Small group number 1615 of order 128

G is the group 128gp1615

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

There are 6 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 4, 4, 4, 4, 4, 4.

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 11 generators:

• y₁ in degree 1, a nilpotent element
• y₂ in degree 1
• y₃ in degree 1
• y₄ in degree 1
• x₁ in degree 2
• x₂ in degree 2
• w in degree 3
• u₁ in degree 5
• u₂ in degree 5
• t in degree 6
• r in degree 8, a regular element

There are 27 minimal relations:

• y₁.y₄ = 0
• y₁² = 0
• y₄.x₂ = y₂.y₃.y₄ + y₂.y₃² + y₂².y₃ + y₁.x₁
• y₁.y₂.y₃ = 0
• y₄.w = y₃.y₄.x₁ + y₂.y₃.x₁ + y₁.y₃.x₁ + y₁.y₂.x₁
• y₂.y₃.x₂ = y₂.y₃².y₄ + y₂.y₃³ + y₂².y₃.y₄ + y₂².y₃² + y₂³.y₃ + y₁.w + y₁.y₂.x₁
• y₂.y₃.y₄² = y₂.y₃².y₄ + y₂².y₃.y₄ + y₂².y₃²
• y₂.y₃.w = y₂.y₃.y₄.x₁ + y₂².y₃.x₁ + y₁.y₃.w + y₁.y₂.w + y₁.y₂².x₁
• y₁.x₁.x₂ = y₁.x₁² + y₁.y₃.w + y₁.y₂.w + y₁.y₂².x₁
• w² = x₁.x₂² + x₁².x₂ + y₃².x₁.x₂ + y₃².x₁² + y₂².x₁.x₂ + y₂².y₃².x₁ + y₁.x₂.w + y₁.x₁.w + y₁.y₃².w
• y₄.u₂ = y₃².y₄².x₁ + y₂.y₃².y₄.x₁ + y₂.y₃³.x₁ + y₂².y₃.y₄.x₁ + y₂².y₃².x₁ + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂³.y₃.x₁ + y₂³.y₃².y₄ + y₂³.y₃³ + y₂⁴.y₃² + y₁.y₃.x₁² + y₁.y₂.x₁²
• y₁.u₂ = y₁.x₂.w + y₁.x₁.w + y₁.y₃.x₁² + y₁.y₂.x₁² + y₁.y₂².w + y₁.y₂³.x₁
• y₁.u₁ = y₁.x₂.w + y₁.y₃³.x₁ + y₁.y₂.x₁²
• x₂.u₁ = x₂².w + x₁.u₂ + y₃.t + y₃.x₁².x₂ + y₃.y₄².x₁² + y₃².u₂ + y₃².u₁ + y₃².x₂.w + y₃².x₁.w + y₃².y₄.x₁² + y₃².y₄³.x₁ + y₃³.x₁.x₂ + y₃³.y₄².x₁ + y₃⁴.w + y₃⁴.y₄.x₁ + y₂.t + y₂.y₄².x₁² + y₂.y₃⁴.x₁ + y₂.y₃⁵.y₄ + y₂.y₃⁶ + y₂².x₂.w + y₂².y₃².y₄.x₁ + y₂².y₃³.x₁ + y₂³.x₁.x₂ + y₂³.y₃².x₁ + y₂⁴.y₃³ + y₁.x₁³ + y₁.y₃.x₁.w + y₁.y₂².x₁² + y₁.y₂³.w + y₁.y₂⁴.x₁
• y₄.t = y₄³.x₁² + y₃.y₄.u₁ + y₃.y₄⁴.x₁ + y₃².y₄.x₁² + y₃².y₄³.x₁ + y₃⁴.y₄.x₁ + y₂.y₃.u₁ + y₂.y₃.y₄.x₁² + y₂.y₃².x₁² + y₂.y₃⁴.x₁ + y₂².y₃.x₁² + y₂².y₃².y₄.x₁ + y₂³.y₃³.y₄ + y₂³.y₃⁴ + y₂⁴.y₃.x₁ + y₁.y₃.x₂.w + y₁.y₃.x₁.w + y₁.y₃³.w + y₁.y₃⁴.x₁ + y₁.y₂.x₂.w + y₁.y₂².x₁²
• y₂.y₃.u₂ = y₂.y₃⁴.x₁ + y₂².y₃².y₄.x₁ + y₂².y₃³.x₁ + y₂².y₃⁴.y₄ + y₂².y₃⁵ + y₂³.y₃.y₄.x₁ + y₂³.y₃².x₁ + y₂³.y₃³.y₄ + y₂³.y₃⁴ + y₂⁴.y₃.x₁ + y₂⁴.y₃².y₄ + y₂⁴.y₃³ + y₂⁵.y₃² + y₁.y₃.x₂.w + y₁.y₃³.w + y₁.y₂.x₂.w + y₁.y₂².x₁²
• y₁.t = y₁.y₃.x₁.w + y₁.y₃².x₁² + y₁.y₃⁴.x₁ + y₁.y₂.x₁.w + y₁.y₂².x₁² + y₁.y₂³.w + y₁.y₂⁴.x₁
• w.u₂ = x₂.t + y₃.x₂.u₂ + y₃.x₁.u₂ + y₃².x₁².x₂ + y₃³.x₂.w + y₃⁴.x₁.x₂ + y₂.x₂².w + y₂.x₁.u₂ + y₂.x₁.x₂.w + y₂.y₃⁶.y₄ + y₂.y₃⁷ + y₂².t + y₂².x₁.x₂² + y₂².x₁².x₂ + y₂².y₄².x₁² + y₂².y₃.y₄.x₁² + y₂².y₃².x₁² + y₂².y₃³.y₄.x₁ + y₂³.x₂.w + y₂³.y₃.x₁² + y₂³.y₃³.x₁ + y₂³.y₃⁴.y₄ + y₂³.y₃⁵ + y₂⁴.x₁.x₂ + y₂⁴.y₃.y₄.x₁ + y₂⁴.y₃².x₁ + y₂⁵.y₃.x₁ + y₂⁵.y₃².y₄ + y₂⁵.y₃³ + y₂⁶.y₃² + y₁.y₃.x₁³ + y₁.y₂².x₂.w + y₁.y₂².x₁.w + y₁.y₂⁴.w + y₁.y₂⁵.x₁
• w.u₁ = x₁.t + x₁.x₂³ + x₁².x₂² + y₄².x₁³ + y₃.x₁².w + y₃.y₄³.x₁² + y₃².x₁.x₂² + y₃².y₄².x₁² + y₃⁴.x₁² + y₂.x₁.u₂ + y₂.x₁.x₂.w + y₂.x₁².w + y₂.y₃².y₄.x₁² + y₂.y₃³.x₁² + y₂².x₁.x₂² + y₂².y₃².x₁² + y₂².y₃³.y₄.x₁ + y₂².y₃⁴.x₁ + y₂³.y₃.x₁² + y₂⁴.y₃.y₄.x₁ + y₂⁵.y₃.x₁ + y₁.x₂².w + y₁.y₃.x₁³ + y₁.y₃².x₂.w + y₁.y₂³.x₁²
• y₂.y₃.t = y₂.y₃.y₄.u₁ + y₂.y₃⁴.y₄.x₁ + y₂².y₃.u₁ + y₂².y₃⁴.x₁ + y₂³.y₃.x₁² + y₂³.y₃².y₄.x₁ + y₂³.y₃⁴.y₄ + y₂³.y₃⁵ + y₂⁴.y₃.y₄.x₁ + y₂⁴.y₃².x₁ + y₂⁵.y₃.x₁ + y₁.y₃⁴.w + y₁.y₂².x₂.w + y₁.y₂³.x₁²
• w.t = x₁.x₂.u₂ + x₁².u₂ + y₃.x₂.t + y₃.x₁.t + y₃².x₂.u₂ + y₃².x₁.u₂ + y₃².x₁².w + y₃².y₄.x₁³ + y₃³.x₁.x₂² + y₃³.x₁³ + y₃⁴.x₂.w + y₃⁴.y₄.x₁² + y₂.x₁.t + y₂.x₁.x₂³ + y₂.x₁³.x₂ + y₂.y₄².x₁³ + y₂.y₃.y₄.x₁³ + y₂.y₃².x₁³ + y₂.y₃⁷.y₄ + y₂.y₃⁸ + y₂².y₃.y₄.u₁ + y₂².y₃³.x₁² + y₂².y₃⁵.x₁ + y₂³.x₁.x₂² + y₂³.x₁².x₂ + y₂³.y₃.u₁ + y₂³.y₃².x₁² + y₂³.y₃³.y₄.x₁ + y₂³.y₃⁴.x₁ + y₂³.y₃⁵.y₄ + y₂³.y₃⁶ + y₂⁴.y₃².y₄.x₁ + y₂⁴.y₃⁴.y₄ + y₂⁴.y₃⁵ + y₂⁵.y₃².x₁ + y₂⁵.y₃³.y₄ + y₂⁵.y₃⁴ + y₂⁶.y₃³ + y₁.y₃.x₁².w + y₁.y₃².x₁³ + y₁.y₃⁴.x₁² + y₁.y₂.x₂².w + y₁.y₂.x₁².w + y₁.y₂³.x₂.w + y₁.y₂³.x₁.w
• u₂² = x₁.x₂⁴ + x₁³.x₂² + y₃².x₁³.x₂ + y₃⁴.x₁.x₂² + y₃⁴.x₁².x₂ + y₃⁴.y₄².x₁² + y₂².x₁³.x₂ + y₂².y₃².x₁³ + y₂².y₃³.y₄.x₁² + y₂².y₃⁷.y₄ + y₂².y₃⁸ + y₂³.y₃².y₄.x₁² + y₂³.y₃³.x₁² + y₂³.y₃⁶.y₄ + y₂³.y₃⁷ + y₂⁴.x₁.x₂² + y₂⁴.x₁².x₂ + y₂⁴.y₃⁴.x₁ + y₂⁴.y₃⁵.y₄ + y₂⁴.y₃⁶ + y₂⁵.y₃⁴.y₄ + y₂⁵.y₃⁵ + y₂⁶.y₃³.y₄ + y₂⁶.y₃⁴ + y₂⁷.y₃².y₄ + y₂⁷.y₃³ + y₂⁸.y₃² + y₁.x₂³.w + y₁.x₁³.w + y₁.y₃².x₂².w + y₁.y₃⁴.x₂.w + y₁.y₃⁶.w
• u₁.u₂ = x₂².t + x₁².x₂³ + x₁⁴.x₂ + y₃.x₂².u₂ + y₃.x₁.x₂.u₂ + y₃.x₁.x₂².w + y₃.x₁².u₂ + y₃.x₁³.w + y₃².x₁.t + y₃².x₁².x₂² + y₃².x₁⁴ + y₃².y₄.x₁.u₁ + y₃².y₄².x₁³ + y₃³.x₂².w + y₃³.x₁.u₁ + y₃³.y₄.x₁³ + y₃³.y₄³.x₁² + y₃⁴.x₁.x₂² + y₃⁴.x₁².x₂ + y₃⁴.x₁³ + y₃⁴.y₄².x₁² + y₃⁵.x₁.w + y₂.x₂³.w + y₂.x₁.x₂.u₂ + y₂.x₁³.w + y₂.y₃.y₄.x₁.u₁ + y₂.y₃³.y₄.u₁ + y₂.y₃⁴.u₁ + y₂.y₃⁴.y₄.x₁² + y₂.y₃⁵.x₁² + y₂.y₃⁸.y₄ + y₂.y₃⁹ + y₂².x₂.t + y₂².x₁.t + y₂².x₁.x₂³ + y₂².y₄².x₁³ + y₂².y₃.x₁.u₁ + y₂².y₃².y₄.u₁ + y₂².y₃³.u₁ + y₂².y₃⁵.y₄.x₁ + y₂².y₃⁶.x₁ + y₂³.x₂².w + y₂³.x₁².w + y₂³.y₃.y₄.u₁ + y₂³.y₃².u₁ + y₂³.y₃².y₄.x₁² + y₂³.y₃³.x₁² + y₂³.y₃⁶.y₄ + y₂³.y₃⁷ + y₂⁴.x₁.x₂² + y₂⁴.x₁².x₂ + y₂⁴.y₃.u₁ + y₂⁴.y₃².x₁² + y₂⁴.y₃³.y₄.x₁ + y₂⁵.y₃².y₄.x₁ + y₂⁵.y₃⁴.y₄ + y₂⁵.y₃⁵ + y₂⁶.y₃².x₁ + y₂⁶.y₃³.y₄ + y₂⁶.y₃⁴ + y₂⁷.y₃².y₄ + y₂⁷.y₃³ + y₂⁸.y₃² + y₁.y₃².x₁².w + y₁.y₃⁴.x₂.w + y₁.y₃⁴.x₁.w + y₁.y₃⁶.w + y₁.y₂.x₁⁴ + y₁.y₂².x₂².w + y₁.y₂⁴.x₂.w + y₁.y₂⁴.x₁.w
• u₁² = x₁.x₂⁴ + x₁².x₂³ + x₁³.x₂² + x₁⁵ + y₄.x₁².u₁ + y₄².x₁⁴ + y₄³.x₁.u₁ + y₄⁴.x₁³ + y₃.y₄⁴.u₁ + y₃.y₄⁷.x₁ + y₃².x₁.x₂³ + y₃².x₁³.x₂ + y₃².y₄.x₁.u₁ + y₃².y₄².x₁³ + y₃².y₄³.u₁ + y₃³.y₄.x₁³ + y₃³.y₄².u₁ + y₃⁴.x₁².x₂ + y₃⁴.x₁³ + y₃⁴.y₄².x₁² + y₃⁵.y₄.x₁² + y₃⁵.y₄³.x₁ + y₃⁶.x₁² + y₃⁶.y₄².x₁ + y₂.y₄.x₁⁴ + y₂.y₄².x₁.u₁ + y₂.y₄³.x₁³ + y₂.y₄⁴.u₁ + y₂.y₃.y₄.x₁.u₁ + y₂.y₃⁸.y₄ + y₂².x₁.x₂³ + y₂².x₁².x₂² + y₂².x₁³.x₂ + y₂².x₁⁴ + y₂².y₄.x₁.u₁ + y₂².y₃.y₄.x₁³ + y₂².y₃⁴.x₁² + y₂².y₃⁷.y₄ + y₂².y₃⁸ + y₂³.y₄.x₁³ + y₂³.y₄².u₁ + y₂³.y₃.x₁³ + y₂³.y₃⁵.x₁ + y₂³.y₃⁷ + y₂⁴.x₁².x₂ + y₂⁴.x₁³ + y₂⁴.y₃².x₁² + y₂⁴.y₃⁴.x₁ + y₂⁵.y₃.x₁² + y₂⁵.y₃².y₄.x₁ + y₂⁵.y₃⁴.y₄ + y₂⁵.y₃⁵ + y₂⁶.y₃.y₄.x₁ + y₂⁶.y₃³.y₄ + y₂⁸.y₃.y₄ + y₂⁸.y₃² + y₁.x₂³.w + y₁.y₃².x₁².w + y₁.y₃⁴.x₂.w + y₁.y₃⁴.x₁.w + y₁.y₂².x₂².w + y₁.y₂².x₁².w + y₁.y₂³.x₁³ + y₁.y₂⁴.x₂.w + y₁.y₂⁵.x₁² + y₄².r
• u₂.t = x₁.x₂³.w + x₁³.x₂.w + y₃.x₁.x₂⁴ + y₃.x₁².x₂³ + y₃.x₁³.x₂² + y₃.x₁⁴.x₂ + y₃².x₁².u₂ + y₃².x₁³.w + y₃².y₄³.x₁³ + y₃³.x₂.t + y₃³.x₁³.x₂ + y₃³.x₁⁴ + y₃³.y₄.x₁.u₁ + y₃³.y₄⁴.x₁² + y₃⁴.x₂.u₂ + y₃⁴.x₁.x₂.w + y₃⁴.x₁².w + y₃⁴.y₄³.x₁² + y₃⁵.x₁.x₂² + y₃⁵.x₁².x₂ + y₃⁵.x₁³ + y₃⁶.x₂.w + y₃⁶.y₄.x₁² + y₃⁷.x₁.x₂ + y₂.x₂².t + y₂.x₁.x₂.t + y₂.x₁².x₂³ + y₂.x₁⁴.x₂ + y₂.y₃².x₁⁴ + y₂.y₃³.x₁.u₁ + y₂.y₃⁴.x₁³ + y₂.y₃⁶.x₁² + y₂.y₃⁷.y₄.x₁ + y₂.y₃⁸.x₁ + y₂.y₃⁹.y₄ + y₂.y₃¹⁰ + y₂².x₂³.w + y₂².x₁.x₂².w + y₂².x₁².x₂.w + y₂².y₃².y₄.x₁³ + y₂².y₃³.x₁³ + y₂².y₃³.y₄.u₁ + y₂².y₃⁵.x₁² + y₂³.x₂.t + y₂³.x₁.x₂³ + y₂³.x₁².x₂² + y₂³.x₁³.x₂ + y₂³.y₃².x₁³ + y₂³.y₃³.u₁ + y₂³.y₃³.y₄.x₁² + y₂³.y₃⁷.y₄ + y₂³.y₃⁸ + y₂⁴.x₂².w + y₂⁴.y₃.y₄.u₁ + y₂⁴.y₃².y₄.x₁² + y₂⁴.y₃⁴.y₄.x₁ + y₂⁴.y₃⁶.y₄ + y₂⁴.y₃⁷ + y₂⁵.x₁.x₂² + y₂⁵.y₃.u₁ + y₂⁶.y₃⁵ + y₁.y₃.x₂³.w + y₁.y₃.x₁³.w + y₁.y₃².x₁⁴ + y₁.y₃³.x₂².w + y₁.y₃³.x₁².w + y₁.y₃⁴.x₁³ + y₁.y₂³.x₂².w + y₁.y₂³.x₁².w + y₁.y₂⁵.x₁.w + y₁.y₂⁶.x₁²
• u₁.t = x₁.x₂².u₂ + x₁².x₂.u₂ + x₁².x₂².w + x₁⁴.w + y₄².x₁².u₁ + y₃.x₂².t + y₃.x₁.x₂.t + y₃.x₁².t + y₃.y₄.x₁².u₁ + y₃.y₄⁴.x₁³ + y₃².x₂².u₂ + y₃².x₁.x₂.u₂ + y₃².x₁.x₂².w + y₃².x₁².u₂ + y₃².x₁³.w + y₃².y₄².x₁.u₁ + y₃².y₄³.x₁³ + y₃².y₄⁴.u₁ + y₃².y₄⁷.x₁ + y₃³.x₁.t + y₃³.x₁.x₂³ + y₃³.x₁³.x₂ + y₃³.y₄.x₁.u₁ + y₃³.y₄².x₁³ + y₃³.y₄³.u₁ + y₃⁴.x₂².w + y₃⁴.x₁.u₂ + y₃⁴.x₁.x₂.w + y₃⁴.y₄².u₁ + y₃⁴.y₄³.x₁² + y₃⁵.x₁².x₂ + y₃⁵.x₁³ + y₃⁶.y₄³.x₁ + y₃⁷.y₄².x₁ + y₂.x₁.x₂⁴ + y₂.x₁².t + y₂.x₁².x₂³ + y₂.x₁³.x₂² + y₂.x₁⁴.x₂ + y₂.y₄².x₁⁴ + y₂.y₃.x₁².u₁ + y₂.y₃.y₄.x₁⁴ + y₂.y₃².x₁⁴ + y₂.y₃².y₄.x₁.u₁ + y₂.y₃³.x₁.u₁ + y₂.y₃⁴.x₁³ + y₂.y₃⁸.x₁ + y₂.y₃¹⁰ + y₂².x₁.x₂.u₂ + y₂².x₁.x₂².w + y₂².x₁².u₂ + y₂².x₁².x₂.w + y₂².x₁³.w + y₂².y₃.y₄.x₁.u₁ + y₂².y₃².y₄.x₁³ + y₂².y₃³.y₄.u₁ + y₂².y₃⁵.x₁² + y₂².y₃⁶.y₄.x₁ + y₂².y₃⁷.x₁ + y₂².y₃⁹ + y₂³.x₁.t + y₂³.x₁.x₂³ + y₂³.y₄².x₁³ + y₂³.y₃.x₁.u₁ + y₂³.y₃².x₁³ + y₂³.y₃³.u₁ + y₂³.y₃³.y₄.x₁² + y₂³.y₃⁴.x₁² + y₂³.y₃⁵.y₄.x₁ + y₂³.y₃⁸ + y₂⁴.x₁.x₂.w + y₂⁴.y₃.x₁³ + y₂⁴.y₃.y₄.u₁ + y₂⁴.y₃².u₁ + y₂⁴.y₃⁴.y₄.x₁ + y₂⁴.y₃⁵.x₁ + y₂⁴.y₃⁶.y₄ + y₂⁵.x₁².x₂ + y₂⁵.y₃.u₁ + y₂⁵.y₃².x₁² + y₂⁵.y₃⁵.y₄ + y₂⁵.y₃⁶ + y₂⁶.y₃².y₄.x₁ + y₂⁶.y₃³.x₁ + y₂⁶.y₃⁵ + y₂⁷.y₃.y₄.x₁ + y₂⁷.y₃³.y₄ + y₂⁸.y₃.x₁ + y₂⁸.y₃³ + y₁.y₃.x₁³.w + y₁.y₃⁴.x₁³ + y₁.y₃⁵.x₁.w + y₁.y₃⁷.w + y₁.y₂.x₂³.w + y₁.y₂.x₁³.w + y₁.y₂³.x₂².w + y₁.y₂⁴.x₁³ + y₁.y₂⁵.x₂.w + y₁.y₂⁶.x₁² + y₃.y₄².r + y₂.y₃.y₄.r
• t² = x₁².x₂⁴ + x₁³.x₂³ + x₁⁴.x₂² + x₁⁵.x₂ + y₄⁴.x₁⁴ + y₃².x₁.x₂⁴ + y₃².x₁².x₂³ + y₃².x₁³.x₂² + y₃².x₁⁵ + y₃².y₄.x₁².u₁ + y₃².y₄².x₁⁴ + y₃².y₄³.x₁.u₁ + y₃².y₄⁴.x₁³ + y₃².y₄⁶.x₁² + y₃³.y₄⁴.u₁ + y₃³.y₄⁷.x₁ + y₃⁴.x₁².x₂² + y₃⁴.x₁³.x₂ + y₃⁴.x₁⁴ + y₃⁴.y₄.x₁.u₁ + y₃⁴.y₄².x₁³ + y₃⁴.y₄³.u₁ + y₃⁴.y₄⁴.x₁² + y₃⁵.y₄.x₁³ + y₃⁵.y₄².u₁ + y₃⁶.x₁².x₂ + y₃⁶.x₁³ + y₃⁶.y₄².x₁² + y₃⁷.y₄.x₁² + y₃⁷.y₄³.x₁ + y₃⁸.x₁.x₂ + y₃⁸.y₄².x₁ + y₂.y₃.y₄.x₁².u₁ + y₂.y₃².x₁².u₁ + y₂.y₃².y₄.x₁⁴ + y₂.y₃³.y₄.x₁.u₁ + y₂.y₃⁴.x₁.u₁ + y₂.y₃⁵.x₁³ + y₂.y₃⁵.y₄.u₁ + y₂.y₃⁶.y₄.x₁² + y₂.y₃⁷.x₁² + y₂.y₃¹⁰.y₄ + y₂².x₁.x₂⁴ + y₂².x₁⁴.x₂ + y₂².y₃.x₁².u₁ + y₂².y₃.y₄.x₁⁴ + y₂².y₃².x₁⁴ + y₂².y₃³.x₁.u₁ + y₂².y₃⁴.x₁³ + y₂².y₃⁴.y₄.u₁ + y₂².y₃⁵.y₄.x₁² + y₂².y₃⁹.y₄ + y₂².y₃¹⁰ + y₂³.y₃.x₁⁴ + y₂³.y₃.y₄.x₁.u₁ + y₂³.y₃².x₁.u₁ + y₂³.y₃².y₄.x₁³ + y₂³.y₃⁴.y₄.x₁² + y₂³.y₃⁵.x₁² + y₂³.y₃⁷.x₁ + y₂³.y₃⁸.y₄ + y₂³.y₃⁹ + y₂⁴.x₁.x₂³ + y₂⁴.x₁².x₂² + y₂⁴.y₃.x₁.u₁ + y₂⁴.y₃².y₄.u₁ + y₂⁴.y₃³.u₁ + y₂⁴.y₃⁴.x₁² + y₂⁴.y₃⁶.x₁ + y₂⁵.y₃².u₁ + y₂⁵.y₃².y₄.x₁² + y₂⁵.y₃⁶.y₄ + y₂⁶.y₃.y₄.x₁² + y₂⁶.y₃³.y₄.x₁ + y₂⁶.y₃⁵.y₄ + y₂⁷.y₃.x₁² + y₂⁷.y₃².y₄.x₁ + y₂⁷.y₃⁴.y₄ + y₂⁷.y₃⁵ + y₂⁸.y₃².x₁ + y₂⁸.y₃⁴ + y₁.y₃².x₂³.w + y₁.y₃².x₁³.w + y₁.y₃⁴.x₁².w + y₁.y₃⁶.x₂.w + y₁.y₂².x₂³.w + y₁.y₂².x₁³.w + y₁.y₂⁴.x₁².w + y₁.y₂⁵.x₁³ + y₁.y₂⁶.x₁.w + y₁.y₂⁷.x₁² + y₃².y₄².r + y₂².y₃².r

This minimal generating set constitutes a Gröbner basis for the relations ideal.

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 12 onwards.

This cohomology ring has dimension 4 and depth 3. Here is a homogeneous system of parameters:

• h₁ = r in degree 8
• h₂ = y₃² + y₂² in degree 2
• h₃ = x₂ + y₄² + y₂.y₃ in degree 2
• h₄ = x₁ in degree 2

The first 3 terms h₁, h₂, h₃ 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, -1, 8, 10.
• 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
• y₃ in degree 1
• y₂ in degree 1
• y₁ in degree 1
• y₄² in degree 2
• y₃.y₄ in degree 2
• y₂.y₄ in degree 2
• y₂.y₃ in degree 2
• y₂² in degree 2
• y₁.y₃ in degree 2
• y₁.y₂ in degree 2
• w in degree 3
• y₃.y₄² in degree 3
• y₂.y₄² in degree 3
• y₂.y₃.y₄ in degree 3
• y₂².y₄ in degree 3
• y₂².y₃ in degree 3
• y₂³ in degree 3
• y₁.y₂² in degree 3
• y₃.w in degree 4
• y₂.w in degree 4
• y₂².y₄² in degree 4
• y₂².y₃.y₄ in degree 4
• y₂³.y₄ in degree 4
• y₂³.y₃ in degree 4
• y₁.w in degree 4
• u₂ in degree 5
• u₁ in degree 5
• y₂².w in degree 5
• y₂³.y₃.y₄ in degree 5
• y₁.y₂.w in degree 5
• t in degree 6
• y₄.u₁ in degree 6
• y₃.u₂ in degree 6
• y₃.u₁ in degree 6
• y₂.u₂ in degree 6
• y₂.u₁ in degree 6
• y₃.t in degree 7
• y₃.y₄.u₁ in degree 7
• y₂.t in degree 7
• y₂.y₄.u₁ in degree 7
• y₂.y₃.u₁ in degree 7
• y₂².u₂ in degree 7
• y₂².u₁ in degree 7
• y₂.y₃.y₄.u₁ in degree 8
• y₂².t in degree 8
• y₂².y₄.u₁ in degree 8
• y₂².y₃.u₁ in degree 8
• y₂³.u₁ in degree 8
• y₂².y₃.y₄.u₁ in degree 9
• y₂³.y₄.u₁ in degree 9
• y₂³.y₃.u₁ in degree 9
• y₂³.y₃.y₄.u₁ in degree 10

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

• y₁.y₃.h + y₁.y₂.h in degree 4
• y₁.y₂².h in degree 5
• y₁.w.h + y₁.y₂.h² in degree 6
• y₁.y₂.w.h + y₁.y₂².h² in degree 7
• y₁.h³ in degree 7
• y₁.y₂.h³ in degree 8

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
• w restricts to 0
• u₁ restricts to 0
• u₂ restricts to 0
• t restricts to 0
• r restricts to y⁸

Restriction to special subgroup number 2, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₃
• y₃ restricts to 0
• y₄ restricts to y₂
• x₁ restricts to y₄² + y₂.y₄
• x₂ restricts to 0
• w restricts to 0
• u₁ restricts to y₄⁵ + y₃.y₄⁴ + y₃².y₄³ + y₂.y₄⁴ + y₂.y₃.y₄³ + y₂².y₄³ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + y₁⁴.y₂
• u₂ restricts to 0
• t restricts to y₂².y₄⁴ + y₂⁴.y₄²
• r restricts to y₄⁸ + y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₂.y₄⁷ + y₂.y₃².y₄⁵ + y₂.y₃³.y₄⁴ + y₂.y₃⁴.y₄³ + y₂².y₄⁶ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂³.y₄⁵ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂⁴.y₃³.y₄ + y₂⁵.y₃².y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂².y₃⁵ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃³ + y₁.y₂⁵.y₃.y₄ + y₁.y₂⁵.y₃² + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂.y₃⁵ + y₁².y₂².y₄⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂³.y₃².y₄ + y₁².y₂⁵.y₄ + y₁².y₂⁵.y₃ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂.y₃³ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₄ + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 3, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₃
• y₄ restricts to y₂
• x₁ restricts to y₄² + y₂.y₄
• x₂ restricts to 0
• w restricts to y₃.y₄² + y₂.y₃.y₄
• u₁ restricts to y₄⁵ + y₃².y₄³ + y₃³.y₄² + y₂.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₄³ + y₂³.y₄² + y₂³.y₃.y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + y₁⁴.y₂
• u₂ restricts to y₂.y₃².y₄² + y₂².y₃².y₄
• t restricts to y₃.y₄⁵ + y₃².y₄⁴ + y₃³.y₄³ + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂².y₄⁴ + y₂².y₃.y₄³ + y₂⁴.y₄² + y₂⁴.y₃.y₄ + y₁.y₂².y₃³ + y₁.y₂³.y₃² + y₁².y₂.y₃³ + y₁².y₂².y₃² + y₁².y₂³.y₃ + y₁⁴.y₂.y₃
• r restricts to y₄⁸ + y₃.y₄⁷ + y₃².y₄⁶ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₂.y₄⁷ + y₂.y₃³.y₄⁴ + y₂.y₃⁵.y₄² + y₂².y₄⁶ + y₂².y₃².y₄⁴ + y₂².y₃⁴.y₄² + y₂³.y₄⁵ + y₂³.y₃².y₄³ + y₂⁴.y₃.y₄³ + y₂⁴.y₃³.y₄ + y₂⁵.y₃.y₄² + y₂⁵.y₃².y₄ + y₂⁶.y₃.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂².y₃⁵ + y₁.y₂³.y₃³.y₄ + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁.y₂⁵.y₃² + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃³.y₄² + y₁².y₂.y₃⁴.y₄ + y₁².y₂.y₃⁵ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃³.y₄ + y₁².y₂².y₃⁴ + y₁².y₂³.y₃³ + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₄ + y₁².y₂⁵.y₃ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂.y₃³ + y₁⁴.y₂³.y₄ + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₂
• y₃ restricts to 0
• y₄ restricts to 0
• x₁ restricts to y₃²
• x₂ restricts to y₄²
• w restricts to y₃.y₄² + y₃².y₄ + y₂.y₃.y₄
• u₁ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁵ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂.y₃⁴ + y₂².y₃².y₄ + y₂².y₃³
• u₂ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄
• t restricts to y₃².y₄⁴ + y₃³.y₄³ + y₃⁴.y₄² + y₃⁵.y₄ + y₂.y₃.y₄⁴ + y₂.y₃⁴.y₄ + y₂².y₃.y₄³ + y₂².y₃².y₄²
• r restricts to y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁷.y₄ + y₃⁸ + y₂.y₃.y₄⁶ + y₂.y₃⁴.y₄³ + y₂².y₃³.y₄³ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + 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

Restriction to special subgroup number 5, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₂
• y₄ restricts to 0
• x₁ restricts to y₃²
• x₂ restricts to y₄²
• w restricts to y₃.y₄² + y₃².y₄ + y₂.y₃.y₄ + y₂.y₃²
• u₁ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁵ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₃².y₄ + y₂².y₃³ + y₂³.y₃²
• u₂ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄
• t restricts to y₃².y₄⁴ + y₃³.y₄³ + y₃⁴.y₄² + y₃⁵.y₄ + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁵ + y₂².y₃².y₄² + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂³.y₃².y₄ + y₂³.y₃³ + y₂⁴.y₃.y₄
• r restricts to y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁷.y₄ + y₃⁸ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂.y₃⁷ + y₂².y₃.y₄⁵ + y₂².y₃².y₄⁴ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂².y₃⁶ + y₂³.y₃².y₄³ + y₂³.y₃⁴.y₄ + y₂⁴.y₃⁴ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + 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

Restriction to special subgroup number 6, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₃
• y₃ restricts to y₄
• y₄ restricts to y₃
• x₁ restricts to y₂.y₃ + y₂²
• x₂ restricts to y₄²
• w restricts to 0
• u₁ restricts to y₃².y₄³ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂².y₄³ + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₂⁵ + y₁.y₃².y₄² + y₁.y₃³.y₄ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁².y₃³ + y₁⁴.y₃
• u₂ restricts to y₃.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂².y₄³ + y₂².y₃.y₄²
• t restricts to y₃².y₄⁴ + y₃³.y₄³ + y₂².y₃⁴ + y₂⁴.y₃²
• r restricts to y₃⁵.y₄³ + y₃⁶.y₄² + y₃⁷.y₄ + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂².y₄⁶ + y₂².y₃³.y₄³ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂³.y₄⁵ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁵.y₄³ + y₂⁵.y₃³ + y₂⁶.y₃² + y₂⁷.y₃ + y₂⁸ + y₁.y₃².y₄⁵ + y₁.y₃⁵.y₄² + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁵.y₄ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃³.y₄² + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₃.y₄⁵ + y₁².y₃².y₄⁴ + y₁².y₃³.y₄³ + y₁².y₃⁴.y₄² + y₁².y₃⁵.y₄ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃³.y₄² + y₁².y₂.y₃⁵ + y₁².y₂².y₄⁴ + y₁².y₂².y₃³.y₄ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃.y₄³ + y₁⁴.y₃³.y₄ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂.y₃³ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 7, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₄
• y₃ restricts to y₃
• y₄ restricts to y₃
• x₁ restricts to y₂.y₃ + y₂²
• x₂ restricts to y₄²
• w restricts to y₂.y₃.y₄ + y₂.y₃² + y₂².y₄ + y₂².y₃
• u₁ restricts to y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₄³ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂².y₃³ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₂⁴.y₄ + y₂⁴.y₃ + y₂⁵ + y₁.y₃².y₄² + y₁.y₃³.y₄ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁².y₃³ + y₁⁴.y₃
• u₂ restricts to y₃.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃⁴ + y₂².y₄³ + y₂².y₃³
• t restricts to y₃⁴.y₄² + y₃⁵.y₄ + y₂.y₃².y₄³ + y₂.y₃⁵ + y₂².y₃³.y₄ + y₂³.y₄³ + y₂³.y₃³ + y₂⁴.y₃² + y₂⁵.y₄ + y₂⁵.y₃ + y₁.y₃².y₄³ + y₁.y₃⁴.y₄ + y₁².y₃.y₄³ + y₁².y₃⁴ + y₁⁴.y₃.y₄ + y₁⁴.y₃²
• r restricts to y₃².y₄⁶ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₂.y₃.y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃³.y₄⁴ + y₂.y₃⁴.y₄³ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂.y₃⁷ + y₂².y₄⁶ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂².y₃⁶ + y₂³.y₄⁵ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + y₂⁶.y₃.y₄ + y₂⁷.y₄ + y₂⁸ + y₁.y₃².y₄⁵ + y₁.y₃³.y₄⁴ + y₁.y₃⁴.y₄³ + y₁.y₃⁶.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃³.y₄³ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃².y₄³ + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₃.y₄⁵ + y₁².y₃⁴.y₄² + y₁².y₃⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃².y₄³ + y₁².y₂.y₃³.y₄² + y₁².y₂².y₄⁴ + y₁².y₂².y₃.y₄³ + y₁².y₂².y₃³.y₄ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃.y₄³ + y₁⁴.y₃².y₄² + y₁⁴.y₃³.y₄ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

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Poincaré series

(1 + 4t + 7t² + 8t³ + 7t⁴ + 5t⁵ + 5t⁶ + 6t⁷ + 4t⁸ + t⁹) / (1 - t²)³ (1 - t⁸)

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Back to the groups of order 128