Small group number 219 of order 64

G is the group 64gp219

The Hall-Senior number of this group is 175.

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

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

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 10 generators:

• y₁ in degree 1
• y₂ in degree 1
• y₃ in degree 1
• y₄ in degree 1
• w₁ in degree 3
• w₂ in degree 3
• v₁ in degree 4
• v₂ in degree 4, a regular element
• v₃ in degree 4, a regular element
• t in degree 6

There are 21 minimal relations:

• y₃.y₄ = y₁.y₄ + y₁.y₂
• y₂.y₄ = y₁²
• y₁.y₂.y₃ = y₁².y₃
• y₁².y₄ = y₁².y₂
• y₄.w₁ = y₁.w₂ + y₁³.y₂
• y₃.w₂ = y₂.w₂ + y₂.w₁ + y₁.w₂ + y₁.w₁ + y₁³.y₂ + y₁⁴
• y₄.v₁ = y₄².w₂ + y₁².w₁ + y₁⁴.y₂
• y₁.v₁ = y₁.y₄.w₂ + y₁.y₃.w₁ + y₁².w₂ + y₁⁴.y₂ + y₁⁵
• y₁.y₂.w₁ = y₁².w₂ + y₁⁴.y₂
• w₂² = y₄³.w₂ + y₂².y₃.w₁ + y₂³.w₂ + y₄².v₃ + y₄².v₂ + y₂².v₃ + y₁².v₃
• w₁.w₂ = y₂.y₃².w₁ + y₂².y₃.w₁ + y₂³.w₁ + y₁.y₄².w₂ + y₁.y₃².w₁ + y₁³.w₂ + y₁⁵.y₂ + y₂.y₃.v₃ + y₂².v₃ + y₁.y₄.v₃ + y₁.y₄.v₂ + y₁.y₃.v₃ + y₁².v₃
• w₁² = y₃³.w₁ + y₂³.w₁ + y₁³.w₂ + y₁⁵.y₂ + y₁⁶ + y₃².v₃ + y₂².v₃ + y₁².v₂
• w₂.v₁ = y₄⁴.w₂ + y₂.t + y₂.y₃².v₁ + y₂.y₃³.w₁ + y₂².y₃.v₁ + y₂².y₃².w₁ + y₂³.v₁ + y₂³.y₃.w₁ + y₁.y₃³.w₁ + y₁⁶.y₂ + y₁⁷ + y₄³.v₃ + y₄³.v₂ + y₂.y₃².v₂ + y₂².y₃.v₃ + y₂².y₃.v₂ + y₁.y₃².v₃ + y₁².y₃.v₃ + y₁³.v₂
• w₁.v₁ = y₃.t + y₃³.v₁ + y₃⁴.w₁ + y₂.t + y₂³.v₁ + y₂³.y₃.w₁ + y₁⁴.w₁ + y₁⁷ + y₃³.v₂ + y₂.y₃².v₃ + y₂².y₃.v₃ + y₂².y₃.v₂ + y₁².y₃.v₃ + y₁².y₂.v₃ + y₁².y₂.v₂ + y₁³.v₂
• y₄.t = y₄⁴.w₂ + y₁.y₄³.w₂ + y₁.y₄⁶ + y₁⁴.w₁ + y₄³.v₃ + y₄³.v₂ + y₁².y₂.v₃ + y₁².y₂.v₂ + y₁³.v₃
• y₁.t = y₁.y₄³.w₂ + y₁.y₃³.w₁ + y₁.y₄².v₃ + y₁.y₄².v₂ + y₁.y₃².v₃ + y₁.y₃².v₂ + y₁².y₃.v₂ + y₁².y₂.v₃ + y₁³.v₃ + y₁³.v₂
• v₁² = y₄⁵.w₂ + y₃⁵.w₁ + y₂².y₃².v₁ + y₂².y₃³.w₁ + y₂³.y₃.v₁ + y₂⁴.y₃.w₁ + y₂⁵.w₁ + y₁⁵.w₂ + y₁⁵.w₁ + y₁⁷.y₂ + y₁⁸ + y₄⁴.v₃ + y₄⁴.v₂ + y₃⁴.v₃ + y₂⁴.v₂ + y₁⁴.v₃
• w₂.t = y₄⁶.w₂ + y₂.y₃⁵.w₁ + y₂³.y₃³.w₁ + y₂⁵.y₃.w₁ + y₁.y₃⁵.w₁ + y₁⁶.w₂ + y₄².w₂.v₃ + y₄².w₂.v₂ + y₄⁵.v₃ + y₄⁵.v₂ + y₂.v₁.v₃ + y₂.y₃.w₁.v₂ + y₂.y₃⁴.v₃ + y₂².w₂.v₃ + y₂².w₁.v₃ + y₂⁴.y₃.v₃ + y₁.y₄⁴.v₃ + y₁.y₄⁴.v₂ + y₁.y₃.w₁.v₃ + y₁.y₃.w₁.v₂ + y₁.y₃⁴.v₃ + y₁².w₂.v₃ + y₁².w₂.v₂ + y₁².w₁.v₃ + y₁⁴.y₂.v₃ + y₁⁵.v₂
• w₁.t = y₃⁶.w₁ + y₂.y₃⁵.w₁ + y₂².y₃⁴.w₁ + y₂³.y₃³.w₁ + y₂⁴.y₃².w₁ + y₂⁵.y₃.w₁ + y₁.y₄⁵.w₂ + y₃.v₁.v₃ + y₃².w₁.v₂ + y₃⁵.v₃ + y₂.v₁.v₃ + y₂.y₃.w₁.v₃ + y₂.y₃.w₁.v₂ + y₂.y₃⁴.v₃ + y₂³.y₃².v₃ + y₂⁴.y₃.v₃ + y₁.y₄.w₂.v₂ + y₁.y₄⁴.v₃ + y₁.y₄⁴.v₂ + y₁².w₁.v₃ + y₁².w₁.v₂ + y₁⁵.v₂
• v₁.t = y₄⁷.w₂ + y₃⁴.t + y₃⁶.v₁ + y₃⁷.w₁ + y₂.y₃⁵.v₁ + y₂.y₃⁶.w₁ + y₂².y₃².t + y₂².y₃⁴.v₁ + y₂³.y₃³.v₁ + y₂³.y₃⁴.w₁ + y₂⁴.y₃².v₁ + y₂⁴.y₃³.w₁ + y₂⁵.y₃.v₁ + y₁⁷.w₂ + y₄³.w₂.v₃ + y₄³.w₂.v₂ + y₄⁶.v₃ + y₄⁶.v₂ + y₃².v₁.v₂ + y₃³.w₁.v₃ + y₃⁶.v₂ + y₂.y₃.v₁.v₃ + y₂.y₃.v₁.v₂ + y₂.y₃².w₁.v₃ + y₂.y₃⁵.v₃ + y₂.y₃⁵.v₂ + y₂².y₃.w₁.v₃ + y₂².y₃⁴.v₂ + y₂³.w₂.v₃ + y₂³.w₂.v₂ + y₂³.w₁.v₃ + y₂³.y₃³.v₂ + y₂⁴.y₃².v₃ + y₂⁴.y₃².v₂ + y₂⁵.y₃.v₂ + y₂⁶.v₃ + y₂⁶.v₂ + y₁.y₄⁵.v₃ + y₁.y₄⁵.v₂ + y₁³.w₁.v₃ + y₁⁶.v₃ + y₁⁶.v₂
• t² = y₄⁹.w₂ + y₃⁹.w₁ + y₂².y₃⁴.t + y₂⁵.y₃.t + y₂⁶.y₃³.w₁ + y₁⁹.w₁ + y₁¹² + y₄⁸.v₃ + y₄⁸.v₂ + y₃⁸.v₃ + y₂².y₃².v₁.v₃ + y₂².y₃⁶.v₂ + y₂³.y₃.v₁.v₃ + y₂³.y₃².w₁.v₃ + y₂³.y₃⁵.v₂ + y₂⁴.y₃.w₁.v₂ + y₂⁴.y₃⁴.v₂ + y₂⁵.w₂.v₃ + y₂⁵.w₂.v₂ + y₂⁵.y₃³.v₂ + y₂⁷.y₃.v₃ + y₂⁸.v₃ + y₂⁸.v₂ + y₁⁵.w₁.v₃ + y₁⁵.w₁.v₂ + y₄⁴.v₃² + y₄⁴.v₂² + y₃⁴.v₃² + y₃⁴.v₂² + y₂².y₃².v₃² + y₂².y₃².v₂² + y₂⁴.v₂.v₃ + y₁⁴.v₂.v₃ + y₁⁴.v₂²

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

• y₁.y₂² = y₁².y₃ + y₁³
• y₁².y₃² = 0
• y₁³.y₃ = 0
• y₁.y₂.w₂ = y₁².w₁ + y₁⁵
• y₁².y₃.w₁ = 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 = v₂ in degree 4
• h₂ = v₃ in degree 4
• h₃ = y₄² + y₃² + y₂.y₃ + y₂² + y₁.y₄ 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
• y₃² in degree 2
• y₁.y₄ in degree 2
• y₁.y₃ in degree 2
• w₂ in degree 3
• w₁ in degree 3
• y₁.y₃² in degree 3
• v in degree 4
• y₄.w₂ in degree 4
• y₃.w₁ in degree 4
• y₁.w₂ in degree 4
• y₁.w₁ in degree 4
• y₃.v in degree 5
• y₁.y₄.w₂ in degree 5
• y₁.y₃.w₁ in degree 5
• t in degree 6
• y₃.t in degree 7

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

• y₃² + y₁.y₃ + y₄.h + y₃.h + y₁.h + h² in degree 2
• y₁.y₄ + y₁.y₃ + y₁.h in degree 2
• y₁.y₃² + y₁².y₃ + y₁.y₄.h + y₁.y₃.h + y₁².h + y₁.h² in degree 3
• y₁.y₃.h in degree 3
• y₄.h² in degree 3
• y₁.h² in degree 3
• y₄.w₂ + y₁.w₁ in degree 4
• y₁.y₄.w₂ + y₁.y₃.w₂ + y₁.w₂.h in degree 5
• y₁.y₃.w₁ in degree 5
• y₃.w₁.h + y₁.w₂.h + y₁.w₁.h + w₂.h² in degree 5
• y₁.w₂.h² in degree 6
• y₁.w₁.h² in degree 6

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

• y₁².y₃ 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
• w₁ restricts to 0
• w₂ restricts to 0
• v₁ restricts to 0
• v₂ restricts to y₂⁴ + y₁⁴
• v₃ restricts to y₂⁴
• t restricts to 0

Restriction to special subgroup number 2, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to 0
• y₄ restricts to y₃
• w₁ restricts to 0
• w₂ restricts to y₁.y₃² + y₁².y₃
• v₁ restricts to y₁.y₃³ + y₁².y₃²
• v₂ restricts to y₂².y₃² + y₂⁴ + y₁².y₃² + y₁⁴
• v₃ restricts to y₂².y₃² + y₂⁴ + y₁.y₃³ + y₁².y₃²
• t restricts to y₁².y₃⁴ + y₁⁴.y₃²

Restriction to special subgroup number 3, which is 8gp5

• y₁ restricts to y₃
• y₂ restricts to y₃
• y₃ restricts to 0
• y₄ restricts to y₃
• w₁ restricts to y₃³ + y₁.y₃² + y₁².y₃
• w₂ restricts to y₁.y₃² + y₁².y₃
• v₁ restricts to 0
• v₂ restricts to y₃⁴ + y₂².y₃² + y₂⁴ + y₁.y₃³ + y₁⁴
• v₃ restricts to y₃⁴ + y₂².y₃² + y₂⁴ + y₁.y₃³ + y₁².y₃²
• t restricts to y₃⁶ + y₂².y₃⁴ + y₂⁴.y₃²

Restriction to special subgroup number 4, which is 16gp14

• y₁ restricts to 0
• y₂ restricts to y₃
• y₃ restricts to y₄
• y₄ restricts to 0
• w₁ restricts to y₂.y₃.y₄ + y₂.y₃² + y₂².y₄ + y₂².y₃
• w₂ restricts to y₂.y₃² + y₂².y₃
• v₁ restricts to y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂².y₃² + y₁.y₃³ + y₁².y₃²
• v₂ restricts to y₂.y₃.y₄² + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂².y₃.y₄ + y₂⁴ + y₁.y₃.y₄² + y₁.y₃².y₄ + y₁².y₄² + y₁².y₃.y₄ + y₁².y₃² + y₁⁴
• v₃ restricts to y₂.y₃.y₄² + y₂.y₃².y₄ + y₂.y₃³ + y₂².y₄² + y₂².y₃.y₄ + y₂⁴
• t restricts to y₂.y₃³.y₄² + y₂.y₃⁴.y₄ + y₂.y₃⁵ + y₂².y₄⁴ + y₂².y₃³.y₄ + y₂³.y₃.y₄² + y₂³.y₃².y₄ + y₂⁴.y₃² + y₁.y₃.y₄⁴ + y₁.y₃⁴.y₄ + y₁.y₃⁵ + y₁.y₂.y₃⁴ + y₁.y₂².y₃³ + y₁².y₄⁴ + y₁².y₃².y₄² + y₁².y₃⁴ + y₁².y₂.y₃³ + y₁².y₂².y₃² + y₁⁴.y₄² + y₁⁴.y₃.y₄

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Poincaré series

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

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