Small group number 23 of order 32

G = 16gp4xC2 is Direct product 16gp4 x C_2

The Hall-Senior number of this group is 12.

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

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 5 generators:

• y₁ in degree 1, a nilpotent element
• y₂ in degree 1, a nilpotent element
• y₃ in degree 1, a regular element
• x₁ in degree 2, a regular element
• x₂ in degree 2, a regular element

There are 2 minimal relations:

• y₂² = y₁.y₂
• y₁² = 0

This minimal generating set constitutes a Gröbner basis for the relations ideal.

Completion information

This cohomology ring was obtained from a calculation out to degree 8. The cohomology ring approximation is stable from degree 2 onwards, and Benson's tests detect stability from degree 3 onwards.

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

• h₁ = y₃ in degree 1
• h₂ = x₁ in degree 2
• h₃ = x₂ in degree 2

The first 3 terms h₁, h₂, h₃ form a regular sequence of maximum length.

The first 3 terms h₁, h₂, h₃ 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, -1, 2.
• 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
• y₁.y₂ in degree 2

Restriction information

Restriction to special subgroup number 1, which is 8gp5

• y₁ restricts to 0
• y₂ restricts to 0
• y₃ restricts to y₁
• x₁ restricts to y₃²
• x₂ restricts to y₂²

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Poincaré series

(1 + 2t + t²) / (1 - t) (1 - t²)²

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