree 3
• y₄.x₁.h + y₄.h³ in degree 4
• y₄³.h in degree 4
• y₄².x₁.h in degree 5
• y₄⁴.h in degree 5
• y₄³.x₁.h in degree 6
• y₄⁵.h in degree 6
• y₄.h⁵ in degree 6
• y₄.u₂.h in degree 7
• y₄⁴.x₁.h in degree 7
• y₂⁶.h + y₂⁵.h² + y₂⁴.h³ + y₂³.h⁴ + y₂².h⁵ + y₂.h⁶ + h⁷ in degree 7
• y₄².u₂.h in degree 8
• y₄⁵.x₁.h in degree 8
• y₄.x₁.u₂.h + y₄.u₂.h³ in degree 9
• y₄³.u₂.h in degree 9
• y₂⁶.x₁.h + y₂⁵.x₁.h² + y₂⁴.x₁.h³ + y₂³.x₁.h⁴ + y₂².x₁.h⁵ + y₂.x₁.h⁶ + x₁.h⁷ in degree 9
• y₄².x₁.u₂.h in degree 10
• y₄⁴.u₂.h in degree 10
• y₄³.x₁.u₂.h in degree 11
• y₄⁵.u₂.h in degree 11
• y₄⁴.x₁.u₂.h in degree 12
• y₁⁶.y₂³.x₂.h + y₁⁵.y₂³.x₂.h² + y₁⁴.y₂³.x₂.h³ + y₁⁶.y₂.x₂.h³ + y₁³.y₂³.x₂.h⁴ + y₁⁵.y₂.x₂.h⁴ + y₂⁵.x₂.h⁵ + y₂⁴.x₂.h⁶ + y₁³.y₂.x₂.h⁶ + y₁⁴.x₂.h⁶ + u₂.h⁷ + y₂.x₁.x₂.h⁷ + y₂².x₂.h⁸ + y₂⁴.h⁸ + y₁².x₂.h⁸ + y₁².y₂².h⁸ + y₁⁴.h⁸ + y₂.x₂.h⁹ + y₁.y₂².h⁹ + y₁².y₂.h⁹ + x₂.h¹⁰ + y₂².h¹⁰ + y₁.y₂.h¹⁰ + y₁².h¹⁰ + h¹² in degree 12
• y₄⁵.x₁.u₂.h in degree 13
• y₂⁵.x₁.x₂.h⁵ + y₂⁴.x₁.x₂.h⁶ + x₁.u₂.h⁷ + y₂⁵.x₂.h⁷ + y₁².y₂³.x₂.h⁷ + y₁⁴.y₂.x₂.h⁷ + y₂⁴.x₁.h⁸ + y₁.y₂³.x₂.h⁸ + y₁².y₂².x₂.h⁸ + y₂³.x₂.h⁹ + y₁.y₂².x₂.h⁹ + y₁².y₂.x₂.h⁹ + x₁.x₂.h¹⁰ + y₂².x₁.h¹⁰ + y₂.x₂.h¹¹ + x₁.h¹² in degree 14

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

• y₁.x₁ in degree 3
• y₁.y₃.x₁ in degree 4
• y₁.y₃².x₁ in degree 5
• y₁.y₃³.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
• y₄ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• u₁ restricts to 0
• u₂ restricts to 0
• u₃ restricts to 0
• t 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
• y₄ restricts to y₂
• x₁ restricts to y₃² + y₂.y₃
• x₂ restricts to 0
• u₁ restricts to 0
• u₂ restricts to y₁².y₂³ + y₁⁴.y₂
• u₃ restricts to 0
• t restricts to 0
• r restricts to 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 y₂
• y₂ restricts to y₃
• y₃ restricts to y₄
• y₄ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• u₁ restricts to y₁.y₂.y₃³ + y₁.y₂³.y₃ + y₁².y₃³ + y₁².y₂³ + y₁⁴.y₃ + y₁⁴.y₂
• u₂ restricts to 0
• u₃ restricts to y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + y₁⁴.y₂
• t restricts to y₁.y₂².y₃³ + y₁.y₂⁴.y₃ + y₁².y₂.y₃³ + y₁².y₂⁴ + y₁⁴.y₂.y₃ + y₁⁴.y₂²
• r restricts to y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂.y₃⁵.y₄ + y₁.y₂.y₃⁶ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃³.y₄ + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁵.y₃.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₃⁵.y₄ + y₁².y₃⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃³.y₄² + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂³.y₃³ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₄ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃³.y₄ + y₁⁴.y₂.y₃³ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₄ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

• y₁ restricts to y₂
• y₂ restricts to y₄
• y₃ restricts to y₃
• y₄ restricts to 0
• x₁ restricts to 0
• x₂ restricts to y₄² + y₂.y₄
• u₁ restricts to y₄⁵ + y₂.y₃².y₄² + y₂².y₄³ + y₂².y₃².y₄ + y₁.y₂.y₄³ + y₁.y₂³.y₄ + y₁².y₄³ + y₁².y₂³ + y₁⁴.y₄ + y₁⁴.y₂
• u₂ restricts to y₃².y₄³ + y₂.y₃².y₄²
• u₃ restricts to y₃⁴.y₄ + y₂.y₄⁴ + y₂².y₄³ + y₂².y₃².y₄ + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• t restricts to y₄⁶ + y₃².y₄⁴ + y₃⁴.y₄² + y₂.y₄⁵ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃³.y₄ + y₂³.y₄³ + y₁.y₂.y₄⁴ + y₁.y₂².y₄³ + y₁.y₂³.y₄² + y₁.y₂⁴.y₄ + y₁².y₄⁴ + y₁².y₂.y₄³ + y₁².y₂³.y₄ + y₁².y₂⁴ + y₁⁴.y₄² + y₁⁴.y₂²
• r restricts to y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁶.y₄² + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂².y₃².y₄⁴ + y₂².y₃⁴.y₄² + y₂³.y₃.y₄⁴ + y₂³.y₃³.y₄² + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁵.y₃.y₄² + y₁.y₂.y₄⁶ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃².y₄³ + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₄⁴ + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁².y₄⁶ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃².y₄³ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃⁴ + y₁².y₂³.y₄³ + y₁².y₂³.y₃.y₄² + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₄³ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 5, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₂
• y₃ restricts to y₃
• y₄ restricts to 0
• x₁ restricts to y₄² + y₂.y₄
• x₂ restricts to 0
• u₁ restricts to y₃².y₄³ + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂².y₃.y₄² + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• u₂ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄
• u₃ restricts to 0
• t restricts to y₃².y₄⁴ + y₂.y₃³.y₄² + y₂².y₃³.y₄ + y₂³.y₃².y₄
• r restricts to y₃².y₄⁶ + y₂.y₃.y₄⁶ + y₂.y₃⁴.y₄³ + y₂².y₃³.y₄³ + y₂².y₃⁵.y₄ + y₂³.y₃².y₄³ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₄⁴ + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁.y₂⁶.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₄ + y₁⁴.y₂³.y₃ + y₁⁸

Restriction to special subgroup number 6, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₄
• y₃ restricts to y₂
• y₄ restricts to 0
• x₁ restricts to y₃.y₄ + y₃²
• x₂ restricts to y₄²
• u₁ restricts to y₄⁵ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₄³ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂².y₃³ + y₂⁴.y₄ + y₂⁴.y₃ + y₁.y₃.y₄³ + y₁.y₃².y₄² + y₁².y₄³ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁⁴.y₄
• u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃⁴ + y₂².y₄³ + y₂³.y₃.y₄ + y₂³.y₃²
• u₃ restricts to y₂².y₃.y₄² + y₂².y₃².y₄ + y₂⁴.y₄
• t restricts to y₄⁶ + y₃².y₄⁴ + y₃⁴.y₄² + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₃².y₄² + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂⁴.y₃.y₄ + y₁.y₃.y₄⁴ + y₁.y₃².y₄³ + y₁².y₄⁴ + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁⁴.y₄²
• r restricts to y₃.y₄⁷ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁶.y₄² + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃³.y₄⁴ + y₂.y₃⁶.y₄ + y₂².y₃².y₄⁴ + y₂².y₃⁴.y₄² + y₂².y₃⁶ + y₂³.y₄⁵ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁵.y₃.y₄² + y₂⁵.y₃².y₄ + y₂⁶.y₃.y₄ + y₁.y₃.y₄⁶ + y₁.y₃⁴.y₄³ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃².y₄³ + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₄⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₃.y₄³ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₃.y₄³ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

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Poincaré series

(1 + 3t + 6t² + 9t³ + 11t⁴ + 14t⁵ + 16t⁶ + 17t⁷ + 17t⁸ + 15t⁹ + 13t¹⁰ + 10t¹¹ + 7t¹² + 4t¹³ + t¹⁴) / (1 - t) (1 - t⁴) (1 - t⁶) (1 - t⁸)

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Back to the groups of order 128