Small group number 232 of order 64

G is the group 64gp232

The Hall-Senior number of this group is 163.

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

There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 4.

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 10 generators:

• y₁ in degree 1, a nilpotent element
• y₂ in degree 1
• y₃ in degree 1
• y₄ 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:

• y₄² = y₂.y₄ + y₁.y₂
• y₃.y₄ = y₁.y₄ + y₁²
• y₁².y₄ = y₁².y₃ + y₁³
• y₁².y₂ = 0
• y₄.v₂ = y₁.y₄.w + y₁.y₂.w
• y₄.v₁ = y₂.y₄.w + y₁.y₂.w + y₁.y₂³.y₄
• y₃.v₁ = y₂.v₂ + y₂.y₄.w + y₂.y₃.w + y₂².w + y₁.y₂.w
• y₁.v₂ = y₁.y₄.w + y₁.y₂.w
• y₁.v₁ = y₁.y₄.w
• y₁².w = 0
• w² = y₃².v₂ + y₂.y₃.v₂ + y₂³.w + y₂⁵.y₄ + y₁.y₂.y₄.w + y₁.y₂².w + y₁.y₂⁴.y₄ + y₁.y₂⁵ + y₃².v₄ + y₂.y₄.v₄ + y₂².v₃ + y₁.y₂.v₄ + y₁².v₄
• w.v₂ = y₃.t + y₃⁴.w + y₂³.v₂ + y₁.y₂³.w + y₃³.v₄ + y₂.y₃².v₃ + y₂².y₄.v₃ + y₂².y₃.v₄ + y₂².y₃.v₃ + y₂³.v₃ + y₁.y₃².v₄ + y₁.y₃².v₃ + y₁.y₂².v₃ + y₁².y₃.v₄ + y₁².y₃.v₃
• w.v₁ = y₂.t + y₂.y₃².v₂ + y₂.y₃³.w + y₂³.v₂ + y₂³.v₁ + y₂³.y₄.w + y₂⁶.y₄ + y₁.y₂⁶ + y₂².y₄.v₄ + y₂².y₃.v₄ + y₂².y₃.v₃ + y₂³.v₄ + y₁.y₂².v₄ + y₁².y₃.v₃ + y₁³.v₄ + y₁³.v₃
• y₄.t = y₂³.y₄.w + y₁.y₂⁵.y₄ + y₂².y₄.v₄ + y₂².y₄.v₃ + y₁.y₂.y₄.v₃ + y₁.y₂².v₃ + y₁².y₃.v₄ + y₁³.v₄ + y₁³.v₃
• y₁.t = y₁.y₃³.w + y₁.y₂³.w + y₁.y₃².v₄ + y₁.y₂.y₄.v₃ + y₁.y₂².v₄ + y₁³.v₄
• v₂² = y₂².y₃².v₂ + y₂².y₃³.w + y₂³.y₃.v₂ + y₂³.y₃².w + y₂⁴.y₄.w + y₂⁵.w + y₁.y₂³.y₄.w + y₂².y₃².v₄ + y₂².y₃².v₃ + y₂³.y₄.v₃ + y₂⁴.v₃ + y₁.y₂³.v₃
• v₁.v₂ = y₂.y₃.t + y₂.y₃⁴.w + y₂².t + y₂².y₃³.w + y₂³.y₃.v₂ + y₂³.y₃².w + y₂⁴.v₁ + y₂⁴.y₄.w + y₂⁴.y₃.w + y₂⁵.w + y₂.y₃³.v₄ + y₂².y₃².v₄ + y₂².y₃².v₃ + y₂³.y₄.v₃ + y₂³.y₃.v₃ + y₂⁴.v₄ + y₁.y₂².y₄.v₄ + y₁.y₂³.v₃
• v₁² = y₂².y₃².v₂ + y₂³.y₃.v₂ + y₂⁴.y₄.w + y₂⁴.y₃.w + y₂⁷.y₄ + y₁.y₂³.y₄.w + y₁.y₂⁴.w + y₁.y₂⁶.y₄ + y₁.y₂⁷ + y₂².y₃².v₄ + y₂³.y₄.v₄ + y₂³.y₄.v₃ + y₁.y₂³.v₄ + y₁.y₂³.v₃
• w.t = y₃⁵.v₂ + y₂.y₃⁴.v₂ + y₂².y₃³.v₂ + y₂².y₃⁴.w + y₂³.y₃³.w + y₂⁴.y₃.v₂ + y₂⁴.y₃².w + y₂⁵.y₃.w + y₂⁶.w + y₂⁸.y₄ + y₁.y₂⁷.y₄ + y₁.y₂⁸ + y₃.v₂.v₄ + y₃².w.v₄ + y₃⁵.v₄ + y₂.v₁.v₃ + y₂.y₃.w.v₃ + y₂².w.v₄ + y₂².y₃³.v₄ + y₂³.y₃².v₄ + y₂³.y₃².v₃ + y₂⁴.y₄.v₄ + y₂⁴.y₃.v₃ + y₂⁵.v₃ + y₁.y₃.w.v₄ + y₁.y₃.w.v₃ + y₁.y₂³.y₄.v₃ + y₁.y₂⁴.v₄ + y₁.y₂⁴.v₃
• v₂.t = y₃⁴.t + y₃⁷.w + y₂².y₃².t + y₂².y₃⁴.v₂ + y₂².y₃⁵.w + y₂³.y₃.t + y₂³.y₃³.v₂ + y₂³.y₃⁴.w + y₂⁴.y₃².v₂ + y₂⁵.y₃.v₂ + y₂⁶.v₂ + y₁.y₂⁵.y₄.w + y₃².v₂.v₄ + y₃⁶.v₄ + y₂.y₃.v₂.v₃ + y₂.y₃⁵.v₃ + y₂².v₂.v₄ + y₂².v₂.v₃ + y₂².v₁.v₃ + y₂².y₃.w.v₄ + y₂².y₃.w.v₃ + y₂².y₃⁴.v₄ + y₂².y₃⁴.v₃ + y₂³.w.v₃ + y₂⁴.y₃².v₄ + y₂⁴.y₃².v₃ + y₂⁵.y₄.v₃ + y₂⁵.y₃.v₄ + y₂⁶.v₃ + y₁.y₃⁵.v₄ + y₁.y₃⁵.v₃ + y₁.y₂.y₄.w.v₄ + y₁.y₂⁴.y₄.v₃
• v₁.t = y₂.y₃³.t + y₂.y₃⁵.v₂ + y₂.y₃⁶.w + y₂³.y₃.t + y₂³.y₃³.v₂ + y₂⁴.t + y₂⁴.y₃³.w + y₂⁶.v₁ + y₂⁷.w + y₂⁹.y₄ + y₁.y₂⁶.w + y₁.y₂⁹ + y₂.y₃².w.v₄ + y₂².v₂.v₄ + y₂².v₂.v₃ + y₂².v₁.v₄ + y₂².y₄.w.v₄ + y₂².y₃.w.v₄ + y₂².y₃.w.v₃ + y₂².y₃⁴.v₄ + y₂².y₃⁴.v₃ + y₂³.w.v₄ + y₂³.w.v₃ + y₂³.y₃³.v₃ + y₂⁴.y₃².v₃ + y₂⁵.y₄.v₄ + y₂⁵.y₃.v₃ + y₂⁶.v₄ + y₁.y₂.y₄.w.v₄ + y₁.y₂².w.v₄ + y₁.y₂⁵.v₄
• t² = y₃⁸.v₂ + y₂.y₃⁷.v₂ + y₂².y₃⁴.t + y₂².y₃⁷.w + y₂³.y₃⁶.w + y₂⁴.y₃².t + y₂⁴.y₃⁴.v₂ + y₂⁵.y₃³.v₂ + y₂⁵.y₃⁴.w + y₂⁶.y₃².v₂ + y₂⁶.y₃³.w + y₂⁷.y₃.v₂ + y₂⁹.w + y₂¹¹.y₄ + y₁.y₂⁷.y₄.w + y₁.y₂⁸.w + y₁.y₂¹⁰.y₄ + y₁.y₂¹¹ + y₃⁸.v₄ + y₂².y₃².v₂.v₃ + y₂².y₃³.w.v₄ + y₂².y₃⁶.v₄ + y₂².y₃⁶.v₃ + y₂³.y₃.v₂.v₃ + y₂³.y₃².w.v₄ + y₂³.y₃⁵.v₃ + y₂⁴.y₃.w.v₃ + y₂⁴.y₃⁴.v₄ + y₂⁵.y₃³.v₄ + y₂⁶.y₃².v₄ + y₂⁶.y₃².v₃ + y₂⁷.y₄.v₄ + y₂⁸.v₃ + y₁.y₂³.y₄.w.v₃ + y₁.y₂⁷.v₄ + y₃⁴.v₄² + y₂².y₃².v₄² + y₂².y₃².v₃.v₄ + y₂².y₃².v₃² + y₂³.y₄.v₃² + y₂⁴.v₄² + y₁.y₂³.v₃²

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

• y₁.y₂.y₃ = y₁².y₄
• y₁².y₃² = 0
• y₁³.y₃ = 0
• y₁⁴ = 0

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 4 and depth 2. Here is a homogeneous system of parameters:

• h₁ = v₃ in degree 4
• h₂ = v₄ in degree 4
• h₃ = y₃² + y₂.y₃ + y₂² in degree 2
• h₄ = y₂ in degree 1

The first 2 terms h₁, h₂ 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, 3, 6, 7.
• 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
• y₁ in degree 1
• y₁.y₄ in degree 2
• y₁.y₃ in degree 2
• y₁² in degree 2
• w in degree 3
• y₁³ in degree 3
• v₂ in degree 4
• v₁ in degree 4
• y₄.w in degree 4
• y₃.w in degree 4
• y₁.w in degree 4
• y₃.v₂ in degree 5
• y₁.y₄.w in degree 5
• y₁.y₃.w in degree 5
• t in degree 6
• y₃.t in degree 7

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

• y₁.y₄ + y₄.h in degree 2
• y₁² in degree 2
• y₁³ in degree 3
• y₁.y₃.h in degree 3
• y₄.h² in degree 3
• y₁.h² in degree 3
• y₃.v₂ + v₂.h + v₁.h + y₃.w.h + y₁.w.h in degree 5
• y₁.y₄.w + y₄.w.h in degree 5
• y₁.y₃.w in degree 5
• y₄.w.h² in degree 6
• y₁.w.h² in degree 6

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

• y₁² in degree 2
• y₁².y₃ in degree 3
• y₁³ in degree 3

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
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• v₃ restricts to y₁⁴
• v₄ restricts to y₂⁴
• t restricts to 0

Restriction to special subgroup number 2, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to y₃
• y₃ restricts to 0
• y₄ restricts to y₃
• w restricts to y₂.y₃² + y₂².y₃ + y₁.y₃² + y₁².y₃
• v₁ restricts to y₂.y₃³ + y₂².y₃² + y₁.y₃³ + y₁².y₃²
• v₂ restricts to 0
• v₃ restricts to y₁².y₃² + y₁⁴
• v₄ restricts to y₃⁴ + y₂.y₃³ + y₂⁴ + y₁.y₃³ + y₁².y₃²
• t restricts to y₃⁶ + y₂².y₃⁴ + y₂⁴.y₃² + y₁².y₃⁴ + y₁⁴.y₃²

Restriction to special subgroup number 3, which is 16gp14

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

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Poincaré series

(1 + 3t + 3t² + 2t⁵ - t⁷) / (1 - t) (1 - t²) (1 - t⁴)²

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