Small group number 224 of order 64

G is the group 64gp224

The Hall-Senior number of this group is 173.

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

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 4.

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 12 generators:

• y₁ in degree 1, a nilpotent element
• y₂ in degree 1, a nilpotent element
• y₃ in degree 1
• y₄ in degree 1
• v₁ in degree 4, a nilpotent element
• v₂ in degree 4
• v₃ in degree 4
• v₄ in degree 4
• v₅ in degree 4
• v₆ in degree 4, a regular element
• v₇ in degree 4, a regular element
• t in degree 6

There are 38 minimal relations:

• y₂.y₄ = y₁.y₄ + y₁.y₃
• y₂² = y₁.y₂ + y₁²
• y₁.y₃.y₄ = y₁.y₃² + y₁².y₄
• y₁³ = 0
• y₄.v₅ = y₄.v₃ + y₃.v₄ + y₂.v₃ + y₁.v₃
• y₄.v₄ = y₃.v₅ + y₃.v₄ + y₂.v₃
• y₄.v₁ = y₁.v₃
• y₃.v₁ = y₂.v₃ + y₁.v₃
• y₂.v₅ = y₁.v₃ + y₁.v₂
• y₂.v₄ = 0
• y₂.v₂ = 0
• y₁.v₅ = y₁.v₃ + y₁.v₂
• y₁.v₄ = y₁.v₂
• y₂.v₁ = 0
• y₁.v₁ = 0
• y₂.t = y₂.y₃².v₃ + y₂.y₃².v₇ + y₁.y₄².v₇ + y₁.y₄².v₆ + y₁.y₃².v₇ + y₁.y₃².v₆ + y₁².y₃.v₆ + y₁².y₂.v₆
• y₁.t = y₁.y₃².v₂ + y₁.y₄².v₇ + y₁.y₄².v₆ + y₁².y₄.v₇ + y₁².y₄.v₆
• v₅² = y₄⁴.v₇ + y₃².y₄².v₇
• v₄.v₅ = y₃.y₄³.v₇ + y₃².y₄².v₇ + y₁.y₄³.v₇ + y₁.y₃³.v₇
• v₄² = y₃².y₄².v₇
• v₃.v₅ = y₄⁴.v₇ + y₃³.y₄.v₇
• v₃.v₄ = y₃.y₄³.v₇ + y₃².y₄².v₇ + y₃³.y₄.v₇ + y₂.y₃³.v₇ + y₁.y₄³.v₇
• v₃² = y₄⁴.v₇ + y₃².y₄².v₇ + y₃⁴.v₇
• v₂.v₅ = y₄².t + y₃.y₄.t + y₃⁴.v₅ + y₄⁴.v₇ + y₄⁴.v₆ + y₃.y₄³.v₇ + y₃.y₄³.v₆ + y₃².y₄².v₇ + y₃³.y₄.v₇
• v₂.v₄ = y₃.y₄.t + y₃⁴.v₄ + y₂.y₃³.v₃ + y₃.y₄³.v₇ + y₃.y₄³.v₆ + y₃³.y₄.v₇ + y₁.y₃³.v₇
• v₂.v₃ = y₄².t + y₃.y₄.t + y₃².t + y₃⁴.v₃ + y₁.y₃³.v₂ + y₄⁴.v₇ + y₄⁴.v₆ + y₃.y₄³.v₇ + y₃.y₄³.v₆ + y₃².y₄².v₆ + y₃³.y₄.v₇ + y₃⁴.v₇ + y₂.y₃³.v₆
• v₂² = y₃².y₄².v₃ + y₃³.y₄.v₃ + y₃⁴.v₅ + y₃².y₄².v₇
• v₁.v₅ = y₁.y₄³.v₇ + y₁.y₃³.v₇
• v₁.v₄ = y₁.y₃³.v₇
• v₁.v₃ = y₂.y₃³.v₇ + y₁.y₄³.v₇
• v₁.v₂ = y₁.y₃³.v₇
• v₁² = 0
• v₅.t = y₄².v₃.v₇ + y₄².v₃.v₆ + y₄².v₂.v₇ + y₃.y₄.v₂.v₇ + y₃².v₅.v₆ + y₃².v₄.v₇ + y₃².v₄.v₆ + y₃⁴.y₄².v₇ + y₃⁵.y₄.v₇ + y₂.y₃.v₃.v₇ + y₂.y₃.v₃.v₆ + y₁.y₃.v₂.v₇ + y₁.y₃.v₂.v₆
• v₄.t = y₃.y₄.v₃.v₇ + y₃.y₄.v₃.v₆ + y₃.y₄.v₂.v₇ + y₃².v₅.v₇ + y₃².v₅.v₆ + y₃².v₄.v₇ + y₃⁵.y₄.v₇ + y₂.y₃⁵.v₇ + y₁.y₄.v₃.v₇ + y₁.y₄.v₃.v₆ + y₁.y₃.v₂.v₇
• v₃.t = y₄².v₃.v₇ + y₄².v₃.v₆ + y₄².v₂.v₇ + y₃.y₄.v₂.v₇ + y₃².v₃.v₇ + y₃².v₂.v₇ + y₃⁴.y₄².v₇ + y₃⁵.y₄.v₇ + y₃⁶.v₇ + y₂.y₃.v₃.v₆ + y₁.y₃⁵.v₇
• v₂.t = y₃⁴.t + y₃⁶.v₅ + y₃⁶.v₃ + y₁.y₃⁵.v₂ + y₄².v₂.v₇ + y₄².v₂.v₆ + y₃².v₅.v₇ + y₃².v₄.v₇ + y₃².v₂.v₇ + y₃².y₄⁴.v₇ + y₃⁴.y₄².v₆ + y₃⁶.v₇ + y₂.y₃.v₃.v₇ + y₂.y₃⁵.v₆ + y₁.y₃.v₂.v₇ + y₁.y₃⁵.v₇
• v₁.t = y₂.y₃.v₃.v₇ + y₂.y₃⁵.v₇ + y₁.y₄.v₃.v₇ + y₁.y₄.v₃.v₆ + y₁.y₃⁵.v₇
• t² = y₃².y₄².v₃.v₇ + y₃³.y₄.v₃.v₇ + y₃⁴.v₅.v₇ + y₃⁸.v₇ + y₄⁴.v₇² + y₄⁴.v₆² + y₃².y₄².v₇² + y₃⁴.v₇²

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.y₄² = 0
• y₁².y₃² = 0
• y₁.y₄.v₂ = y₁.y₃.v₂
• y₁.y₃.v₃ = y₁.y₃.v₂
• y₁.y₂.v₃ = 0
• y₁².v₃ = 0
• y₁².v₂ = 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 = v₆ in degree 4
• h₂ = v₇ in degree 4
• h₃ = y₄² + y₃.y₄ + y₃² in degree 2
• h₄ = y₄ 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
• y₃ in degree 1
• y₂ in degree 1
• y₁ in degree 1
• y₂.y₃ in degree 2
• y₁.y₂ in degree 2
• y₁² in degree 2
• y₁².y₂ in degree 3
• v₅ in degree 4
• v₄ in degree 4
• v₃ in degree 4
• v₂ in degree 4
• v₁ in degree 4
• y₃.v₃ in degree 5
• y₃.v₂ in degree 5
• y₂.v₃ in degree 5
• y₁.v₂ in degree 5
• t in degree 6
• y₃.t in degree 7

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

• y₂.y₃ + y₁.h in degree 2
• y₁² + y₁.h in degree 2
• y₁².y₂ + y₁.y₂.h in degree 3
• y₁.y₂.h in degree 3
• y₂.h² in degree 3
• y₁.h² in degree 3
• y₃.v₃ in degree 5
• y₂.v₃ + y₁.v₂ + v₁.h in degree 5
• v₃.h in degree 5
• y₁.v₂.h in degree 6
• v₁.h² in degree 6

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

• y₁.y₂ in degree 2
• y₁² in degree 2
• y₁².y₄ in degree 3
• y₁².y₃ in degree 3
• y₁².y₂ 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
• v₁ restricts to 0
• v₂ restricts to 0
• v₃ restricts to 0
• v₄ restricts to 0
• v₅ restricts to 0
• v₆ restricts to y₁⁴
• v₇ restricts to y₂⁴
• t restricts to 0

Restriction to special subgroup number 2, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₄
• y₄ restricts to y₃
• v₁ restricts to 0
• v₂ restricts to y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₃.y₄
• v₃ restricts to y₂².y₄² + y₂².y₃.y₄ + y₂².y₃²
• v₄ restricts to y₂².y₃.y₄
• v₅ restricts to y₂².y₃.y₄ + y₂².y₃²
• v₆ restricts to y₂.y₃.y₄² + y₂.y₃².y₄ + y₂².y₃.y₄ + y₁.y₃.y₄² + y₁.y₃².y₄ + y₁².y₄² + y₁².y₃.y₄ + y₁².y₃² + y₁⁴
• v₇ restricts to y₂⁴
• t restricts to y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂².y₄⁴ + y₂².y₃³.y₄ + y₂³.y₃.y₄² + y₂³.y₃².y₄ + y₂⁴.y₄² + y₂⁴.y₃.y₄ + y₂⁴.y₃² + y₁.y₃³.y₄² + y₁.y₃⁴.y₄ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₃⁴ + y₁⁴.y₃²

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Poincaré series

(1 + 3t + 3t² - t³ - t⁴ + 3t⁵ + t⁶ - t⁷) / (1 - t) (1 - t²) (1 - t⁴)²

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