Small group number 111 of order 64

G is the group 64gp111

The Hall-Senior number of this group is 121.

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

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 3.

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, a nilpotent element
• y₃ in degree 1
• x in degree 2
• v in degree 4, a nilpotent element
• u₁ in degree 5
• u₂ in degree 5
• u₃ in degree 5
• r₁ in degree 8, a nilpotent element
• r₂ in degree 8, a regular element

There are 29 minimal relations:

• y₂² = y₁.y₂
• y₁² = 0
• y₂.y₃² = y₁.x
• y₁.x² = y₁.y₃².x
• y₂.v = 0
• y₁.v = 0
• x.v = y₂.u₂
• y₃².v = y₁.u₃
• y₂.u₃ = y₁.u₁
• y₂.u₁ = y₁.u₁
• y₁.u₂ = y₁.u₁
• x.u₃ = y₃².u₁ + y₁.y₃.u₃ + y₁.y₃⁴.x
• v² = 0
• v.u₃ = y₁.y₃⁶.x
• v.u₂ = y₂.x⁴
• v.u₁ = y₁.y₃⁶.x
• y₂.r₁ = 0
• y₁.r₁ = 0
• u₃² = y₃⁸.x + y₁.y₃⁴.u₃ + y₁.y₃⁷.x
• u₂.u₃ = y₃⁴.x³ + y₃².r₁ + y₁.y₃⁴.u₃ + y₁.y₃⁴.u₁ + y₁.y₂.r₂
• u₂² = x⁵ + y₁.y₃⁴.u₁ + y₁.y₃⁷.x + y₁.y₂.r₂
• u₁.u₃ = y₃⁶.x² + y₁.y₂.r₂
• u₁.u₂ = y₃².x⁴ + x.r₁ + y₂.y₃.x⁴ + y₁.y₃⁴.u₁
• u₁² = y₃⁴.x³ + y₁.y₃⁴.u₁ + y₁.y₃⁷.x + y₁.y₂.r₂
• v.r₁ = 0
• x⁴.u₁ = y₃².x³.u₂ + u₂.r₁ + y₂.y₃.x³.u₂ + y₁.y₃⁷.u₁
• y₃².x³.u₁ = y₃⁴.x².u₂ + u₁.r₁
• y₃⁴.x².u₁ = y₃⁶.x.u₂ + u₃.r₁
• r₁² = 0

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

• y₁.y₂.x = 0
• y₁.x.u₁ = y₁.y₃².u₁
• x.u₁.r₁ = y₃².u₂.r₁

Completion information

This cohomology ring was obtained from a calculation out to degree 16. The cohomology ring approximation is stable from degree 16 onwards, and Benson's tests detect stability from degree 16 onwards.

This cohomology ring has dimension 3 and depth 1. Here is a homogeneous system of parameters:

• h₁ = r₂ in degree 8
• h₂ = x² + y₃².x + y₃⁴ in degree 4
• h₃ = y₃ in degree 1

The first term h₁ forms 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, 3, 9, 10.
• Filter degree type: -1, -2, -3, -3.
• 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) is as follows.

• 1 in degree 0
• y₂ in degree 1
• y₁ in degree 1
• x in degree 2
• y₁.y₂ in degree 2
• y₂.x in degree 3
• v in degree 4
• u₃ in degree 5
• u₂ in degree 5
• u₁ in degree 5
• y₂.u₂ in degree 6
• y₁.u₁ in degree 6
• x.u₂ in degree 7
• x.u₁ in degree 7
• r in degree 8
• x.r in degree 10

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

• y₁.y₂.h in degree 3
• y₂.x.h + y₂.h³ in degree 4
• y₁.h³ in degree 4
• y₂.h⁵ in degree 6
• y₂.u₂.h + y₁.u₁.h in degree 7
• x.u₁.h + y₁.u₁.h² + u₃.h³ + u₁.h³ + v.h⁴ in degree 8
• y₁.u₁.h³ in degree 9
• v.h⁵ in degree 9

A basis for Ann_R/(h1)(h₂) is as follows.

• y₁.y₂ in degree 2
• y₁.y₂.y₃ in degree 3

Restriction information

Restriction to special subgroup number 1, which is 2gp1

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to 0
• x restricts to 0
• v restricts to 0
• u₁ restricts to 0
• u₂ restricts to 0
• u₃ 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
• x restricts to y₃²
• v restricts to 0
• u₁ restricts to y₂².y₃³
• u₂ restricts to y₃⁵
• u₃ restricts to y₂⁴.y₃
• r₁ restricts to 0
• r₂ restricts to y₃⁸ + y₂.y₃⁷ + y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁴.y₃⁴ + y₂⁶.y₃² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

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Poincaré series

(1 + 2t + 2t² + t³ + t⁵ + t⁶ + t⁷ + t⁸ - t⁹ - t¹⁰) / (1 - t) (1 - t⁴) (1 - t⁸)

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