Small group number 2264 of order 128

G is the group 128gp2264

G has 5 minimal generators, rank 5 and exponent 4. The centre has rank 2.

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

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
• y₅ in degree 1
• w in degree 3
• v₁ in degree 4
• v₂ in degree 4
• v₃ in degree 4, a regular element
• v₄ in degree 4, a regular element
• t in degree 6

There are 22 minimal relations:

• y₄.y₅ = y₁.y₅ + y₁.y₃
• y₃.y₅ = y₂.y₅ + y₁.y₂
• y₁.y₃.y₄ = y₁.y₂.y₄ + y₁².y₂
• y₁.y₂.y₅ = y₁.y₂.y₃ + y₁².y₂
• y₅.v₂ = y₅².w + y₂².y₅³ + y₁.y₅.w + y₁.y₅⁴ + y₁².y₅³ + y₁³.y₂.y₃
• y₅.v₁ = y₅².w + y₂.y₅.w + y₁.y₅.w + y₁.y₂.w + y₁².y₅³ + y₁³.y₅² + y₁³.y₂.y₃ + y₁³.y₂² + y₁⁴.y₂
• y₄.v₂ = y₄².w + y₃.v₂ + y₃.y₄.w + y₃².w + y₂.v₁ + y₂³.y₅² + y₁.y₅.w + y₁².y₅³ + y₁³.y₅²
• y₁.v₂ = y₁.y₅.w + y₁.y₄.w + y₁.y₂³.y₃ + y₁².y₅³ + y₁³.y₅² + y₁³.y₂.y₃ + y₁³.y₂²
• y₁.v₁ = y₁.y₅.w + y₁.y₄.w + y₁³.y₅² + y₁³.y₂² + y₁⁴.y₅ + y₁⁴.y₃ + y₁⁴.y₂
• y₁.y₃.w = y₁³.y₂.y₃
• w² = y₃.y₄².w + y₃³.w + y₂.y₄².w + y₂.y₃.y₄.w + y₂³.w + y₁.y₅².w + y₁.y₂².w + y₁².y₅⁴ + y₁⁴.y₂² + y₁⁵.y₅ + y₁⁵.y₃ + y₅².v₄ + y₄².v₃ + y₃².v₃ + y₂².v₄ + y₂².v₃
• w.v₂ = y₃.y₄³.w + y₃⁴.w + y₂.t + y₂.y₄³.w + y₂.y₃².v₂ + y₂².y₅².w + y₂².y₅⁵ + y₂².y₄².w + y₂².y₃.v₁ + y₂².y₃².w + y₂³.v₂ + y₂³.v₁ + y₂³.y₅⁴ + y₂³.y₄.w + y₂³.y₃.w + y₂⁴.w + y₂⁴.y₅³ + y₁.y₂³.w + y₁².y₅⁵ + y₁².y₂².w + y₁².y₂⁵ + y₁³.y₅⁴ + y₁³.y₂⁴ + y₁⁴.y₂³ + y₁⁵.y₅² + y₁⁶.y₅ + y₁⁶.y₃ + y₅³.v₄ + y₄³.v₃ + y₃³.v₃ + y₂.y₅².v₃ + y₂.y₃.y₄.v₄ + y₂².y₅.v₃ + y₂².y₄.v₄ + y₂².y₄.v₃ + y₂³.v₃ + y₁.y₅².v₄ + y₁.y₂.y₄.v₄ + y₁.y₂.y₃.v₄ + y₁.y₂.y₃.v₃ + y₁.y₂².v₃ + y₁².y₅.v₃ + y₁².y₃.v₃ + y₁².y₂.v₃
• w.v₁ = y₄.t + y₃.t + y₃.y₄³.w + y₃².y₄².w + y₃⁴.w + y₂.y₄³.w + y₂.y₃.y₄.v₁ + y₂.y₃.y₄².w + y₂.y₃².y₄.w + y₂.y₃³.w + y₂².y₅⁵ + y₂².y₄.v₁ + y₂².y₄².w + y₂².y₃.v₁ + y₂².y₃².w + y₂³.v₁ + y₂³.y₅⁴ + y₂³.y₄.w + y₂³.y₃.w + y₂⁴.y₅³ + y₁.y₅³.w + y₁.y₂⁵.y₃ + y₁².y₅².w + y₁².y₂⁵ + y₁³.y₄.w + y₁⁴.y₂³ + y₁⁵.y₂.y₃ + y₁⁵.y₂² + y₁⁶.y₂ + y₅³.v₄ + y₃.y₄².v₄ + y₃².y₄.v₄ + y₂.y₅².v₄ + y₂.y₅².v₃ + y₂.y₃².v₄ + y₂.y₃².v₃ + y₂².y₅.v₃ + y₂².y₄.v₃ + y₂².y₃.v₃ + y₁.y₅².v₄ + y₁.y₅².v₃ + y₁.y₄².v₄ + y₁.y₂.y₄.v₄ + y₁.y₂.y₄.v₃ + y₁.y₂².v₃ + y₁².y₅.v₃ + y₁².y₃.v₃ + y₁².y₂.v₄ + y₁².y₂.v₃
• y₅.t = y₂.y₅⁶ + y₂².y₅⁵ + y₂³.y₅⁴ + y₁.y₅³.w + y₁.y₅⁶ + y₁.y₂³.w + y₁.y₂⁵.y₃ + y₁².y₅².w + y₁².y₅⁵ + y₁³.y₅.w + y₁³.y₂⁴ + y₁⁴.y₅³ + y₁⁵.y₅² + y₁⁵.y₂² + y₅³.v₃ + y₂.y₅².v₄ + y₂.y₅².v₃ + y₁.y₂.y₄.v₄ + y₁.y₂.y₄.v₃ + y₁.y₂².v₃ + y₁².y₅.v₄ + y₁².y₅.v₃ + y₁².y₃.v₄ + y₁².y₃.v₃ + y₁².y₂.v₄ + y₁².y₂.v₃
• y₁.t = y₁.y₂³.w + y₁.y₂⁵.y₃ + y₁².y₅².w + y₁².y₅⁵ + y₁³.y₅.w + y₁³.y₅⁴ + y₁³.y₂⁴ + y₁⁴.w + y₁⁴.y₂³ + y₁⁵.y₅² + y₁⁶.y₅ + y₁⁶.y₃ + y₁⁶.y₂ + y₁.y₅².v₃ + y₁.y₄².v₃ + y₁.y₂.y₄.v₄ + y₁.y₂.y₄.v₃ + y₁.y₂.y₃.v₄ + y₁.y₂².v₃ + y₁².y₄.v₄ + y₁².y₃.v₃
• v₂² = y₃.y₄⁴.w + y₃³.y₄².w + y₃⁵.w + y₂.y₄⁴.w + y₂.y₃.y₄³.w + y₂.y₃³.v₂ + y₂.y₃³.y₄.w + y₂.y₃⁴.w + y₂².y₃.y₄².w + y₂².y₃².v₂ + y₂².y₃².v₁ + y₂².y₃².y₄.w + y₂³.y₃.v₁ + y₂⁴.y₅⁴ + y₁.y₅⁴.w + y₁.y₂⁶.y₃ + y₁³.y₅².w + y₁⁴.y₂.w + y₁⁵.y₅³ + y₁⁶.y₂.y₃ + y₁⁷.y₅ + y₁⁷.y₃ + y₅⁴.v₄ + y₄⁴.v₃ + y₃².y₄².v₃ + y₃⁴.v₃ + y₂².y₅².v₄ + y₂².y₄².v₄ + y₂².y₃².v₃ + y₁².y₅².v₄ + y₁².y₅².v₃
• v₁.v₂ = y₄².t + y₃.y₄.t + y₃.y₄⁴.w + y₃².t + y₃².y₄³.w + y₃³.y₄².w + y₃⁴.v₂ + y₂.y₄⁴.w + y₂.y₃.y₄².v₁ + y₂.y₃².y₄.v₁ + y₂.y₃².y₄².w + y₂.y₃³.v₁ + y₂.y₃³.y₄.w + y₂².y₅³.w + y₂².y₄².v₁ + y₂².y₃.y₄².w + y₂².y₃².v₂ + y₂².y₃².v₁ + y₂².y₃².y₄.w + y₂².y₃³.w + y₂³.y₅².w + y₂³.y₅⁵ + y₂³.y₄.v₁ + y₂³.y₄².w + y₂³.y₃.y₄.w + y₂³.y₃².w + y₂⁴.y₅⁴ + y₂⁵.y₅³ + y₁².y₅³.w + y₁².y₅⁶ + y₁².y₂³.w + y₁³.y₅².w + y₁³.y₄².w + y₁³.y₂².w + y₁⁴.y₂.w + y₁⁴.y₂⁴ + y₁⁶.y₅² + y₁⁶.y₂.y₃ + y₁⁷.y₂ + y₅⁴.v₄ + y₃.y₄³.v₄ + y₃².y₄².v₄ + y₃³.y₄.v₄ + y₂.y₅³.v₄ + y₂.y₄³.v₃ + y₂.y₃.y₄².v₃ + y₂.y₃².y₄.v₄ + y₂.y₃³.v₃ + y₂².y₅².v₄ + y₂².y₅².v₃ + y₂².y₄².v₃ + y₂².y₃.y₄.v₃ + y₂².y₃².v₃ + y₂³.y₅.v₃ + y₁.y₄³.v₄ + y₁.y₂².y₃.v₃ + y₁.y₂³.v₃ + y₁².y₅².v₄ + y₁².y₂.y₃.v₃ + y₁².y₂².v₄ + y₁².y₂².v₃
• v₁² = y₃.y₄⁴.w + y₃².y₄².v₁ + y₃³.y₄.v₁ + y₃³.y₄².w + y₃⁴.y₄.w + y₃⁵.w + y₂.y₄⁴.w + y₂.y₃.y₄².v₁ + y₂.y₃².y₄.v₁ + y₂.y₃².y₄².w + y₂.y₃⁴.w + y₁.y₅⁴.w + y₁².y₅⁶ + y₁³.y₅².w + y₁⁴.y₂.w + y₁⁴.y₂⁴ + y₁⁵.y₅³ + y₁⁶.y₅² + y₁⁶.y₂² + y₁⁷.y₅ + y₁⁷.y₃ + y₅⁴.v₄ + y₄⁴.v₃ + y₃⁴.v₄ + y₁².y₅².v₄ + y₁².y₅².v₃
• w.t = y₃.y₄².t + y₃³.t + y₂.y₅⁵.w + y₂.y₄².t + y₂.y₃².y₄².v₁ + y₂.y₃².y₄³.w + y₂.y₃³.y₄.v₁ + y₂.y₃⁴.v₂ + y₂.y₃⁴.y₄.w + y₂.y₃⁵.w + y₂².y₅⁴.w + y₂².y₄.t + y₂².y₃.t + y₂².y₃².y₄.v₁ + y₂².y₃².y₄².w + y₂².y₃³.y₄.w + y₂².y₃⁴.w + y₂³.y₅³.w + y₂³.y₄².v₁ + y₂³.y₄³.w + y₂³.y₃.y₄².w + y₂³.y₃².v₂ + y₂³.y₃².v₁ + y₂³.y₃².y₄.w + y₂³.y₃³.w + y₂⁴.y₃.y₄.w + y₂⁵.v₁ + y₂⁶.w + y₁.y₅⁵.w + y₁².y₂⁴.w + y₁².y₂⁷ + y₁³.y₅³.w + y₁³.y₅⁶ + y₁³.y₂³.w + y₁⁴.y₅⁵ + y₁⁴.y₂².w + y₁⁵.y₅.w + y₁⁵.y₅⁴ + y₁⁶.y₅³ + y₁⁶.y₂³ + y₁⁷.y₅² + y₁⁸.y₅ + y₁⁸.y₃ + y₅².w.v₃ + y₄.v₁.v₃ + y₃.v₁.v₃ + y₃.y₄.w.v₄ + y₃.y₄⁴.v₃ + y₃².y₄³.v₄ + y₃³.y₄².v₃ + y₃⁴.y₄.v₄ + y₃⁴.y₄.v₃ + y₃⁵.v₃ + y₂.v₂.v₄ + y₂.v₂.v₃ + y₂.y₅.w.v₃ + y₂.y₄.w.v₄ + y₂.y₄.w.v₃ + y₂.y₄⁴.v₃ + y₂.y₃.y₄³.v₄ + y₂.y₃³.y₄.v₄ + y₂.y₃³.y₄.v₃ + y₂.y₃⁴.v₄ + y₂².w.v₃ + y₂².y₄³.v₃ + y₂².y₃.y₄².v₃ + y₂².y₃³.v₄ + y₂².y₃³.v₃ + y₂³.y₅².v₄ + y₂³.y₅².v₃ + y₂³.y₄².v₄ + y₂³.y₄².v₃ + y₂³.y₃².v₃ + y₂⁴.y₄.v₄ + y₂⁴.y₃.v₄ + y₂⁵.v₄ + y₂⁵.v₃ + y₁.y₅.w.v₃ + y₁.y₅⁴.v₄ + y₁.y₄.w.v₄ + y₁.y₂.w.v₃ + y₁.y₂³.y₃.v₄ + y₁.y₂⁴.v₄ + y₁.y₂⁴.v₃ + y₁².y₅³.v₄ + y₁².y₂³.v₄ + y₁³.y₅².v₄ + y₁³.y₄².v₃ + y₁³.y₂.y₃.v₃ + y₁³.y₂².v₄ + y₁⁴.y₂.v₄
• v₂.t = y₃.y₄³.t + y₃⁴.t + y₂.y₅⁶.w + y₂.y₄³.t + y₂.y₃.y₄².t + y₂.y₃².y₄³.v₁ + y₂.y₃².y₄⁴.w + y₂.y₃³.t + y₂.y₃³.y₄².v₁ + y₂.y₃³.y₄³.w + y₂.y₃⁴.y₄.v₁ + y₂.y₃⁵.y₄.w + y₂².y₅⁵.w + y₂².y₃².y₄³.w + y₂².y₃³.y₄.v₁ + y₂².y₃⁴.v₂ + y₂².y₃⁴.v₁ + y₂².y₃⁴.y₄.w + y₂².y₃⁵.w + y₂³.y₅⁴.w + y₂³.y₅⁷ + y₂³.y₄.t + y₂³.y₄³.v₁ + y₂³.y₄⁴.w + y₂³.y₃.t + y₂³.y₃.y₄³.w + y₂³.y₃².y₄.v₁ + y₂³.y₃².y₄².w + y₂³.y₃³.v₁ + y₂³.y₃³.y₄.w + y₂⁴.t + y₂⁴.y₅⁶ + y₂⁴.y₄².v₁ + y₂⁴.y₄³.w + y₂⁴.y₃.y₄².w + y₂⁴.y₃².v₁ + y₂⁴.y₃³.w + y₂⁵.y₅⁵ + y₂⁵.y₄.v₁ + y₂⁵.y₄².w + y₂⁵.y₃.v₁ + y₂⁵.y₃.y₄.w + y₂⁵.y₃².w + y₂⁶.v₂ + y₂⁷.w + y₁.y₅⁶.w + y₁.y₂⁶.w + y₁.y₂⁸.y₃ + y₁².y₅⁸ + y₁³.y₅⁴.w + y₁³.y₅⁷ + y₁³.y₂⁴.w + y₁³.y₂⁷ + y₁⁴.y₅³.w + y₁⁴.y₅⁶ + y₁⁴.y₂³.w + y₁⁵.y₅⁵ + y₁⁶.y₅.w + y₁⁷.y₅³ + y₁⁸.y₂² + y₁⁹.y₅ + y₁⁹.y₃ + y₅³.w.v₃ + y₄².v₁.v₃ + y₃.y₄.v₁.v₃ + y₃.y₄².w.v₄ + y₃.y₄⁵.v₃ + y₃².v₂.v₄ + y₃².v₁.v₃ + y₃².y₄.w.v₄ + y₃².y₄⁴.v₄ + y₃³.w.v₄ + y₃⁵.y₄.v₄ + y₃⁵.y₄.v₃ + y₃⁶.v₃ + y₂.y₅².w.v₄ + y₂.y₅².w.v₃ + y₂.y₄².w.v₃ + y₂.y₄⁵.v₃ + y₂.y₃.v₂.v₄ + y₂.y₃.v₂.v₃ + y₂.y₃.v₁.v₄ + y₂.y₃.y₄.w.v₄ + y₂.y₃.y₄.w.v₃ + y₂.y₃.y₄⁴.v₄ + y₂.y₃.y₄⁴.v₃ + y₂.y₃².w.v₄ + y₂.y₃².y₄³.v₄ + y₂.y₃².y₄³.v₃ + y₂.y₃³.y₄².v₄ + y₂.y₃³.y₄².v₃ + y₂.y₃⁵.v₄ + y₂.y₃⁵.v₃ + y₂².v₂.v₃ + y₂².v₁.v₄ + y₂².v₁.v₃ + y₂².y₅⁴.v₃ + y₂².y₃.y₄³.v₄ + y₂².y₃².y₄².v₄ + y₂².y₃³.y₄.v₄ + y₂².y₃³.y₄.v₃ + y₂².y₃⁴.v₃ + y₂³.y₅³.v₄ + y₂³.y₅³.v₃ + y₂³.y₄³.v₄ + y₂³.y₃.y₄².v₃ + y₂³.y₃².y₄.v₄ + y₂³.y₃².y₄.v₃ + y₂³.y₃³.v₄ + y₂⁴.y₄².v₃ + y₂⁴.y₃.y₄.v₃ + y₂⁴.y₃².v₃ + y₂⁵.y₄.v₄ + y₂⁵.y₃.v₃ + y₂⁶.v₃ + y₁.y₅².w.v₃ + y₁.y₅⁵.v₄ + y₁.y₅⁵.v₃ + y₁.y₄².w.v₄ + y₁.y₂⁴.y₃.v₄ + y₁.y₂⁴.y₃.v₃ + y₁².y₅.w.v₄ + y₁².y₅⁴.v₃ + y₁².y₂.w.v₄ + y₁².y₂⁴.v₄ + y₁².y₂⁴.v₃ + y₁³.y₅³.v₄ + y₁³.y₄³.v₃ + y₁³.y₂³.v₄ + y₁⁴.y₂².v₃ + y₁⁵.y₅.v₃ + y₁⁵.y₃.v₃ + y₁⁵.y₂.v₄
• v₁.t = y₃.y₄³.t + y₃³.y₄.t + y₃³.y₄⁴.w + y₃⁴.t + y₃⁶.v₂ + y₃⁷.w + y₂.y₅⁶.w + y₂.y₄³.t + y₂.y₃².y₄³.v₁ + y₂.y₃³.t + y₂.y₃³.y₄³.w + y₂.y₃⁴.y₄².w + y₂.y₃⁵.v₁ + y₂.y₃⁵.y₄.w + y₂.y₃⁶.w + y₂².y₄².t + y₂².y₃².t + y₂².y₃².y₄².v₁ + y₂².y₃³.y₄.v₁ + y₂².y₃⁴.v₂ + y₂².y₃⁴.v₁ + y₂².y₃⁵.w + y₂³.y₄.t + y₂³.y₄³.v₁ + y₂³.y₄⁴.w + y₂³.y₃.t + y₂³.y₃.y₄².v₁ + y₂³.y₃.y₄³.w + y₂³.y₃³.v₁ + y₂³.y₃⁴.w + y₂⁴.y₅³.w + y₂⁴.y₄³.w + y₂⁴.y₃.y₄.v₁ + y₂⁴.y₃³.w + y₂⁶.v₁ + y₂⁶.y₄.w + y₂⁶.y₃.w + y₁.y₅⁶.w + y₁².y₅⁵.w + y₁².y₂⁸ + y₁⁴.y₅³.w + y₁⁴.y₂⁶ + y₁⁵.y₅².w + y₁⁵.y₅⁵ + y₁⁵.y₂⁵ + y₁⁶.y₅⁴ + y₁⁶.y₂.w + y₁⁸.y₅² + y₁⁸.y₂.y₃ + y₁⁹.y₅ + y₁⁹.y₃ + y₅³.w.v₃ + y₄³.w.v₃ + y₃.y₄.v₁.v₄ + y₃.y₄².w.v₃ + y₃.y₄⁵.v₃ + y₃².v₂.v₄ + y₃².v₂.v₃ + y₃².y₄.w.v₄ + y₃².y₄⁴.v₄ + y₃².y₄⁴.v₃ + y₃³.w.v₃ + y₃³.y₄³.v₃ + y₃⁴.y₄².v₄ + y₃⁵.y₄.v₄ + y₃⁶.v₃ + y₂.y₅².w.v₄ + y₂.y₄⁵.v₃ + y₂.y₃.y₄⁴.v₄ + y₂.y₃³.y₄².v₄ + y₂.y₃³.y₄².v₃ + y₂.y₃⁴.y₄.v₃ + y₂.y₃⁵.v₄ + y₂².v₁.v₃ + y₂².y₅.w.v₃ + y₂².y₄⁴.v₃ + y₂².y₃⁴.v₄ + y₂².y₃⁴.v₃ + y₂³.y₄³.v₄ + y₂³.y₄³.v₃ + y₂³.y₃³.v₄ + y₂³.y₃³.v₃ + y₂⁴.y₅².v₄ + y₂⁴.y₅².v₃ + y₂⁴.y₄².v₃ + y₂⁴.y₃².v₄ + y₂⁵.y₄.v₃ + y₂⁵.y₃.v₃ + y₁.y₅².w.v₃ + y₁.y₅⁵.v₄ + y₁.y₄².w.v₄ + y₁.y₂².w.v₃ + y₁.y₂⁴.y₃.v₃ + y₁².y₅.w.v₄ + y₁².y₅.w.v₃ + y₁².y₅⁴.v₃ + y₁².y₂.w.v₄ + y₁³.y₄³.v₃ + y₁⁴.y₅².v₃ + y₁⁴.y₂.y₃.v₄ + y₁⁴.y₂².v₃ + y₁⁵.y₅.v₄ + y₁⁵.y₅.v₃ + y₁⁵.y₃.v₄ + y₁⁵.y₃.v₃ + y₁⁵.y₂.v₃
• t² = y₃³.y₄³.t + y₃⁴.y₄⁵.w + y₃⁵.y₄.t + y₃⁵.y₄⁴.w + y₃⁷.y₄².w + y₂.y₃³.y₄².t + y₂.y₃⁴.y₄.t + y₂.y₃⁴.y₄³.v₁ + y₂.y₃⁴.y₄⁴.w + y₂.y₃⁵.y₄².v₁ + y₂.y₃⁵.y₄³.w + y₂.y₃⁷.v₂ + y₂.y₃⁷.y₄.w + y₂².y₅¹⁰ + y₂².y₃.y₄³.t + y₂².y₃².y₄².t + y₂².y₃².y₄⁵.w + y₂².y₃³.y₄³.v₁ + y₂².y₃⁴.y₄³.w + y₂².y₃⁵.y₄².w + y₂².y₃⁶.v₂ + y₂².y₃⁶.v₁ + y₂³.y₃².y₄.t + y₂³.y₃².y₄³.v₁ + y₂³.y₃³.y₄².v₁ + y₂³.y₃⁴.y₄.v₁ + y₂³.y₃⁴.y₄².w + y₂³.y₃⁵.v₂ + y₂³.y₃⁵.v₁ + y₂⁴.y₅⁸ + y₂⁴.y₃.y₄.t + y₂⁴.y₃.y₄³.v₁ + y₂⁴.y₃.y₄⁴.w + y₂⁴.y₃³.y₄.v₁ + y₂⁴.y₃³.y₄².w + y₂⁴.y₃⁴.v₂ + y₂⁴.y₃⁴.v₁ + y₂⁴.y₃⁵.w + y₂⁵.y₄⁴.w + y₂⁵.y₃.y₄³.w + y₂⁵.y₃².y₄.v₁ + y₂⁵.y₃³.v₁ + y₂⁵.y₃³.y₄.w + y₂⁶.y₅⁶ + y₂⁶.y₃.y₄.v₁ + y₂⁶.y₃³.w + y₂⁷.y₃².w + y₂⁹.w + y₁.y₂⁸.w + y₁².y₅¹⁰ + y₁².y₂⁷.w + y₁³.y₅⁶.w + y₁³.y₂⁶.w + y₁⁴.y₂⁵.w + y₁⁵.y₅⁴.w + y₁⁶.y₅⁶ + y₁⁶.y₂³.w + y₁⁷.y₅².w + y₁⁷.y₅⁵ + y₁⁷.y₂².w + y₁⁸.y₂.w + y₁⁸.y₂⁴ + y₁⁹.y₅³ + y₁¹⁰.y₅² + y₁¹⁰.y₂² + y₁¹¹.y₅ + y₁¹¹.y₃ + y₃².y₄².v₁.v₃ + y₃³.y₄.v₁.v₃ + y₃³.y₄².w.v₃ + y₃⁴.y₄.w.v₃ + y₃⁴.y₄⁴.v₄ + y₃⁴.y₄⁴.v₃ + y₃⁵.w.v₄ + y₃⁵.w.v₃ + y₃⁵.y₄³.v₃ + y₃⁶.y₄².v₄ + y₃⁷.y₄.v₃ + y₂.y₃.y₄².v₁.v₃ + y₂.y₃.y₄³.w.v₃ + y₂.y₃².y₄.v₁.v₃ + y₂.y₃².y₄⁵.v₃ + y₂.y₃³.v₂.v₄ + y₂.y₃³.v₂.v₃ + y₂.y₃³.y₄.w.v₄ + y₂.y₃³.y₄⁴.v₃ + y₂.y₃⁴.w.v₄ + y₂.y₃⁴.y₄³.v₄ + y₂.y₃⁵.y₄².v₃ + y₂.y₃⁶.y₄.v₄ + y₂.y₃⁷.v₃ + y₂².y₃.y₄².w.v₄ + y₂².y₃.y₄².w.v₃ + y₂².y₃.y₄⁵.v₃ + y₂².y₃².v₂.v₄ + y₂².y₃².v₂.v₃ + y₂².y₃².v₁.v₄ + y₂².y₃².v₁.v₃ + y₂².y₃².y₄.w.v₄ + y₂².y₃².y₄.w.v₃ + y₂².y₃².y₄⁴.v₄ + y₂².y₃².y₄⁴.v₃ + y₂².y₃³.y₄³.v₄ + y₂².y₃⁴.y₄².v₄ + y₂².y₃⁶.v₃ + y₂³.y₄².w.v₄ + y₂³.y₃.v₁.v₄ + y₂³.y₃.v₁.v₃ + y₂³.y₃.y₄⁴.v₃ + y₂³.y₃².w.v₃ + y₂³.y₃².y₄³.v₄ + y₂³.y₃².y₄³.v₃ + y₂³.y₃³.y₄².v₄ + y₂³.y₃⁴.y₄.v₄ + y₂³.y₃⁴.y₄.v₃ + y₂³.y₃⁵.v₄ + y₂⁴.y₄⁴.v₃ + y₂⁴.y₃.y₄³.v₄ + y₂⁴.y₃².y₄².v₄ + y₂⁴.y₃⁴.v₃ + y₂⁵.y₃.y₄².v₄ + y₂⁵.y₃.y₄².v₃ + y₂⁵.y₃².y₄.v₃ + y₂⁵.y₃³.v₄ + y₂⁵.y₃³.v₃ + y₂⁶.y₃.y₄.v₃ + y₂⁶.y₃².v₄ + y₂⁶.y₃².v₃ + y₂⁸.v₄ + y₂⁸.v₃ + y₁.y₂⁶.y₃.v₄ + y₁.y₂⁶.y₃.v₃ + y₁².y₅⁶.v₄ + y₁².y₂³.w.v₄ + y₁².y₂⁶.v₄ + y₁².y₂⁶.v₃ + y₁⁴.y₅⁴.v₄ + y₁⁴.y₅⁴.v₃ + y₁⁴.y₂⁴.v₃ + y₁⁶.y₅².v₄ + y₁⁶.y₅².v₃ + y₁⁶.y₄².v₃ + y₁⁶.y₂.y₃.v₃ + y₁⁶.y₂².v₄ + y₁⁶.y₂².v₃ + y₅⁴.v₃² + y₄⁴.v₃² + y₃².y₄².v₄² + y₃⁴.v₃.v₄ + y₂².y₅².v₄² + y₂².y₅².v₃² + y₂².y₄².v₃.v₄ + y₂².y₄².v₃² + y₂².y₃².v₃.v₄ + y₂².y₃².v₃² + y₂⁴.v₃² + y₁².y₄².v₄² + y₁².y₂².v₃.v₄ + y₁².y₂².v₃²

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

• y₁.y₃² = y₁.y₂.y₄ + y₁.y₂.y₃ + y₁².y₂
• y₁.y₂.y₄² = y₁³.y₂
• y₁.y₂².y₄ = y₁².y₂.y₃ + y₁².y₂²
• y₁².y₂.y₄ = y₁².y₂.y₃ + y₁³.y₂
• y₁².y₂².y₃ = y₁³.y₂.y₃
• y₁.y₂.y₄.w = y₁².y₂.w + y₁⁴.y₂.y₃

Completion information

This cohomology ring was obtained from a calculation out to degree 12. 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 5 and depth 3. Here is a homogeneous system of parameters:

• h₁ = v₃ in degree 4
• h₂ = v₄ in degree 4
• h₃ = y₅⁴ + y₄⁴ + y₃².y₄² + y₃⁴ + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₅² + y₂².y₄² + y₂².y₃.y₄ + y₂².y₃² + y₂⁴ + y₁².y₅² + y₁².y₄² + y₁².y₂² + y₁⁴ in degree 4
• h₄ = y₃² + y₂.y₃ + y₂² + y₁.y₃ + y₁² in degree 2
• h₅ = y₃ in degree 1

The first 3 terms h₁, h₂, h₃ form a regular sequence of maximum length.

The first 2 terms h₁, h₂ form a complete Duflot regular sequence. That is, their restrictions to the greatest central elementary abelian subgroup form a regular sequence of maximal length.

Data for Benson's test:

• Raw filter degree type: -1, -1, -1, 6, 9, 10.
• Filter degree type: -1, -2, -3, -4, -5, -5.
• 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, h₅) is as follows.

• 1 in degree 0
• y₅ in degree 1
• y₄ in degree 1
• y₂ in degree 1
• y₁ in degree 1
• y₅² in degree 2
• y₄² in degree 2
• y₂.y₄ in degree 2
• y₁.y₅ in degree 2
• y₁.y₄ in degree 2
• y₁.y₂ in degree 2
• y₁² in degree 2
• w in degree 3
• y₅³ in degree 3
• y₄³ in degree 3
• y₂.y₄² in degree 3
• y₁.y₅² in degree 3
• y₁.y₄² in degree 3
• y₁².y₄ in degree 3
• y₁².y₂ in degree 3
• y₁³ in degree 3
• v₂ in degree 4
• v₁ in degree 4
• y₅.w in degree 4
• y₄.w in degree 4
• y₄⁴ in degree 4
• y₂.w in degree 4
• y₂.y₄³ in degree 4
• y₁.w in degree 4
• y₁.y₅³ in degree 4
• y₁.y₄³ in degree 4
• y₁².y₄² in degree 4
• y₁³.y₂ in degree 4
• y₁⁴ in degree 4
• y₅².w in degree 5
• y₄.v₁ in degree 5
• y₄².w in degree 5
• y₂.v₂ in degree 5
• y₂.v₁ in degree 5
• y₂.y₄.w in degree 5
• y₁.y₅.w in degree 5
• y₁.y₄.w in degree 5
• y₁.y₄⁴ in degree 5
• y₁.y₂.w in degree 5
• y₁².w in degree 5
• y₁².y₄³ in degree 5
• y₁⁴.y₂ in degree 5
• y₁⁵ in degree 5
• t in degree 6
• y₅³.w in degree 6
• y₄².v₁ in degree 6
• y₄³.w in degree 6
• y₂.y₄.v₁ in degree 6
• y₂.y₄².w in degree 6
• y₁.y₅².w in degree 6
• y₁.y₄².w in degree 6
• y₁².y₄.w in degree 6
• y₁².y₂.w in degree 6
• y₁³.w in degree 6
• y₁⁶ in degree 6
• y₄.t in degree 7
• y₄³.v₁ in degree 7
• y₄⁴.w in degree 7
• y₂.t in degree 7
• y₂.y₄².v₁ in degree 7
• y₂.y₄³.w in degree 7
• y₁.y₅³.w in degree 7
• y₁.y₄³.w in degree 7
• y₁².y₄².w in degree 7
• y₁³.y₂.w in degree 7
• y₁⁴.w in degree 7
• y₄².t in degree 8
• y₂.y₄.t in degree 8
• y₁.y₄⁴.w in degree 8
• y₁².y₄³.w in degree 8
• y₁⁴.y₂.w in degree 8
• y₁⁵.w in degree 8
• y₄³.t in degree 9
• y₂.y₄².t in degree 9
• y₂.y₄³.t in degree 10

A basis for Ann_{R/(h1, h2, h3, h4)}(h₅) is as follows.

• y₁.y₅ + y₁.y₂ in degree 2
• y₁.y₄ + y₁.y₂ + y₁.h in degree 2
• y₁.y₅² + y₁.y₂.y₅ in degree 3
• y₁.y₄² + y₁.y₂.y₄ + y₁.y₄.h in degree 3
• y₁².y₄ + y₁².y₂ + y₁².h in degree 3
• y₁².y₂ + y₁³ + y₁².h + y₅.h² + y₁.h² in degree 3
• y₁.y₂.h + y₅.h² + y₁.h² in degree 3
• y₁².h in degree 3
• y₅.h² in degree 3
• y₁.w + y₁⁴ in degree 4
• y₁.y₅³ + y₁.y₂.y₅² in degree 4
• y₁.y₄³ + y₁.y₂.y₄² + y₁.y₄².h in degree 4
• y₁².y₄² + y₁².y₂.y₄ + y₁².y₄.h in degree 4
• y₁³.y₂ + y₁⁴ + y₁³.h + y₁.y₅.h² + y₁².h² in degree 4
• y₁³.h in degree 4
• y₅².h² in degree 4
• y₁.h³ in degree 4
• y₁.y₅.w + y₁.y₂.w in degree 5
• y₁.y₄.w + y₁.y₂.w + y₁.w.h in degree 5
• y₁.y₄⁴ + y₁.y₂.y₄³ + y₁.y₄³.h in degree 5
• y₁.y₂.w + y₁⁴.y₂ in degree 5
• y₁².w + y₁⁵ in degree 5
• y₁².y₄³ + y₁².y₂.y₄² + y₁².y₄².h in degree 5
• y₁⁴.y₂ + y₁⁵ + y₁⁴.h + y₁².y₅.h² + y₁³.h² in degree 5
• y₁⁵ in degree 5
• y₁⁴.h in degree 5
• y₅³.h² in degree 5
• y₁.y₅².w + y₁.y₂.y₅.w in degree 6
• y₁.y₄².w + y₁.y₂.y₄.w + y₁.y₄.w.h in degree 6
• y₁².y₄.w + y₁².y₂.w + y₁².w.h in degree 6
• y₁².y₂.w + y₁³.w + y₁².w.h + y₅.w.h² + y₁.w.h² in degree 6
• y₁³.w + y₁⁶ in degree 6
• y₁⁶ in degree 6
• y₅.w.h² in degree 6
• y₄⁴.h² + y₄.h⁵ in degree 6
• y₁.y₅³.w + y₁.y₂.y₅².w in degree 7
• y₁.y₄³.w + y₁.y₂.y₄².w + y₁.y₄².w.h in degree 7
• y₁².y₄².w + y₁².y₂.y₄.w + y₁².y₄.w.h in degree 7
• y₁³.y₂.w + y₁⁴.w + y₁³.w.h + y₁.y₅.w.h² + y₁².w.h² in degree 7
• y₁⁴.w + y₁⁷ in degree 7
• y₅².w.h² in degree 7
• y₁.y₄⁴.w + y₁.y₂.y₄³.w + y₁.y₄³.w.h in degree 8
• y₁².y₄³.w + y₁².y₂.y₄².w + y₁².y₄².w.h in degree 8
• y₁⁴.y₂.w + y₁⁵.w + y₁⁴.w.h + y₁².y₅.w.h² + y₁³.w.h² in degree 8
• y₁⁵.w + y₁⁸ in degree 8
• y₅³.w.h² in degree 8
• y₄³.v₁.h² + y₂.y₄³.w.h² + y₄².v₁.h³ + y₄³.w.h³ + y₂.y₄².w.h³ + y₄.v₁.h⁴ + y₄².w.h⁴ + y₂.y₄.w.h⁴ + y₄.w.h⁵ in degree 9
• y₄⁴.w.h² + y₄.w.h⁵ in degree 9

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

• y₁.y₂² + y₁².y₃ + y₁³ + y₁.h in degree 3
• y₁³.y₅ + y₁⁴ + y₁.y₅.h + y₁.y₃.h + y₁².h in degree 4
• y₁⁴.y₃ + y₁².y₃.h in degree 5
• y₁.y₂.y₃.h in degree 5
• y₁².y₃.h in degree 5
• y₁³.y₃.h in degree 6
• y₁.y₃.h² in degree 6

Restriction information

Restriction to special subgroup number 1, which is 4gp2

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to 0
• y₄ restricts to 0
• y₅ restricts to 0
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• v₃ restricts to y₂⁴
• v₄ restricts to y₂⁴ + y₁⁴
• t restricts to 0

Restriction to special subgroup number 2, which is 16gp14

• y₁ restricts to y₄
• y₂ restricts to 0
• y₃ restricts to 0
• y₄ restricts to y₄
• y₅ restricts to y₃
• w restricts to y₃².y₄ + y₂.y₄² + y₂.y₃² + y₂².y₄ + y₂².y₃ + y₁.y₃² + y₁².y₃
• v₁ restricts to y₃.y₄³ + y₃³.y₄ + y₂.y₄³ + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂².y₃² + y₁.y₃².y₄ + y₁.y₃³ + y₁².y₃.y₄ + y₁².y₃²
• v₂ restricts to y₂.y₄³ + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂².y₃² + y₁.y₃².y₄ + y₁.y₃³ + y₁².y₃.y₄ + y₁².y₃²
• v₃ restricts to y₃.y₄³ + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂⁴ + y₁.y₃³ + y₁².y₃²
• v₄ restricts to y₃².y₄² + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₄² + y₂².y₃.y₄ + y₂².y₃² + y₂⁴ + y₁.y₃.y₄² + y₁.y₃².y₄ + y₁².y₄² + y₁².y₃.y₄ + y₁².y₃² + y₁⁴
• t restricts to y₃².y₄⁴ + y₃⁵.y₄ + y₂.y₄⁵ + y₂.y₃⁵ + y₂².y₄⁴ + y₂².y₃.y₄³ + y₂².y₃³.y₄ + y₂⁴.y₃² + y₁.y₃.y₄⁴ + y₁.y₃⁴.y₄ + y₁.y₃⁵ + y₁².y₄⁴ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₃⁴ + y₁⁴.y₄²

Restriction to special subgroup number 3, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₄
• y₃ restricts to y₄
• y₄ restricts to 0
• y₅ restricts to y₃
• w restricts to y₂.y₃.y₄ + y₂.y₃² + y₂².y₄ + y₂².y₃ + y₁.y₃.y₄ + y₁.y₃² + y₁².y₄ + y₁².y₃
• v₁ restricts to y₂.y₃.y₄² + y₂.y₃³ + y₂².y₄² + y₂².y₃² + y₁.y₃.y₄² + y₁.y₃³ + y₁².y₄² + y₁².y₃²
• v₂ restricts to y₃².y₄² + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₃.y₄ + y₂².y₃² + y₁.y₃².y₄ + y₁.y₃³ + y₁².y₃.y₄ + y₁².y₃²
• v₃ restricts to y₂.y₃.y₄² + y₂.y₃³ + y₂².y₄² + y₂⁴ + y₁.y₃².y₄ + y₁.y₃³ + y₁².y₃.y₄ + y₁².y₃²
• v₄ restricts to y₂².y₃² + y₂⁴ + y₁².y₃² + y₁⁴
• t restricts to y₃³.y₄³ + y₃⁴.y₄² + y₃⁵.y₄ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂.y₃⁵ + y₂².y₃.y₄³ + y₂².y₃².y₄² + y₂².y₃³.y₄ + y₂⁴.y₃² + y₁.y₃³.y₄² + y₁.y₃⁵ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₃⁴ + y₁⁴.y₃.y₄

Restriction to special subgroup number 4, which is 16gp14

• y₁ restricts to y₃
• y₂ restricts to 0
• y₃ restricts to 0
• y₄ restricts to y₄
• y₅ restricts to 0
• w restricts to y₂.y₃.y₄ + y₂².y₄
• v₁ restricts to y₂.y₃.y₄² + y₂².y₄²
• v₂ restricts to y₂.y₃.y₄² + y₂².y₄²
• v₃ restricts to y₂².y₃² + y₂⁴
• v₄ restricts to y₂².y₃² + y₂⁴ + y₁.y₃.y₄² + y₁.y₃².y₄ + y₁².y₄² + y₁².y₃.y₄ + y₁².y₃² + y₁⁴
• t restricts to y₂.y₃⁴.y₄ + y₂².y₃².y₄² + y₂⁴.y₄² + y₂⁴.y₃.y₄ + y₁.y₃².y₄³ + y₁.y₃³.y₄² + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁⁴.y₃.y₄

Restriction to special subgroup number 5, which is 16gp14

• y₁ restricts to y₄
• y₂ restricts to y₃
• y₃ restricts to 0
• y₄ restricts to y₄
• y₅ restricts to y₄
• w restricts to y₄³ + y₁.y₄² + y₁.y₃.y₄ + y₁².y₄ + y₁².y₃
• v₁ restricts to y₃.y₄³ + y₃².y₄²
• v₂ restricts to y₃².y₄²
• v₃ restricts to y₄⁴ + y₃.y₄³ + y₃².y₄² + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₄² + y₂².y₃.y₄ + y₂².y₃² + y₂⁴ + y₁.y₄³ + y₁.y₃².y₄ + y₁².y₄² + y₁².y₃²
• v₄ restricts to y₄⁴ + y₃².y₄² + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₄² + y₂².y₃.y₄ + y₂².y₃² + y₂⁴ + y₁².y₄² + y₁⁴
• t restricts to y₃.y₄⁵ + y₃².y₄⁴ + y₃³.y₄³ + y₂.y₃.y₄⁴ + y₂.y₃².y₄³ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃.y₄³ + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂⁴.y₄² + y₂⁴.y₃² + y₁.y₄⁵ + y₁.y₃².y₄³ + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁⁴.y₄² + y₁⁴.y₃.y₄

Restriction to special subgroup number 6, which is 32gp51

• y₁ restricts to 0
• y₂ restricts to y₃
• y₃ restricts to y₄
• y₄ restricts to y₅
• y₅ restricts to 0
• w restricts to y₂.y₄.y₅ + y₂.y₄² + y₂.y₃.y₅ + y₂.y₃.y₄ + y₂².y₅ + y₂².y₄ + y₁.y₃.y₅ + y₁².y₃
• v₁ restricts to y₂.y₄².y₅ + y₂.y₄³ + y₂.y₃.y₅² + y₂.y₃.y₄² + y₂².y₅² + y₂².y₄² + y₁.y₄³ + y₁.y₃.y₄.y₅ + y₁.y₃.y₄² + y₁².y₄²
• v₂ restricts to y₂.y₄.y₅² + y₂.y₄².y₅ + y₂.y₄³ + y₂.y₃.y₅² + y₂.y₃.y₄.y₅ + y₂.y₃².y₅ + y₂.y₃².y₄ + y₂².y₅² + y₂².y₄.y₅ + y₂².y₄² + y₂².y₃.y₅ + y₂².y₃.y₄ + y₁.y₃.y₅² + y₁.y₃.y₄² + y₁.y₃².y₄ + y₁².y₃.y₅
• v₃ restricts to y₂.y₄².y₅ + y₂.y₄³ + y₂.y₃.y₄² + y₂.y₃².y₅ + y₂².y₄.y₅ + y₂².y₃.y₅ + y₂².y₃² + y₂⁴ + y₁.y₃.y₄.y₅ + y₁.y₃².y₅ + y₁².y₃.y₄ + y₁².y₃²
• v₄ restricts to y₂.y₄².y₅ + y₂.y₄³ + y₂.y₃.y₄.y₅ + y₂.y₃².y₄ + y₂².y₄.y₅ + y₂².y₃.y₄ + y₂².y₃² + y₂⁴ + y₁.y₄.y₅² + y₁.y₄².y₅ + y₁.y₃.y₄.y₅ + y₁².y₅² + y₁².y₄.y₅ + y₁².y₄² + y₁².y₃.y₄ + y₁⁴
• t restricts to y₂.y₄².y₅³ + y₂.y₄⁴.y₅ + y₂.y₄⁵ + y₂.y₃.y₄².y₅² + y₂.y₃².y₅³ + y₂.y₃².y₄.y₅² + y₂.y₃².y₄².y₅ + y₂.y₃².y₄³ + y₂.y₃³.y₄.y₅ + y₂.y₃⁴.y₅ + y₂².y₄.y₅³ + y₂².y₄³.y₅ + y₂².y₃.y₅³ + y₂².y₃.y₄.y₅² + y₂².y₃.y₄³ + y₂².y₃².y₅² + y₂².y₃³.y₅ + y₂².y₃⁴ + y₂³.y₄.y₅² + y₂³.y₄².y₅ + y₂⁴.y₅² + y₂⁴.y₄.y₅ + y₂⁴.y₄² + y₂⁴.y₃² + y₁.y₄².y₅³ + y₁.y₄³.y₅² + y₁.y₃.y₄.y₅³ + y₁.y₃.y₄².y₅² + y₁.y₃.y₄³.y₅ + y₁.y₃².y₅³ + y₁.y₃².y₄³ + y₁.y₃³.y₅² + y₁.y₃³.y₄.y₅ + y₁.y₃⁴.y₄ + y₁.y₂.y₄⁴ + y₁.y₂.y₃².y₅² + y₁.y₂.y₃².y₄² + y₁.y₂².y₄³ + y₁.y₂².y₃.y₅² + y₁.y₂².y₃.y₄² + y₁².y₄.y₅³ + y₁².y₄².y₅² + y₁².y₄³.y₅ + y₁².y₃.y₄.y₅² + y₁².y₃².y₅² + y₁².y₃².y₄.y₅ + y₁².y₃³.y₅ + y₁².y₂.y₄³ + y₁².y₂.y₃².y₅ + y₁².y₂.y₃².y₄ + y₁².y₂².y₄² + y₁².y₂².y₃.y₅ + y₁².y₂².y₃.y₄ + y₁³.y₃.y₄² + y₁³.y₃².y₄ + y₁⁴.y₄.y₅

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Poincaré series

(1 + 4t + 7t² + 7t³ + 6t⁴ + 5t⁵ + 2t⁶ + t⁷ + t⁸ - t⁹ - t¹⁰) / (1 - t) (1 - t²) (1 - t⁴)³

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Back to the groups of order 128