Small group number 1758 of order 128

G is the group 128gp1758

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
• 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
• u₃ in degree 5
• r in degree 8, a regular element

There are 27 minimal relations:

• y₂.y₄ = 0
• y₁.y₄ = 0
• y₃.y₄² = 0
• y₃².y₄ = y₂.x₂ + y₁.x₁
• y₃.y₄.x₂ = y₂.y₃.x₂ + y₁.w + y₁.y₃.x₂ + y₁.y₃.x₁ + y₁².x₁
• y₃.y₄.x₁ = y₂.w + y₂.y₃.x₂ + y₂.y₃.x₁ + y₁.y₃.x₁ + y₁².x₁
• y₁.y₂.x₁ = y₁².x₁
• y₃.y₄.w = y₂.y₃².x₂ + y₁.y₃².x₁
• y₁.x₁.x₂ = y₁.x₁² + y₁.y₃.w + y₁.y₃².x₂ + y₁².y₃.x₁
• w² = x₁.x₂² + x₁².x₂ + y₄.u₂ + y₄².x₁.x₂ + y₄³.w + y₃².x₂² + y₃².x₁.x₂ + y₃².x₁² + y₁.x₂.w + y₁.y₃.x₂² + y₁.y₃².w + y₁.y₃³.x₂ + y₁².y₃².x₁
• y₄.u₃ = y₄.x₂.w + y₄².x₁.x₂ + y₄³.w + y₂.x₁.w + y₂.y₃.x₁² + y₂.y₃².w + y₂.y₃³.x₂ + y₂.y₃³.x₁ + y₁.x₁.w + y₁.y₃.x₁² + y₁.y₃².w + y₁.y₃³.x₂ + y₁.y₃³.x₁
• y₄.u₁ = y₄.x₂.w + y₄.x₁.w + y₄³.w + y₂.x₁.w + y₂.y₃.x₁² + y₂.y₃².w + y₂.y₃³.x₁ + y₁².x₁² + y₁².y₃².x₁
• y₂.u₃ = y₂.u₁ + y₂.x₁.w + y₂.y₃³.x₂ + y₁.u₁ + y₁.x₂.w + y₁.x₁.w + y₁.y₃².w + y₁.y₃³.x₁ + y₁².y₃².x₁ + y₁³.y₃.x₁ + y₁⁴.x₁
• y₂.u₂ = y₂.x₁.w + y₂.y₃.x₁² + y₂.y₃².w + y₂.y₃³.x₂ + y₁.u₁ + y₁.x₂.w + y₁.x₁.w + y₁.y₃.x₁² + y₁.y₃².w + y₁².x₁² + y₁².y₃².x₁
• y₁.u₃ = y₁.u₂ + y₁.u₁ + y₁.x₂.w + y₁.x₁.w + y₁.y₃².w + y₁.y₃³.x₂ + y₁².x₁² + y₁³.y₃.x₁ + y₁⁴.x₁
• x₂.u₁ = x₂².w + x₁.u₃ + x₁.u₁ + x₁.x₂.w + x₁².w + y₄.x₁².x₂ + y₄².x₂.w + y₃.x₁.x₂² + y₃.x₁².x₂ + y₃².u₃ + y₃².u₂ + y₃².u₁ + y₃².x₂.w + y₃³.x₂² + y₃³.x₁² + y₃⁴.w + y₃⁵.x₂ + y₂.y₃.x₁.w + y₂.y₃².x₁² + y₂.y₃⁴.x₂ + y₁.y₃.x₁.w + y₁.y₃².x₁² + y₁.y₃⁴.x₁ + y₁².y₃.x₁² + y₁².y₃³.x₁ + y₁³.x₁² + y₁³.y₃².x₁
• y₃.y₄.u₂ = y₂.y₃.x₁.w + y₂.y₃².x₁² + y₁.y₃³.w + y₁.y₃⁴.x₂ + y₁².y₃.x₁² + y₁².y₃³.x₁
• w.u₃ = x₁.x₂³ + x₁².x₂² + y₄.x₂.u₂ + y₄.x₁.x₂.w + y₄².x₁².x₂ + y₄³.u₂ + y₄³.x₂.w + y₄⁴.x₁.x₂ + y₄⁵.w + y₃.x₂.u₂ + y₃.x₂².w + y₃.x₁.u₃ + y₃.x₁.u₂ + y₃².x₂³ + y₃².x₁.x₂² + y₃².x₁².x₂ + y₃³.x₂.w + y₃⁴.x₁² + y₂.y₃⁵.x₂ + y₁.x₂².w + y₁.x₁².w + y₁.y₃.x₂³ + y₁.y₃.x₁³ + y₁.y₃².x₂.w + y₁.y₃².x₁.w + y₁.y₃³.x₂² + y₁.y₃³.x₁² + y₁.y₃⁵.x₁ + y₁².x₁³ + y₁³.y₃.x₁² + y₁⁴.x₁²
• w.u₁ = x₁.x₂³ + x₁³.x₂ + y₄.x₂.u₂ + y₄.x₁.u₂ + y₄³.u₂ + y₄³.x₂.w + y₄³.x₁.w + y₄⁴.x₁.x₂ + y₄⁵.w + y₃.x₁.u₃ + y₃.x₁.u₂ + y₃.x₁.x₂.w + y₃².x₂³ + y₃².x₁².x₂ + y₃².x₁³ + y₃³.u₃ + y₃³.u₂ + y₃³.u₁ + y₃⁵.w + y₃⁶.x₂ + y₂.y₃⁴.w + y₂.y₃⁵.x₁ + y₁.x₂².w + y₁.x₁.u₁ + y₁.x₁².w + y₁.y₃.x₂³ + y₁.y₃.x₁³ + y₁.y₃².x₁.w + y₁.y₃³.x₁² + y₁³.y₃.x₁² + y₁³.y₃³.x₁
• y₁.x₁.u₂ = y₁.x₁.u₁ + y₁.x₁².w + y₁.y₃³.x₁² + y₁².x₁³ + y₁².y₃².x₁²
• x₁.x₂.u₃ = x₁.x₂².w + y₄.x₁².x₂² + y₄².x₁.x₂.w + y₃.w.u₂ + y₃.x₁.x₂³ + y₃.x₁².x₂² + y₃².x₂.u₃ + y₃².x₂².w + y₃².x₁.u₂ + y₃².x₁.x₂.w + y₃³.x₂³ + y₃³.x₁².x₂ + y₃⁴.x₁.w + y₃⁵.x₂² + y₃⁵.x₁² + y₂.y₃.x₁².w + y₂.y₃².x₁³ + y₂.y₃³.x₁.w + y₂.y₃⁴.x₁² + y₂.y₃⁶.x₂ + y₁.y₃.x₂².w + y₁.y₃.x₁.u₁ + y₁.y₃.x₁².w + y₁.y₃².x₂³ + y₁.y₃².x₁³ + y₁.y₃³.x₂.w + y₁.y₃⁴.x₂² + y₁.y₃⁶.x₁ + y₁³.x₁³ + y₁³.y₃².x₁²
• u₃² = x₁.x₂⁴ + x₁².x₂³ + y₄.x₂².u₂ + y₄².x₁.x₂³ + y₄².x₁².x₂² + y₄³.x₂².w + y₄⁴.x₁.x₂² + y₄⁴.x₁².x₂ + y₄⁵.u₂ + y₄⁶.x₁.x₂ + y₄⁷.w + y₃².x₂⁴ + y₃².x₁³.x₂ + y₃².x₁⁴ + y₃⁴.x₂³ + y₃⁴.x₁³ + y₃⁶.x₁² + y₃⁸.x₂ + y₃⁸.x₁ + y₂.y₃².x₁.u₁ + y₂.y₃⁴.u₁ + y₂.y₃⁴.x₁.w + y₂.y₃⁷.x₁ + y₂².y₃².x₁³ + y₂².y₃⁴.x₁² + y₂³.x₁.u₁ + y₂³.y₃².u₁ + y₂³.y₃⁵.x₁ + y₂⁴.y₃.u₁ + y₂⁴.y₃⁴.x₁ + y₂⁵.u₁ + y₂⁵.y₃.x₁² + y₂⁵.y₃³.x₁ + y₁.x₂³.w + y₁.y₃.x₂⁴ + y₁.y₃².x₂.u₂ + y₁.y₃².x₂².w + y₁.y₃².x₁².w + y₁.y₃³.x₁³ + y₁.y₃⁴.u₂ + y₁.y₃⁴.x₂.w + y₁.y₃⁴.x₁.w + y₁.y₃⁵.x₂² + y₁.y₃⁶.w + y₁.y₃⁷.x₁ + y₁.y₂.y₃³.u₁ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₁ + y₁².y₃².x₂³ + y₁².y₃³.u₂ + y₁².y₃⁶.x₂ + y₁².y₂.y₃².u₁ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁³.x₂.u₂ + y₁³.y₃.x₂³ + y₁³.y₃³.x₂² + y₁³.y₂².u₁ + y₁⁴.y₃.u₂ + y₁⁴.y₃².x₁² + y₁⁴.y₃⁴.x₂ + y₁⁴.y₃⁴.x₁ + y₁⁵.y₃.x₁² + y₁⁶.x₁² + y₁⁶.y₃².x₂ + y₁⁶.y₃².x₁ + y₁⁷.y₃.x₂ + y₁⁷.y₃.x₁ + y₂².r + y₁².r
• u₂.u₃ = x₂.w.u₂ + y₄.x₁.x₂.u₂ + y₄².w.u₂ + y₃.x₂².u₂ + y₃.x₂³.w + y₃.x₁.x₂.u₂ + y₃.x₁³.w + y₃².w.u₂ + y₃².x₁.x₂³ + y₃².x₁⁴ + y₃³.x₂.u₃ + y₃³.x₂.u₂ + y₃³.x₁.u₃ + y₃³.x₁.u₂ + y₃³.x₁.x₂.w + y₃³.x₁².w + y₃⁴.x₂³ + y₃⁴.x₁.x₂² + y₃⁴.x₁².x₂ + y₃⁴.x₁³ + y₃⁵.x₂.w + y₃⁶.x₂² + y₃⁷.w + y₃⁸.x₁ + y₂.y₃².x₁².w + y₂.y₃³.x₁³ + y₂.y₃⁶.w + y₂.y₃⁷.x₁ + y₁.y₃².x₂.u₂ + y₁.y₃².x₁².w + y₁.y₃³.x₂³ + y₁.y₃⁴.u₂ + y₁.y₃⁴.u₁ + y₁.y₃⁴.x₂.w + y₁.y₃⁵.x₁² + y₁.y₂².y₃².u₁ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₁ + y₁².y₃.x₁.u₁ + y₁².y₃².x₂³ + y₁².y₃³.u₂ + y₁².y₃³.u₁ + y₁².y₃⁴.x₂² + y₁².y₃⁶.x₁ + y₁².y₂.y₃².u₁ + y₁³.x₂.u₂ + y₁³.y₃.x₂³ + y₁³.y₃.x₁³ + y₁³.y₃³.x₂² + y₁³.y₃⁵.x₁ + y₁³.y₂.y₃.u₁ + y₁³.y₂².u₁ + y₁⁴.y₃.u₂ + y₁⁴.y₃.u₁ + y₁⁴.y₃².x₂² + y₁⁴.y₃⁴.x₁ + y₁⁵.y₃³.x₁ + y₁⁶.y₃².x₂ + y₁⁶.y₃².x₁ + y₁⁷.y₃.x₂ + y₁⁷.y₃.x₁ + y₁.y₂.r + y₁².r
• u₂² = x₁.x₂⁴ + x₁⁴.x₂ + y₄.x₂².u₂ + y₄.x₁.x₂.u₂ + y₄.x₁.x₂².w + y₄.x₁².u₂ + y₄.x₁².x₂.w + y₄².x₁.x₂³ + y₄³.x₂.u₂ + y₄³.x₁.u₂ + y₄³.x₁².w + y₄⁴.x₁.x₂² + y₄⁵.x₁.w + y₄⁶.x₁.x₂ + y₃².x₂⁴ + y₃².x₁.x₂³ + y₃⁴.x₂³ + y₃⁶.x₂² + y₃⁶.x₁² + y₃⁸.x₂ + y₂.y₃⁴.x₁.w + y₂.y₃⁵.x₁² + y₁.x₂³.w + y₁.y₃.x₂⁴ + y₁.y₃².x₂.u₂ + y₁.y₃².x₁².w + y₁.y₃³.x₂³ + y₁.y₃⁴.u₂ + y₁.y₃⁷.x₂ + y₁.y₃⁷.x₁ + y₁².y₃².x₂³ + y₁².y₃².x₁³ + y₁².y₃³.u₂ + y₁².y₃⁴.x₁² + y₁².y₃⁶.x₂ + y₁².y₂.y₃².u₁ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁³.x₂.u₂ + y₁³.y₃.x₂³ + y₁³.y₃³.x₂² + y₁³.y₃⁵.x₁ + y₁³.y₂.y₃.u₁ + y₁³.y₂².u₁ + y₁⁴.x₁³ + y₁⁴.y₃.u₂ + y₁⁴.y₃².x₁² + y₁⁴.y₃⁴.x₂ + y₁⁵.y₃.x₁² + y₁⁶.y₃².x₂ + y₁⁶.y₃².x₁ + y₁⁷.y₃.x₂ + y₁⁸.x₁ + y₄².r + y₁².r
• u₁.u₃ = x₁.x₂⁴ + x₁³.x₂² + y₄.x₂².u₂ + y₄.x₁.x₂.u₂ + y₄.x₁.x₂².w + y₄.x₁².x₂.w + y₄².x₁.x₂³ + y₄².x₁³.x₂ + y₄³.x₂².w + y₄³.x₁.u₂ + y₄⁴.x₁.x₂² + y₄⁵.u₂ + y₄⁵.x₁.w + y₄⁶.x₁.x₂ + y₄⁷.w + y₃.x₂².u₂ + y₃.x₂³.w + y₃.x₁.x₂.u₂ + y₃.x₁².u₃ + y₃.x₁².u₂ + y₃².w.u₂ + y₃².x₂⁴ + y₃².x₁.x₂³ + y₃².x₁².x₂² + y₃².x₁³.x₂ + y₃².x₁⁴ + y₃³.x₂.u₃ + y₃³.x₂.u₂ + y₃³.x₁.u₃ + y₃³.x₁.u₁ + y₃³.x₁.x₂.w + y₃⁴.x₂³ + y₃⁴.x₁.x₂² + y₃⁴.x₁³ + y₃⁵.u₃ + y₃⁵.u₂ + y₃⁵.u₁ + y₃⁵.x₂.w + y₃⁵.x₁.w + y₂.y₃².x₁.u₁ + y₂.y₃².x₁².w + y₂.y₃³.x₁³ + y₂.y₃⁴.u₁ + y₂.y₃⁵.x₁² + y₂.y₃⁷.x₁ + y₂².y₃².x₁³ + y₂².y₃⁴.x₁² + y₂³.x₁.u₁ + y₂³.y₃².u₁ + y₂³.y₃⁵.x₁ + y₂⁴.y₃.u₁ + y₂⁴.y₃⁴.x₁ + y₂⁵.u₁ + y₂⁵.y₃.x₁² + y₂⁵.y₃³.x₁ + y₁.x₂³.w + y₁.x₁².u₁ + y₁.y₃.x₂⁴ + y₁.y₃².x₂².w + y₁.y₃³.x₂³ + y₁.y₃³.x₁³ + y₁.y₃⁴.u₁ + y₁.y₃⁴.x₁.w + y₁.y₃⁵.x₂² + y₁.y₃⁵.x₁² + y₁.y₃⁶.w + y₁.y₃⁷.x₁ + y₁.y₂.y₃³.u₁ + y₁.y₂².y₃².u₁ + y₁².x₁⁴ + y₁².y₃.x₁.u₁ + y₁².y₃³.u₁ + y₁².y₃⁴.x₂² + y₁².y₃⁶.x₂ + y₁².y₃⁶.x₁ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁³.y₃.x₁³ + y₁³.y₂.y₃.u₁ + y₁⁴.y₃.u₁ + y₁⁴.y₃².x₂² + y₁⁴.y₃².x₁² + y₁⁴.y₃⁴.x₂ + y₁⁵.y₃.x₁² + y₁⁵.y₃³.x₁ + y₂².r + y₁.y₂.r
• u₁.u₂ = x₂.w.u₂ + x₁.w.u₂ + y₄².w.u₂ + y₃.x₁.x₂².w + y₃.x₁².u₃ + y₃.x₁².u₁ + y₃.x₁².x₂.w + y₃².w.u₂ + y₃².x₁.x₂³ + y₃².x₁².x₂² + y₃².x₁³.x₂ + y₃².x₁⁴ + y₃³.x₂².w + y₃³.x₁.u₃ + y₃⁴.x₂³ + y₃⁴.x₁.x₂² + y₃⁴.x₁².x₂ + y₃⁵.x₁.w + y₃⁷.w + y₃⁸.x₂ + y₃⁸.x₁ + y₁.y₃.x₁⁴ + y₁.y₃².x₂².w + y₁.y₃².x₁.u₁ + y₁.y₃².x₁².w + y₁.y₃³.x₂³ + y₁.y₃³.x₁³ + y₁.y₃⁴.u₁ + y₁.y₃⁴.x₂.w + y₁.y₃⁴.x₁.w + y₁.y₃⁵.x₁² + y₁.y₃⁶.w + y₁.y₃⁷.x₁ + y₁.y₂².y₃².u₁ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₁ + y₁².x₁⁴ + y₁².y₃³.u₁ + y₁².y₃⁴.x₂² + y₁².y₃⁶.x₂ + y₁².y₃⁶.x₁ + y₁².y₂².y₃.u₁ + y₁².y₂³.u₁ + y₁³.x₁.u₁ + y₁³.y₃.x₁³ + y₁³.y₃³.x₁² + y₁⁴.x₁³ + y₁⁴.y₃.u₁ + y₁⁴.y₃².x₂² + y₁⁴.y₃⁴.x₂ + y₁⁴.y₃⁴.x₁ + y₁⁵.y₃.x₁² + y₁⁵.y₃³.x₁ + y₁⁷.y₃.x₁ + y₁⁸.x₁ + y₁.y₂.r
• u₁² = x₁.x₂⁴ + x₁².x₂³ + x₁³.x₂² + x₁⁴.x₂ + y₄.x₂².u₂ + y₄.x₁².u₂ + y₄².x₁.x₂³ + y₄².x₁³.x₂ + y₄³.x₂².w + y₄³.x₁².w + y₄⁴.x₁.x₂² + y₄⁴.x₁².x₂ + y₄⁵.u₂ + y₄⁶.x₁.x₂ + y₄⁷.w + y₃².x₂⁴ + y₃².x₁.x₂³ + y₃².x₁².x₂² + y₃⁴.x₁³ + y₃⁶.x₂² + y₃⁶.x₁.x₂ + y₃⁶.x₁² + y₃⁸.x₁ + y₂.y₃².x₁.u₁ + y₂.y₃⁴.u₁ + y₂.y₃⁴.x₁.w + y₂.y₃⁷.x₁ + y₂².y₃².x₁³ + y₂².y₃⁴.x₁² + y₂³.x₁.u₁ + y₂³.y₃².u₁ + y₂³.y₃⁵.x₁ + y₂⁴.y₃.u₁ + y₂⁴.y₃⁴.x₁ + y₂⁵.u₁ + y₂⁵.y₃.x₁² + y₂⁵.y₃³.x₁ + y₁.x₂³.w + y₁.x₁³.w + y₁.y₃.x₂⁴ + y₁.y₃.x₁⁴ + y₁.y₃².x₂².w + y₁.y₃².x₁².w + y₁.y₃³.x₂³ + y₁.y₃⁴.x₂.w + y₁.y₃⁵.x₂² + y₁.y₃⁵.x₁² + y₁.y₃⁶.w + y₁.y₃⁷.x₂ + y₁.y₂.y₃³.u₁ + y₁.y₂³.y₃.u₁ + y₁.y₂⁴.u₁ + y₁².x₁⁴ + y₁³.y₃⁵.x₁ + y₁³.y₂.y₃.u₁ + y₁⁴.x₁³ + y₁⁴.y₃².x₁² + y₁⁴.y₃⁴.x₁ + y₁⁷.y₃.x₁ + y₁⁸.x₁ + y₂².r

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

• y₂².x₂ = y₁.y₂.x₁
• y₁.y₂.x₂ = y₁².x₁
• y₂.x₂² = y₂.y₃².x₂ + y₁.x₁.x₂ + y₁.y₃.w + y₁.y₃².x₂ + y₁.y₃².x₁ + y₁².y₃.x₁
• y₂.x₁.x₂ = y₂.y₃.w + y₂.y₃².x₂ + y₂.y₃².x₁ + y₁.x₁² + y₁.y₃².x₁ + y₁².y₃.x₁
• y₂².w = y₂².y₃.x₁ + y₁³.x₁
• y₁.y₂.w = y₁².y₃.x₁ + y₁³.x₁
• y₁².w = y₁².y₃.x₂ + y₁³.x₁
• y₂.x₂.w = y₂.y₃³.x₂ + y₁.x₁.w + y₁.y₃³.x₁
• y₁.w.u₂ = y₁.y₃.x₂.u₂ + y₁.y₃.x₁².w + y₁.y₃².x₁³ + y₁.y₃³.x₂.w + y₁.y₃⁴.x₂² + y₁.y₃⁵.w + y₁.y₃⁶.x₂ + y₁².x₁.u₁ + y₁².y₃⁵.x₁ + y₁³.y₃².x₁²

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 10 onwards, and Benson's tests detect stability from degree 17 onwards.

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

• h₁ = r in degree 8
• h₂ = x₂² + x₁.x₂ + x₁² + y₄⁴ + y₃.w + y₃⁴ + y₂.w + y₂².y₃² + y₂⁴ + y₁.w + y₁.y₂.y₃² + y₁.y₂².y₃ + y₁².y₃² + y₁².y₂.y₃ + y₁².y₂² + y₁⁴ in degree 4
• h₃ = x₁.x₂² + x₁².x₂ + y₄².x₂² + y₄².x₁.x₂ + y₄².x₁² + y₃.u₃ + y₃.u₂ + y₃.u₁ + y₃.x₁.w + y₃².x₁.x₂ + y₃².x₁² + y₃⁴.x₂ + y₂.x₁.w + y₂.y₃².w + y₂.y₃³.x₂ + y₂.y₃³.x₁ + y₂².x₁² + y₂².y₃⁴ + y₂³.y₃.x₁ + y₂⁴.y₃² + y₁.x₂.w + y₁.x₁.w + y₁.y₂.y₃⁴ + y₁.y₂⁴.y₃ + y₁².x₂² + y₁².y₃⁴ + y₁².y₂².y₃² + y₁².y₂⁴ + y₁³.y₃.x₂ + y₁⁴.y₃² + y₁⁴.y₂.y₃ + y₁⁴.y₂² in degree 6
• h₄ = y₄ + y₃ in degree 1

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, 13, 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
• y₃ in degree 1
• y₂ in degree 1
• y₁ in degree 1
• x₂ in degree 2
• x₁ in degree 2
• y₃² in degree 2
• y₂² in degree 2
• y₁.y₂ in degree 2
• y₁² in degree 2
• w in degree 3
• y₃.x₂ in degree 3
• y₃.x₁ in degree 3
• y₂.x₁ in degree 3
• y₂³ in degree 3
• y₁.x₂ in degree 3
• y₁.x₁ in degree 3
• y₁.y₂² in degree 3
• y₁².y₂ in degree 3
• y₁³ in degree 3
• x₁.x₂ in degree 4
• x₁² in degree 4
• y₃.w in degree 4
• y₂.w in degree 4
• y₂².x₁ in degree 4
• y₂⁴ in degree 4
• y₁.w in degree 4
• y₁.y₂³ in degree 4
• y₁².x₂ in degree 4
• y₁².x₁ 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
• x₂.w in degree 5
• x₁.w in degree 5
• y₃.x₁.x₂ in degree 5
• y₃.x₁² in degree 5
• y₂³.x₁ in degree 5
• y₂⁵ in degree 5
• y₁.x₁² in degree 5
• y₁.y₂⁴ in degree 5
• y₁².y₂³ in degree 5
• y₁³.x₂ in degree 5
• y₁³.x₁ in degree 5
• y₁³.y₂² in degree 5
• y₁⁴.y₂ in degree 5
• y₁⁵ in degree 5
• x₁².x₂ in degree 6
• y₃.u₂ in degree 6
• y₃.x₂.w in degree 6
• y₃.x₁.w in degree 6
• y₂.u₁ in degree 6
• y₂⁴.x₁ in degree 6
• y₂⁶ in degree 6
• y₁.u₂ in degree 6
• y₁.u₁ in degree 6
• y₁.x₂.w in degree 6
• y₁.x₁.w in degree 6
• y₁².x₁² in degree 6
• y₁².y₂⁴ in degree 6
• y₁³.y₂³ in degree 6
• y₁⁴.x₂ in degree 6
• y₁⁴.x₁ in degree 6
• y₁⁴.y₂² in degree 6
• y₁⁵.y₂ in degree 6
• y₁⁶ in degree 6
• x₂.u₃ in degree 7
• x₂.u₂ in degree 7
• x₁.u₃ in degree 7
• x₁.u₂ in degree 7
• x₁.u₁ in degree 7
• x₁.x₂.w in degree 7
• x₁².w in degree 7
• y₃.x₁².x₂ in degree 7
• y₂².u₁ in degree 7
• y₂⁵.x₁ in degree 7
• y₁.y₂.u₁ in degree 7
• y₁².u₂ in degree 7
• y₁².u₁ in degree 7
• y₁³.x₁² in degree 7
• y₁³.y₂⁴ in degree 7
• y₁⁴.y₂³ in degree 7
• y₁⁵.x₂ in degree 7
• y₁⁵.x₁ in degree 7
• y₁⁵.y₂² in degree 7
• y₁⁶.y₂ in degree 7
• w.u₂ in degree 8
• y₃.x₂.u₂ in degree 8
• y₃.x₁.u₂ in degree 8
• y₃.x₁.x₂.w in degree 8
• y₃.x₁².w in degree 8
• y₂.x₁.u₁ in degree 8
• y₂³.u₁ in degree 8
• y₂⁶.x₁ in degree 8
• y₁.x₂.u₂ in degree 8
• y₁.x₁.u₁ in degree 8
• y₁.x₁².w in degree 8
• y₁.y₂².u₁ in degree 8
• y₁².y₂.u₁ in degree 8
• y₁³.u₂ in degree 8
• y₁³.u₁ in degree 8
• y₁⁴.x₁² in degree 8
• y₁⁴.y₂⁴ in degree 8
• y₁⁵.y₂³ in degree 8
• y₁⁶.x₂ in degree 8
• y₁⁶.x₁ in degree 8
• y₁⁶.y₂² in degree 8
• x₁.x₂.u₂ in degree 9
• x₁².u₃ in degree 9
• x₁².u₂ in degree 9
• x₁².x₂.w in degree 9
• y₃.w.u₂ in degree 9
• y₂².x₁.u₁ in degree 9
• y₂⁴.u₁ in degree 9
• y₁.y₂³.u₁ in degree 9
• y₁².x₂.u₂ in degree 9
• y₁².x₁.u₁ in degree 9
• y₁².y₂².u₁ in degree 9
• y₁³.y₂.u₁ in degree 9
• y₁⁴.u₂ in degree 9
• y₁⁴.u₁ in degree 9
• y₁⁵.x₁² in degree 9
• y₁⁵.y₂⁴ in degree 9
• y₁⁶.y₂³ in degree 9
• x₂.w.u₂ in degree 10
• x₁.w.u₂ in degree 10
• y₃.x₁.x₂.u₂ in degree 10
• y₃.x₁².u₂ in degree 10
• y₃.x₁².x₂.w in degree 10
• y₂³.x₁.u₁ in degree 10
• y₂⁵.u₁ in degree 10
• y₁.y₂⁴.u₁ in degree 10
• y₁².y₂³.u₁ in degree 10
• y₁³.x₂.u₂ in degree 10
• y₁³.x₁.u₁ in degree 10
• y₁³.y₂².u₁ in degree 10
• y₁⁴.y₂.u₁ in degree 10
• y₁⁵.u₂ in degree 10
• y₁⁵.u₁ in degree 10
• y₁⁶.x₁² in degree 10
• y₁⁶.y₂⁴ in degree 10
• x₁².x₂.u₂ in degree 11
• y₂⁴.x₁.u₁ in degree 11
• y₂⁶.u₁ in degree 11
• y₁².y₂⁴.u₁ in degree 11
• y₁³.y₂³.u₁ in degree 11
• y₁⁴.x₂.u₂ in degree 11
• y₁⁴.x₁.u₁ in degree 11
• y₁⁴.y₂².u₁ in degree 11
• y₁⁵.y₂.u₁ in degree 11
• y₁⁶.u₂ in degree 11
• y₁⁶.u₁ in degree 11
• x₁.x₂.w.u₂ in degree 12
• x₁².w.u₂ in degree 12
• y₃.x₁².x₂.u₂ in degree 12
• y₂⁵.x₁.u₁ in degree 12
• y₁³.y₂⁴.u₁ in degree 12
• y₁⁴.y₂³.u₁ in degree 12
• y₁⁵.x₂.u₂ in degree 12
• y₁⁵.x₁.u₁ in degree 12
• y₁⁵.y₂².u₁ in degree 12
• y₁⁶.y₂.u₁ in degree 12
• y₁⁴.y₂⁴.u₁ in degree 13
• y₁⁵.y₂³.u₁ in degree 13
• y₁⁶.x₂.u₂ in degree 13
• y₁⁶.x₁.u₁ in degree 13
• y₁⁶.y₂².u₁ in degree 13
• x₁².x₂.w.u₂ in degree 14
• y₁⁵.y₂⁴.u₁ in degree 14
• y₁⁶.y₂³.u₁ in degree 14
• y₁⁶.y₂⁴.u₁ in degree 15

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

• y₃².h⁵ + y₃.h⁶ in degree 7
• y₂.w.h⁵ + y₁².x₁.h⁵ + y₂.x₁.h⁶ in degree 9
• y₁.w.h⁵ + y₁².x₁.h⁵ + y₁.x₂.h⁶ in degree 9
• y₁.x₂.w.h⁵ + y₁².x₁.x₂.h⁵ + y₁.x₂².h⁶ in degree 11
• y₁.x₁.w.h⁵ + y₁².x₁².h⁵ + y₁.x₁.x₂.h⁶ in degree 11
• y₁.x₁².w.h⁵ + y₁².x₁³.h⁵ + y₁.x₁².x₂.h⁶ in degree 13

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
• u₃ restricts to 0
• r restricts to y⁸

Restriction to special subgroup number 2, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to 0
• y₄ restricts to y₂
• x₁ restricts to y₄² + y₂.y₄
• x₂ restricts to y₃² + y₂.y₃
• w restricts to y₃.y₄² + y₃².y₄ + y₁.y₂² + y₁².y₂
• u₁ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₁.y₂².y₄² + y₁.y₂².y₃² + y₁.y₂³.y₄ + y₁.y₂³.y₃ + y₁.y₂⁴ + y₁².y₂.y₄² + y₁².y₂.y₃² + y₁².y₂².y₄ + y₁².y₂².y₃ + y₁².y₂³
• u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₃².y₄² + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₃.y₄ + y₁.y₂⁴ + y₁⁴.y₂
• u₃ restricts to y₃³.y₄² + y₃⁴.y₄ + y₂.y₃³.y₄ + y₂³.y₃.y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁.y₂⁴ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³
• r restricts to y₃².y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂².y₃².y₄⁴ + y₂².y₃⁴.y₄² + y₂².y₃⁵.y₄ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃⁴.y₄ + y₂⁴.y₃².y₄² + y₂⁵.y₃².y₄ + y₂⁶.y₃.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃⁴ + y₁.y₂⁵.y₃.y₄ + y₁.y₂⁶.y₃ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂³.y₃².y₄ + y₁².y₂⁵.y₄ + y₁².y₂⁶ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂³.y₄ + 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
• w restricts to 0
• u₁ restricts to y₁.y₂.y₃³ + y₁.y₂².y₃² + y₁².y₃³ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁⁴.y₃
• u₂ restricts to y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + y₁⁴.y₂
• u₃ restricts to y₁.y₂.y₃³ + y₁.y₂³.y₃ + y₁².y₃³ + y₁².y₂³ + y₁⁴.y₃ + y₁⁴.y₂
• r restricts to y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂.y₃⁵.y₄ + y₁.y₂.y₃⁶ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃².y₄³ + y₁.y₂².y₃³.y₄² + y₁.y₂³.y₃.y₄³ + y₁.y₂³.y₃³.y₄ + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₃⁵.y₄ + y₁².y₃⁶ + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃².y₄³ + y₁².y₂.y₃³.y₄² + y₁².y₂².y₄⁴ + y₁².y₂².y₃.y₄³ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₄³ + y₁².y₂³.y₃³ + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₄ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃³.y₄ + y₁⁴.y₂.y₄³ + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂.y₃³ + 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 0
• y₃ restricts to y₃
• y₄ restricts to 0
• x₁ restricts to 0
• x₂ restricts to y₄² + y₂.y₄
• w restricts to y₃.y₄² + y₂.y₃.y₄
• u₁ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄²
• u₂ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₃.y₄ + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• u₃ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₃.y₄ + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• r restricts to y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁷.y₄ + y₂.y₃³.y₄⁴ + y₂.y₃⁴.y₄³ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂².y₃.y₄⁵ + y₂².y₃⁵.y₄ + y₂³.y₃.y₄⁴ + y₂³.y₃⁴.y₄ + y₂⁴.y₃.y₄³ + y₂⁴.y₃³.y₄ + y₂⁶.y₃.y₄ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₄⁴ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃³.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁵.y₄² + y₁.y₂⁵.y₃.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃³.y₄² + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₃³.y₄ + y₁².y₂².y₃⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂³.y₃².y₄ + y₁².y₂³.y₃³ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + 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 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
• w restricts to y₃.y₄² + y₂.y₃.y₄
• u₁ restricts to y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• u₂ restricts to y₃³.y₄² + y₂.y₃³.y₄
• u₃ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₁.y₂².y₄² + y₁.y₂³.y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁².y₂³ + y₁⁴.y₂
• r restricts to y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁶.y₄² + y₂.y₃³.y₄⁴ + y₂.y₃⁴.y₄³ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂².y₃.y₄⁵ + y₂².y₃².y₄⁴ + y₂².y₃³.y₄³ + y₂².y₃⁵.y₄ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃⁴.y₄ + y₂⁴.y₃².y₄² + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₄⁴ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁵.y₃.y₄ + y₁.y₂⁶.y₄ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁⁴.y₄⁴ + y₁⁴.y₃².y₄² + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃².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 0
• y₃ restricts to y₂
• y₄ restricts to 0
• x₁ restricts to y₄²
• x₂ restricts to y₃²
• w restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂.y₃.y₄ + y₂.y₃²
• u₁ restricts to y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂.y₃⁴ + y₂².y₄³ + y₂³.y₄² + y₂³.y₃.y₄ + y₂³.y₃² + y₂⁴.y₄
• u₂ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₃³.y₄ + y₂.y₃⁴ + y₂².y₃³ + y₂³.y₄² + y₂³.y₃² + y₂⁴.y₃
• u₃ restricts to y₃³.y₄² + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃⁴ + y₂².y₄³ + y₂².y₃³ + y₂³.y₄² + y₂⁴.y₄ + y₂⁴.y₃
• r restricts to y₃².y₄⁶ + y₂.y₃².y₄⁵ + y₂.y₃⁵.y₄² + y₂³.y₄⁵ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁵ + y₂⁴.y₄⁴ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁴.y₃⁴ + y₂⁵.y₄³ + y₂⁵.y₃.y₄² + y₂⁶.y₄² + y₂⁶.y₃.y₄ + y₂⁷.y₃ + 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 7, which is 16gp14

• y₁ restricts to y₃
• y₂ restricts to y₃
• y₃ restricts to y₂
• y₄ restricts to 0
• x₁ restricts to y₄² + y₃.y₄
• x₂ restricts to y₄² + y₃.y₄
• w restricts to y₃.y₄² + y₃².y₄ + y₂.y₄² + y₂.y₃.y₄
• u₁ restricts to y₂.y₄⁴ + y₂.y₃.y₄³ + y₂.y₃².y₄² + y₂.y₃³.y₄ + y₂².y₄³ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₂⁴.y₄ + y₁.y₃².y₄² + y₁.y₃³.y₄ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁².y₃³ + y₁⁴.y₃
• u₂ restricts to y₂.y₃.y₄³ + y₂.y₃³.y₄ + y₂².y₄³ + y₂⁴.y₄ + y₁.y₃².y₄² + y₁.y₃³.y₄ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁².y₃³ + y₁⁴.y₃
• u₃ restricts to y₃.y₄⁴ + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃³.y₄ + y₂³.y₄² + y₂³.y₃.y₄
• r restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁵.y₄³ + y₃⁶.y₄² + y₃⁷.y₄ + y₂.y₃.y₄⁶ + y₂.y₃³.y₄⁴ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂².y₃².y₄⁴ + y₂².y₃⁵.y₄ + y₂³.y₄⁵ + y₂³.y₃.y₄⁴ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁷.y₄ + y₁.y₃³.y₄⁴ + y₁.y₃⁵.y₄² + y₁.y₂.y₃⁴.y₄² + y₁.y₂.y₃⁵.y₄ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃³.y₄ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁².y₃⁴.y₄² + y₁².y₃⁵.y₄ + y₁².y₂.y₃³.y₄² + y₁².y₂.y₃⁴.y₄ + y₁².y₂.y₃⁵ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂³.y₃.y₄² + y₁².y₂³.y₃².y₄ + y₁².y₂³.y₃³ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁴.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃³.y₄ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃³ + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

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Poincaré series

(1 + 3t + 6t² + 10t³ + 13t⁴ + 17t⁵ + 19t⁶ + 20t⁷ + 20t⁸ + 17t⁹ + 15t¹⁰ + 11t¹¹ + 8t¹² + 5t¹³ + 2t¹⁴ + t¹⁵) / (1 - t) (1 - t⁴) (1 - t⁶) (1 - t⁸)

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Back to the groups of order 128