Small group number 929 of order 128

G is the group 128gp929

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.

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

This cohomology ring calculation is complete.

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
• w in degree 3
• v₁ in degree 4
• v₂ in degree 4
• u₁ in degree 5
• u₂ in degree 5
• u₃ in degree 5
• t in degree 6
• r₁ in degree 8
• r₂ in degree 8, a regular element

There are 53 minimal relations:

• y₂.y₃ = 0
• y₁.y₃ = 0
• y₂.x₂ = y₁³
• y₁.x₁ = y₁³
• y₁².y₂ = y₁³
• x₁.x₂ = 0
• y₃.w = y₁².x₂
• y₁⁴ = 0
• y₃.v₂ = 0
• y₃.v₁ = 0
• y₁.v₂ = 0
• y₁².w = 0
• w² = x₂.v₁ + x₁.v₁ + y₂.u₂ + y₂.x₁.w + y₂².v₁ + y₁.u₂ + y₁.y₂.v₁ + y₁².v₁
• x₂.v₂ = y₁.x₂.w
• y₃.u₃ = y₃.u₁ + y₁².x₂²
• y₃.u₂ = 0
• y₂.u₃ = y₂².v₂ + y₂².v₁ + y₂³.w + y₁.y₂².w
• y₂.u₁ = y₂.x₁.w + y₂².v₁ + y₁.u₂ + y₁.u₁
• y₁.u₃ = y₁.x₂.w + y₁.y₂.v₁ + y₁.y₂².w
• w.v₂ = x₁.u₂ + y₂.t + y₂.x₁.v₂ + y₂.x₁.v₁ + y₂².u₂ + y₂³.v₂ + y₁.x₂.v₁ + y₁².u₁
• x₂.u₃ = x₂.u₂ + x₂.u₁ + x₂².w + y₁².u₁
• x₁.u₃ = y₃.t + y₃².u₁ + y₂.x₁.v₂ + y₂.x₁.v₁ + y₂².x₁.w
• x₁.u₁ = x₁².w + y₃.t + y₃².u₁ + y₂.x₁.v₁ + y₁².u₁
• y₁.t = y₁.x₂.v₁ + y₁.y₂.u₂ + y₁².u₁ + y₁³.v₁
• y₁².u₂ = 0
• v₂² = x₁².v₁ + y₂.x₁².w
• w.u₃ = x₂².v₁ + y₂.w.v₁ + y₂.x₁.u₂ + y₂².t + y₂².x₁.v₂ + y₂³.x₁.w + y₂⁴.v₂ + y₂⁴.v₁
• x₂.t = x₂².v₁ + y₃.x₂.u₁ + y₁.x₂.u₁ + y₁.x₂².w + y₁².x₂³
• y₁³.u₁ = 0
• v₂.u₃ = y₂.v₁.v₂ + y₂.x₁².v₁ + y₂².x₁.u₂ + y₂².x₁².w + y₂³.t + y₂³.x₁.v₂ + y₂³.x₁.v₁ + y₂⁴.u₂ + y₂⁵.v₂ + y₁.x₂².v₁
• v₂.u₂ = x₁.w.v₁ + y₂.r₁ + y₂.v₁² + y₂.x₁².v₂ + y₂⁵.v₂ + y₂⁶.w + y₁.w.u₂ + y₁.y₂.w.v₁ + y₁.y₂⁴.v₁ + y₁.y₂⁵.w
• v₂.u₁ = x₁².u₂ + y₂.v₁.v₂ + y₂.x₁.t + y₂.x₁².v₂ + y₂.x₁².v₁ + y₂².x₁.u₂ + y₂³.x₁.v₂ + y₁.w.u₁ + y₁.y₂.w.v₁
• v₁.u₃ = x₂.w.v₁ + y₂.v₁.v₂ + y₂.v₁² + y₂².w.v₁ + y₁.w.u₁
• y₃.r₁ = y₃.x₂⁴ + y₃³.x₂³ + y₃⁴.u₁ + y₃⁵.x₂²
• y₁.r₁ = y₁.v₁² + y₁.x₂⁴ + y₁.y₂⁵.w
• u₃² = x₂³.v₁ + y₃².x₂⁴ + y₃³.x₂.u₁ + y₂².v₁² + y₂².x₁².v₁ + y₂³.x₁².w + y₂⁴.x₁.v₁ + y₂⁵.u₂ + y₂⁵.x₁.w + y₂⁶.v₁ + y₁.x₂².u₁ + y₁.y₂⁴.u₂ + y₁.y₂⁵.v₁ + y₃².r₂
• u₂.u₃ = x₂.w.u₁ + y₂.v₁.u₂ + y₂.x₁.w.v₁ + y₂².r₁ + y₂².v₁² + y₂².w.u₂ + y₂².x₁².v₂ + y₂⁶.v₂ + y₂⁷.w + y₁.x₂.w.v₁ + y₁.x₂².u₁ + y₁.y₂².w.v₁ + y₁.y₂⁵.v₁ + y₁.y₂⁶.w
• u₂² = x₂.v₁² + x₁.v₁² + y₂.v₁.u₂ + y₂².r₁ + y₂².w.u₂ + y₂².x₁².v₂ + y₂³.x₁.u₂ + y₂³.x₁².w + y₂⁴.x₁.v₂ + y₂⁵.u₂ + y₂⁵.x₁.w + y₂⁶.v₁ + y₁.v₁.u₂ + y₁.y₂.v₁² + y₁.y₂.w.u₂ + y₁.y₂².w.v₁ + y₁.y₂⁴.u₂ + y₁².v₁² + y₁².x₂⁴ + y₂².r₂ + y₁².r₂
• u₁.u₃ = x₂.w.u₁ + y₃².x₂⁴ + y₃³.x₂.u₁ + y₂.x₁.w.v₁ + y₂.x₁².u₂ + y₂².v₁.v₂ + y₂².v₁² + y₂².x₁.t + y₂².x₁².v₂ + y₂³.w.v₁ + y₂³.x₁².w + y₂⁴.x₁.v₂ + y₂⁴.x₁.v₁ + y₁.v₁.u₂ + y₁.v₁.u₁ + y₁.x₂.w.v₁ + y₁.y₂.w.u₂ + y₃².r₂
• u₁.u₂ = x₂.v₁² + x₁.w.u₂ + y₂.v₁.u₂ + y₁.v₁.u₁ + y₁.x₂.w.v₁ + y₁.y₂.w.u₂ + y₁.y₂⁴.u₂ + y₁.y₂⁵.v₁ + y₁.y₂⁶.w + y₁².x₂⁴ + y₁.y₂.r₂ + y₁².r₂
• u₁² = x₂.v₁² + x₁³.v₁ + y₃².x₂⁴ + y₃³.x₂.u₁ + y₂.x₁².u₂ + y₂.x₁³.w + y₂².v₁² + y₂².x₁².v₁ + y₁.v₁.u₁ + y₁.y₂.v₁² + y₁².v₁² + y₁².x₂⁴ + y₃².r₂ + y₁².r₂
• v₂.t = x₁.r₁ + x₁.v₁² + x₁.w.u₂ + x₁³.v₂ + y₃⁴.t + y₃⁵.u₁ + y₂.w.t + y₂.x₁.w.v₁ + y₂.x₁².u₂ + y₂².v₁.v₂ + y₂².w.u₂ + y₂³.x₁.u₂ + y₂³.x₁².w + y₂⁴.t + y₂⁴.x₁.v₁ + y₂⁵.u₂ + y₂⁵.x₁.w + y₂⁶.v₂ + y₁.x₂.w.v₁
• x₂.r₁ = x₂.v₁² + x₂⁵ + y₃².x₂⁴ + y₃³.x₂.u₁ + y₃⁴.x₂³ + y₁.x₂².u₁ + y₁².x₂⁴
• u₃.t = x₂².w.v₁ + y₃³.x₂⁴ + y₃⁴.x₂.u₁ + y₂.v₁.t + y₂.x₁.r₁ + y₂.x₁.v₁² + y₂.x₁.w.u₂ + y₂.x₁³.v₂ + y₂².x₁.w.v₁ + y₂².x₁².u₂ + y₂³.v₁.v₂ + y₂³.w.u₂ + y₂⁴.x₁.u₂ + y₂⁴.x₁².w + y₂⁵.t + y₂⁵.x₁.v₁ + y₂⁶.u₂ + y₂⁶.x₁.w + y₂⁷.v₂ + y₁.x₂³.v₁ + y₁.y₂².w.u₂ + y₃.x₁.r₂ + y₃³.r₂
• u₂.t = w.r₁ + w.v₁² + x₂.v₁.u₁ + x₂⁴.w + x₁.v₁.u₂ + x₁³.u₂ + y₂.x₁.v₁² + y₂.x₁.w.u₂ + y₂.x₁².t + y₂.x₁³.v₂ + y₂.x₁³.v₁ + y₂².v₁.u₂ + y₂².x₁.w.v₁ + y₂².x₁³.w + y₂³.v₁² + y₂³.w.u₂ + y₂⁴.x₁.u₂ + y₂⁵.t + y₂⁵.x₁.v₂ + y₂⁵.x₁.v₁ + y₂⁶.u₂ + y₂⁶.x₁.w + y₂⁸.w + y₁.x₂.v₁² + y₁.x₂.w.u₁ + y₁.y₂².v₁² + y₁.y₂².w.u₂ + y₁.y₂³.w.v₁ + y₁³.v₁² + y₂.x₁.r₂ + y₂³.r₂ + y₁.y₂².r₂ + y₁³.r₂
• u₁.t = x₂.v₁.u₁ + x₁.w.t + y₃³.x₂⁴ + y₃⁴.x₂.u₁ + y₂.v₁.t + y₁.x₂.v₁² + y₁.x₂.w.u₁ + y₁.y₂².v₁² + y₁.y₂².w.u₂ + y₁.y₂⁵.u₂ + y₁.y₂⁶.v₁ + y₁.y₂⁷.w + y₃.x₁.r₂ + y₃³.r₂ + y₁.y₂².r₂
• t² = x₂².v₁² + x₁².r₁ + x₁⁴.v₂ + y₃⁴.x₂⁴ + y₃⁴.x₁.t + y₃⁵.x₂.u₁ + y₃⁶.t + y₃⁷.u₁ + y₂.x₁².w.v₁ + y₂.x₁⁴.w + y₂².x₁.v₁² + y₂².x₁³.v₂ + y₂².x₁³.v₁ + y₂³.v₁.u₂ + y₂³.x₁².u₂ + y₂³.x₁³.w + y₂⁴.r₁ + y₂⁴.w.u₂ + y₂⁴.x₁².v₂ + y₂⁵.x₁.u₂ + y₂⁶.x₁.v₂ + y₂⁷.u₂ + y₂⁷.x₁.w + y₂⁸.v₁ + y₁.y₂².v₁.u₂ + y₁.y₂³.v₁² + y₁.y₂³.w.u₂ + y₁.y₂⁴.w.v₁ + y₁.y₂⁶.u₂ + x₁².r₂ + y₃⁴.r₂ + y₂⁴.r₂
• v₂.r₁ = v₁².v₂ + x₁.v₁.t + x₁².v₁² + x₁⁴.v₁ + y₂.x₁.v₁.u₂ + y₂.x₁.w.t + y₂.x₁⁴.w + y₂².x₁.r₁ + y₂².x₁.v₁² + y₂².x₁.w.u₂ + y₂².x₁³.v₂ + y₂³.x₁².u₂ + y₂⁴.x₁.t + y₂⁴.x₁².v₂ + y₂⁵.x₁².w + y₂⁶.t + y₂⁶.x₁.v₂ + y₂⁶.x₁.v₁ + y₂⁷.u₂ + y₂⁷.x₁.w + y₂⁸.v₂ + y₁.x₂⁴.w
• u₃.r₁ = x₂.w.v₁² + x₂⁴.u₂ + x₂⁴.u₁ + x₂⁵.w + y₃².x₂³.u₁ + y₃⁴.x₂².u₁ + y₃⁵.x₂⁴ + y₃⁶.x₂.u₁ + y₂.v₁.r₁ + y₂.v₁².v₂ + y₂.x₁.v₁.t + y₂.x₁².v₁² + y₂.x₁⁴.v₁ + y₂².w.r₁ + y₂².x₁.v₁.u₂ + y₂².x₁.w.t + y₂².x₁⁴.w + y₂³.x₁.r₁ + y₂³.x₁.v₁² + y₂³.x₁.w.u₂ + y₂³.x₁³.v₂ + y₂⁴.x₁².u₂ + y₂⁵.x₁.t + y₂⁵.x₁².v₂ + y₂⁶.x₁².w + y₂⁷.t + y₂⁷.x₁.v₂ + y₂⁷.x₁.v₁ + y₂⁸.u₂ + y₂⁸.x₁.w + y₂⁹.v₂ + y₁.w.v₁.u₁ + y₁.x₂².w.u₁ + y₁.y₂⁷.u₂ + y₁.y₂⁸.v₁ + y₃⁵.r₂
• u₂.r₁ = v₁².u₂ + w.v₁.t + x₂.w.v₁² + x₂⁴.u₂ + x₁.w.v₁² + x₁³.w.v₁ + y₂.w.v₁.u₂ + y₂.x₁.v₁.t + y₂.x₁².r₁ + y₂.x₁².v₁² + y₂.x₁⁴.v₂ + y₂².w.r₁ + y₂².w.v₁² + y₂².x₁.v₁.u₂ + y₂².x₁.w.t + y₂³.v₁.t + y₂³.x₁³.v₁ + y₂⁴.v₁.u₂ + y₂⁴.w.t + y₂⁴.x₁³.w + y₂⁵.r₁ + y₂⁵.v₁.v₂ + y₂⁵.v₁² + y₂⁵.x₁².v₁ + y₂⁶.x₁.u₂ + y₂⁷.t + y₂⁷.x₁.v₂ + y₂⁸.x₁.w + y₂⁹.v₁ + y₂¹⁰.w + y₁.w.v₁.u₁ + y₁.y₂⁴.v₁² + y₁.y₂⁴.w.u₂ + y₁.y₂⁵.w.v₁ + y₁.y₂⁷.u₂ + y₁.y₂⁹.w + y₂.v₂.r₂ + y₁.y₂.w.r₂
• u₁.r₁ = v₁².u₁ + x₂⁴.u₁ + x₁.w.r₁ + x₁.w.v₁² + y₃².x₂³.u₁ + y₃⁴.x₂².u₁ + y₃⁵.x₂⁴ + y₃⁶.x₂.u₁ + y₂.v₁.r₁ + y₂.v₁³ + y₁.x₂².v₁² + y₁.y₂⁴.w.u₂ + y₁.y₂⁵.w.v₁ + y₃⁵.r₂
• t.r₁ = v₁².t + x₂⁵.v₁ + x₁.v₁.r₁ + x₁.v₁³ + x₁².v₁.t + x₁³.r₁ + x₁⁵.v₂ + y₃.x₂⁴.u₁ + y₃³.x₂³.u₁ + y₃⁴.x₁².t + y₃⁵.x₂².u₁ + y₃⁶.x₂⁴ + y₃⁶.x₁.t + y₃⁷.x₂.u₁ + y₃⁸.t + y₃⁹.u₁ + y₂.w.v₁.t + y₂.x₁.w.r₁ + y₂.x₁².w.t + y₂.x₁³.w.v₁ + y₂.x₁⁴.u₂ + y₂².w.v₁.u₂ + y₂².x₁².v₁² + y₂².x₁⁴.v₁ + y₂³.w.r₁ + y₂³.w.v₁² + y₂³.x₁².w.v₁ + y₂³.x₁³.u₂ + y₂³.x₁⁴.w + y₂⁴.v₁.t + y₂⁴.x₁.r₁ + y₂⁴.x₁.v₁² + y₂⁴.x₁³.v₂ + y₂⁵.v₁.u₂ + y₂⁵.w.t + y₂⁵.x₁.w.v₁ + y₂⁵.x₁².u₂ + y₂⁶.x₁.t + y₂⁹.u₂ + y₂¹⁰.v₁ + y₁.x₂².v₁.u₁ + y₁.x₂⁴.u₁ + y₁.x₂⁵.w + y₁.y₂².w.v₁² + y₁.y₂⁵.v₁² + y₁.y₂⁸.u₂ + y₁.y₂⁹.v₁ + y₁².x₂⁶ + x₁.v₂.r₂ + y₃⁴.x₁.r₂ + y₃⁶.r₂ + y₂².v₂.r₂ + y₂².x₁².r₂ + y₁.y₂².w.r₂
• r₁² = v₁⁴ + x₂⁸ + x₁².v₁.r₁ + x₁².v₁³ + x₁⁴.w.u₂ + x₁⁶.v₁ + y₃⁴.x₂⁶ + y₃⁹.x₂.u₁ + y₂.x₁².w.r₁ + y₂.x₁³.w.t + y₂.x₁⁴.w.v₁ + y₂.x₁⁵.u₂ + y₂.x₁⁶.w + y₂².x₁³.r₁ + y₂².x₁³.w.u₂ + y₂².x₁⁴.t + y₂³.x₁².v₁.u₂ + y₂³.x₁³.w.v₁ + y₂³.x₁⁵.w + y₂⁴.x₁².r₁ + y₂⁴.x₁².v₁² + y₂⁵.x₁².w.v₁ + y₂⁶.x₁³.v₁ + y₂⁷.x₁².u₂ + y₂⁹.x₁².w + y₂¹⁰.x₁.v₁ + y₂¹¹.u₂ + y₂¹¹.x₁.w + y₂¹².v₁ + y₁.y₂¹⁰.u₂ + y₁.y₂¹¹.v₁ + x₁².v₁.r₂ + y₃⁸.r₂ + y₂.x₁².w.r₂ + y₂⁴.x₁².r₂

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

• y₁³.x₂ = 0
• y₁.x₂.u₂ = y₁.x₂.u₁ + y₁³.u₁
• y₁².x₂.v₁ = 0
• y₁².x₂.u₁ = 0
• x₂.w.u₂ = x₂.w.u₁ + y₁.x₂².u₁
• x₁.v₁.v₂ = x₁.w.u₂ + y₂.w.t + y₂².r₁ + y₂².v₁.v₂ + y₂².v₁² + y₂².w.u₂ + y₂².x₁².v₂ + y₂³.x₁.u₂ + y₂⁴.t + y₂⁴.x₁.v₂ + y₂⁴.x₁.v₁ + y₂⁵.u₂ + y₂⁷.w + y₁.y₂.w.u₂ + y₁.y₂².w.v₁ + y₁.y₂⁵.v₁ + y₁.y₂⁶.w
• x₂.v₁.u₂ = x₂.v₁.u₁ + y₁.x₂.w.u₁ + y₁².v₁.u₁ + y₁³.v₁²

Completion information

This cohomology ring was obtained from a calculation out to degree 17. The cohomology ring approximation is stable from degree 16 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 = r₂ in degree 8
• h₂ = v₁ + x₂² + x₁² + y₃⁴ + y₂⁴ in degree 4
• h₃ = x₂.v₁ + x₁.v₁ + y₃².x₂² + y₃².x₁² + y₂².v₁ + y₂².x₁² in degree 6
• h₄ = y₂ 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
• y₃ in degree 1
• y₁ in degree 1
• x₂ in degree 2
• x₁ in degree 2
• y₃² in degree 2
• y₁² in degree 2
• w in degree 3
• y₃.x₂ in degree 3
• y₃.x₁ in degree 3
• y₃³ in degree 3
• y₁.x₂ in degree 3
• v₂ in degree 4
• x₂² in degree 4
• x₁² in degree 4
• y₃².x₂ in degree 4
• y₃².x₁ in degree 4
• y₃⁴ in degree 4
• y₁.w in degree 4
• y₁².x₂ 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.x₁² in degree 5
• y₃³.x₂ in degree 5
• y₃³.x₁ in degree 5
• y₃⁵ in degree 5
• y₁.x₂² in degree 5
• t in degree 6
• x₁.v₂ in degree 6
• x₁³ in degree 6
• y₃.u₁ in degree 6
• y₃².x₁² in degree 6
• y₃⁴.x₂ in degree 6
• y₃⁴.x₁ in degree 6
• y₃⁶ in degree 6
• y₁.u₁ in degree 6
• y₁.x₂.w in degree 6
• y₁².x₂² in degree 6
• x₂.u₂ in degree 7
• x₂.u₁ in degree 7
• x₂².w in degree 7
• x₁.u₂ in degree 7
• x₁².w in degree 7
• y₃.t in degree 7
• y₃².u₁ in degree 7
• y₃³.x₁² in degree 7
• y₃⁵.x₂ in degree 7
• y₃⁵.x₁ in degree 7
• r in degree 8
• w.u₂ in degree 8
• w.u₁ in degree 8
• x₁.t in degree 8
• x₁².v₂ in degree 8
• y₃.x₂.u₁ in degree 8
• y₃².t in degree 8
• y₃³.u₁ in degree 8
• y₃⁴.x₁² in degree 8
• y₃⁶.x₂ in degree 8
• y₃⁶.x₁ in degree 8
• y₁.x₂.u₁ in degree 8
• y₁.x₂².w in degree 8
• w.t in degree 9
• x₂².u₁ in degree 9
• x₁².u₂ in degree 9
• x₁³.w in degree 9
• y₃.x₁.t in degree 9
• y₃².x₂.u₁ in degree 9
• y₃³.t in degree 9
• y₃⁴.u₁ in degree 9
• y₃⁵.x₁² in degree 9
• y₁.w.u₁ in degree 9
• x₂.w.u₁ in degree 10
• x₁.r in degree 10
• x₁².t in degree 10
• y₃².x₁.t in degree 10
• y₃³.x₂.u₁ in degree 10
• y₃⁴.t in degree 10
• y₃⁵.u₁ in degree 10
• y₃⁶.x₁² in degree 10
• y₁.x₂².u₁ in degree 10
• w.r in degree 11
• x₁.w.t in degree 11
• y₃³.x₁.t in degree 11
• y₃⁴.x₂.u₁ in degree 11
• y₃⁵.t in degree 11
• y₁.x₂.w.u₁ in degree 11
• x₂².w.u₁ in degree 12
• x₁².r in degree 12
• y₃⁴.x₁.t in degree 12
• y₃⁵.x₂.u₁ in degree 12
• y₃⁶.t in degree 12
• x₁.w.r in degree 13
• x₁².w.t in degree 13
• y₃⁵.x₁.t in degree 13
• y₁.x₂².w.u₁ in degree 13
• y₃⁶.x₁.t in degree 14
• x₁².w.r in degree 15

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

• y₃ in degree 1
• x₂ + y₁² in degree 2
• y₃² in degree 2
• y₃.x₂ in degree 3
• y₃.x₁ in degree 3
• y₃³ in degree 3
• y₁.x₂ + y₁³ in degree 3
• y₁².h in degree 3
• x₂² + y₁².x₂ in degree 4
• y₃².x₂ in degree 4
• y₃².x₁ in degree 4
• y₃⁴ in degree 4
• y₁².x₂ + y₁⁴ in degree 4
• u₃ + v₂.h + x₁².h + y₁.w.h + w.h² + h⁵ in degree 5
• x₂.w + y₁².w in degree 5
• y₃.x₁² in degree 5
• y₃³.x₂ in degree 5
• y₃³.x₁ in degree 5
• y₃⁵ in degree 5
• y₁.x₂² + y₁³.x₂ in degree 5
• x₁³ + x₁.h⁴ + h⁶ in degree 6
• y₃.u₁ in degree 6
• y₃².x₁² in degree 6
• y₃⁴.x₂ in degree 6
• y₃⁴.x₁ in degree 6
• y₃⁶ in degree 6
• y₁.x₂.w + y₁³.w in degree 6
• y₁².x₂² + y₁⁴.x₂ in degree 6
• x₂.u₂ + y₁².u₂ in degree 7
• x₂.u₁ + y₁².u₁ in degree 7
• x₂².w + y₁².x₂.w in degree 7
• y₃.t in degree 7
• y₃².u₁ in degree 7
• y₃³.x₁² in degree 7
• y₃⁵.x₂ in degree 7
• y₃⁵.x₁ in degree 7
• y₁.u₁.h in degree 7
• y₁.h⁶ in degree 7
• y₃.x₂.u₁ in degree 8
• y₃².t in degree 8
• y₃³.u₁ in degree 8
• y₃⁴.x₁² in degree 8
• y₃⁶.x₂ in degree 8
• y₃⁶.x₁ in degree 8
• y₁.x₂.u₁ + y₁³.u₁ in degree 8
• y₁.x₂².w + y₁³.x₂.w in degree 8
• x₂².u₁ + y₁².x₂.u₁ in degree 9
• x₁³.w + x₁.w.h⁴ + w.h⁶ in degree 9
• y₃.x₁.t in degree 9
• y₃².x₂.u₁ in degree 9
• y₃³.t in degree 9
• y₃⁴.u₁ in degree 9
• y₃⁵.x₁² in degree 9
• y₁.w.u₁ + y₁.w.h⁵ in degree 9
• x₂.w.u₁ + y₁².w.u₁ in degree 10
• y₃².x₁.t in degree 10
• y₃³.x₂.u₁ in degree 10
• y₃⁴.t in degree 10
• y₃⁵.u₁ in degree 10
• y₃⁶.x₁² in degree 10
• y₁.x₂².u₁ + y₁³.x₂.u₁ in degree 10
• y₁.w.h⁶ in degree 10
• y₃³.x₁.t in degree 11
• y₃⁴.x₂.u₁ in degree 11
• y₃⁵.t in degree 11
• y₁.x₂.w.u₁ + y₁³.w.u₁ in degree 11
• u₁.h⁶ + x₁.w.h⁶ + x₁².h⁷ + h¹¹ in degree 11
• x₂².w.u₁ + y₁².x₂.w.u₁ in degree 12
• y₃⁴.x₁.t in degree 12
• y₃⁵.x₂.u₁ in degree 12
• y₃⁶.t in degree 12
• x₁².w.t + x₁².r.h + x₁.w.t.h² + x₁.r.h³ + x₁².t.h³ + w.u₂.h⁵ + w.u₁.h⁵ + x₁.t.h⁵ + x₁².v₂.h⁵ + x₁.u₂.h⁶ + x₁².w.h⁶ + x₁.v₂.h⁷ + u₂.h⁸ + x₁.w.h⁸ + v₂.h⁹ + w.h¹⁰ + h¹³ in degree 13
• y₃⁵.x₁.t in degree 13
• y₁.x₂².w.u₁ + y₁³.x₂.w.u₁ in degree 13
• y₃⁶.x₁.t in degree 14
• w.u₁.h⁶ + x₁.w².h⁶ + x₁².w.h⁷ + w.h¹¹ in degree 14

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

• y₁² in degree 2
• y₁³ in degree 3
• y₁².x₂ in degree 4
• y₁².x₂² in degree 6
• y₁².u₁ 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
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u₁ restricts to 0
• u₂ restricts to 0
• u₃ restricts to 0
• t restricts to 0
• r₁ restricts to 0
• r₂ restricts to y⁸

Restriction to special subgroup number 2, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₂
• x₁ restricts to 0
• x₂ restricts to y₃² + y₂.y₃
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u₁ restricts to y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + 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.y₂³.y₃² + y₁.y₂⁴.y₃ + y₁².y₂².y₃² + y₁².y₂³.y₃ + y₁².y₂⁴ + y₁⁴.y₂²
• r₁ restricts to y₃⁸ + y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁵.y₃³ + y₂⁶.y₃² + y₁.y₂⁵.y₃² + y₁.y₂⁶.y₃ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁⁴.y₂⁴
• r₂ restricts to y₃⁸ + y₂⁴.y₃⁴ + y₁.y₂³.y₃⁴ + y₁.y₂⁵.y₃² + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁⁴.y₃⁴ + y₁⁴.y₂³.y₃ + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 3, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₂
• x₁ restricts to y₃² + y₂.y₃
• x₂ restricts to 0
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u₁ restricts to y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁².y₂³ + 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.y₂.y₃⁴ + y₁.y₂⁴.y₃ + y₁².y₃⁴ + y₁².y₂².y₃² + y₁².y₂⁴ + y₁⁴.y₃² + y₁⁴.y₂.y₃ + y₁⁴.y₂²
• r₁ restricts to y₁.y₂⁵.y₃² + y₁.y₂⁶.y₃ + y₁².y₂⁴.y₃² + y₁².y₂⁵.y₃ + y₁².y₂⁶ + y₁⁴.y₂⁴
• r₂ restricts to y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to 0
• x₁ restricts to 0
• x₂ restricts to y₂²
• w restricts to y₂.y₃² + y₂².y₃
• v₁ restricts to y₃⁴ + y₂².y₃²
• v₂ restricts to 0
• u₁ restricts to y₂.y₃⁴ + y₂³.y₃²
• u₂ restricts to y₂.y₃⁴ + y₂³.y₃²
• u₃ restricts to y₂³.y₃² + y₂⁴.y₃
• t restricts to y₂².y₃⁴ + y₂⁴.y₃²
• r₁ restricts to y₃⁸ + y₂⁴.y₃⁴ + y₂⁸
• r₂ restricts to y₃⁸ + y₂⁶.y₃² + y₂⁸ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 5, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₂
• y₃ restricts to 0
• x₁ restricts to y₃² + y₂.y₃
• x₂ restricts to 0
• w restricts to y₃.y₄² + y₃².y₄ + y₂.y₃.y₄ + y₁².y₂
• v₁ restricts to y₄⁴ + y₃².y₄² + y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₄² + y₂².y₃.y₄
• v₂ restricts to y₃².y₄² + y₃³.y₄ + y₂.y₃.y₄² + y₂².y₃.y₄ + y₁.y₂.y₃² + y₁.y₂².y₃
• u₁ restricts to y₃³.y₄² + y₃⁴.y₄ + y₂.y₄⁴ + y₂².y₃.y₄² + y₂³.y₄² + y₂³.y₃.y₄ + y₁².y₂.y₃² + y₁².y₂².y₃
• u₂ restricts to y₃.y₄⁴ + y₃³.y₄² + y₂.y₄⁴ + y₂.y₃².y₄² + y₂².y₃².y₄ + y₂³.y₄² + y₂³.y₃.y₄ + y₁².y₂.y₃² + y₁².y₂².y₃ + y₁⁴.y₂
• u₃ restricts to y₂.y₄⁴ + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂³.y₄² + y₂³.y₃.y₄ + y₁.y₂².y₃² + y₁.y₂³.y₃ + y₁².y₂³
• t restricts to y₃⁴.y₄² + y₃⁵.y₄ + y₂.y₃.y₄⁴ + y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃².y₄² + y₂³.y₃.y₄² + y₂⁴.y₄² + y₁.y₃³.y₄² + y₁.y₃⁴.y₄ + y₁.y₂.y₃².y₄² + y₁.y₂.y₃⁴ + y₁.y₂².y₃².y₄ + y₁.y₂⁴.y₃ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₃⁴ + y₁².y₂.y₃.y₄² + y₁².y₂².y₃.y₄ + y₁².y₂³.y₃ + y₁³.y₂.y₃² + y₁³.y₂².y₃ + y₁⁴.y₃² + y₁⁴.y₂.y₃ + y₁⁴.y₂²
• r₁ restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁴.y₄⁴ + y₃⁵.y₄³ + y₃⁷.y₄ + y₂.y₃.y₄⁶ + y₂.y₃³.y₄⁴ + y₂.y₃⁵.y₄² + y₂².y₃.y₄⁵ + y₂².y₃².y₄⁴ + y₂³.y₃.y₄⁴ + y₂³.y₃³.y₄² + y₂⁴.y₄⁴ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁵.y₃².y₄ + y₁.y₃³.y₄⁴ + y₁.y₃⁵.y₄² + y₁.y₂.y₃⁶ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂².y₃⁵ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁴.y₃³ + y₁.y₂⁵.y₃² + y₁.y₂⁶.y₃ + y₁².y₃².y₄⁴ + y₁².y₃⁵.y₄ + y₁².y₂.y₃.y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃³.y₄ + y₁².y₂³.y₃.y₄² + y₁².y₂⁶ + y₁³.y₂.y₃⁴ + y₁³.y₂³.y₃² + y₁⁴.y₃².y₄² + y₁⁴.y₃³.y₄ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁵.y₂.y₃² + y₁⁵.y₂².y₃
• r₂ restricts to y₄⁸ + y₃².y₄⁶ + y₃³.y₄⁵ + y₃⁵.y₄³ + y₂.y₃².y₄⁵ + y₂.y₃⁴.y₄³ + y₂.y₃⁵.y₄² + y₂.y₃⁶.y₄ + y₂².y₃³.y₄³ + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂⁴.y₄⁴ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₁.y₃³.y₄⁴ + y₁.y₃⁵.y₄² + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃³.y₄² + y₁.y₂².y₃⁴.y₄ + y₁.y₂³.y₃².y₄² + y₁.y₂³.y₃⁴ + y₁.y₂⁴.y₃.y₄² + y₁.y₂⁴.y₃².y₄ + y₁.y₂⁶.y₃ + y₁².y₃⁴.y₄² + y₁².y₃⁵.y₄ + y₁².y₂.y₃.y₄⁴ + y₁².y₂².y₄⁴ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃³.y₄ + y₁².y₂³.y₃².y₄ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃.y₄ + y₁².y₂⁶ + y₁³.y₂.y₃⁴ + y₁³.y₂³.y₃² + y₁⁴.y₄⁴ + y₁⁴.y₃³.y₄ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₃.y₄² + y₁⁴.y₂².y₄² + y₁⁴.y₂².y₃.y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁵.y₂.y₃² + y₁⁵.y₂².y₃ + y₁⁶.y₂² + y₁⁸

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Poincaré series

(1 + 2t + 3t² + 3t³ + 3t⁴ + 5t⁵ + 4t⁶ + 2t⁷ + 2t⁸ + 2t⁹ + t¹⁰ - t¹¹ - t¹² - t¹⁴ - t¹⁵) / (1 - t) (1 - t⁴) (1 - t⁶) (1 - t⁸)

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Back to the groups of order 128