Small group number 642 of order 128

G is the group 128gp642

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 1.

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

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 12 generators:

y₁ in degree 1, a nilpotent element
y₂ in degree 1
y₃ in degree 1
x₁ in degree 2
x₂ in degree 2
x₃ in degree 2
x₄ in degree 2
w in degree 3
u₁ in degree 5
u₂ in degree 5
t in degree 6
r in degree 8, a regular element

There are 32 minimal relations:

y₂.y₃ = y₁.y₂
y₁.y₃ = 0
y₁² = 0
y₃.x₄ = y₃.x₁ + y₁.x₁
y₂.x₄ = y₁.x₁
y₂.x₂ = y₂.x₁ + y₁.x₃
y₁.x₄ = 0
x₄² = y₃².x₂ + y₃².x₁
x₃.x₄ = x₁.x₃ + y₂.w + y₂².x₃
x₂.x₄ = x₁.x₂ + y₂².x₁ + y₁.w
x₁.x₄ = y₃².x₂ + y₃².x₁ + y₁.y₂.x₁
x₁² = y₃².x₂ + y₃².x₁ + y₂².x₁ + y₁.y₂.x₃ + y₁.y₂.x₁
x₄.w = x₁.w
y₁.x₃² = y₁.x₂.x₃ + y₁.y₂.w + y₁.y₂².x₃
w² = x₂.x₃² + x₂².x₃ + y₃.u₂ + y₃.x₃.w + y₃.x₂.w + y₃².x₂² + y₃⁴.x₁ + y₃⁶ + y₂².x₃² + y₂³.w + y₂⁴.x₃ + y₁.x₃.w + y₁.x₂.w + y₁.y₂².w
x₁.x₃² = x₁.x₂.x₃ + y₃.u₁ + y₃.x₃.w + y₃.x₁.w + y₃².x₃² + y₃².x₂² + y₃².x₁.x₂ + y₃³.w + y₃⁴.x₃ + y₃⁶ + y₂.x₃.w + y₂².x₃² + y₂³.w + y₂⁴.x₃ + y₁.x₃.w + y₁.y₂².w + y₁.y₂³.x₁
y₂.u₂ = y₂².x₃² + y₂³.w + y₁.x₃.w + y₁.y₂³.x₃ + y₁.y₂³.x₁
y₁.u₂ = y₁.x₃.w + y₁.y₂².w
y₁.u₁ = y₁.x₃.w + y₁.y₂².w + y₁.y₂³.x₁
x₄.u₂ = x₁.x₃.w + y₃.t + y₃.x₃³ + y₃.x₂.x₃² + y₃.x₁.x₂.x₃ + y₃.x₁.x₂² + y₃².u₂ + y₃².u₁ + y₃².x₃.w + y₃².x₂.w + y₃².x₁.w + y₃³.x₃² + y₃³.x₂.x₃ + y₃³.x₁.x₃ + y₃³.x₁.x₂ + y₃⁵.x₁ + y₁.y₂³.w + y₁.y₂⁴.x₃ + y₁.y₂⁴.x₁
x₄.u₁ = x₁.x₃.w + y₃.x₂.x₃² + y₃.x₂².x₃ + y₃.x₁.x₂.x₃ + y₃.x₁.x₂² + y₃².x₂.w + y₃³.x₂² + y₃³.x₁.x₃ + y₃³.x₁.x₂ + y₃⁵.x₁
x₁.u₂ = x₁.x₃.w + y₃.t + y₃.x₃³ + y₃.x₂.x₃² + y₃.x₁.x₂.x₃ + y₃.x₁.x₂² + y₃².u₂ + y₃².u₁ + y₃².x₃.w + y₃².x₂.w + y₃².x₁.w + y₃³.x₃² + y₃³.x₂.x₃ + y₃³.x₁.x₃ + y₃³.x₁.x₂ + y₃⁵.x₁ + y₂².x₃.w + y₂³.x₃²
x₁.u₁ = x₁.x₃.w + y₃.x₂.x₃² + y₃.x₂².x₃ + y₃.x₁.x₂.x₃ + y₃.x₁.x₂² + y₃².x₂.w + y₃³.x₂² + y₃³.x₁.x₃ + y₃³.x₁.x₂ + y₃⁵.x₁ + y₂.t + y₂.x₃³ + y₂².u₁ + y₂².x₃.w + y₂⁴.w + y₁.x₂².x₃ + y₁.y₂⁴.x₃ + y₁.y₂⁴.x₁
y₁.t = 0
x₄.t = y₃.x₃.u₁ + y₃.x₃².w + y₃.x₂.u₂ + y₃.x₂.x₃.w + y₃.x₁.x₂.w + y₃².x₃³ + y₃².x₂.x₃² + y₃².x₂².x₃ + y₃².x₂³ + y₃².x₁.x₂.x₃ + y₃³.u₁ + y₃⁴.x₂.x₃ + y₃⁴.x₂² + y₃⁴.x₁.x₂ + y₃⁵.w + y₃⁶.x₂ + y₃⁸ + y₁.y₂⁴.w + y₁.y₂⁵.x₃
x₁.t = y₃.x₃.u₁ + y₃.x₃².w + y₃.x₂.u₂ + y₃.x₂.x₃.w + y₃.x₁.x₂.w + y₃².x₃³ + y₃².x₂.x₃² + y₃².x₂².x₃ + y₃².x₂³ + y₃².x₁.x₂.x₃ + y₃³.u₁ + y₃⁴.x₂.x₃ + y₃⁴.x₂² + y₃⁴.x₁.x₂ + y₃⁵.w + y₃⁶.x₂ + y₃⁸ + y₂.x₃².w + y₂².x₃³ + y₁.x₂.x₃.w + y₁.y₂⁵.x₁
u₂² = x₂².x₃³ + x₂³.x₃² + y₃.x₂.x₃.u₂ + y₃².w.u₁ + y₃².x₃.t + y₃².x₃⁴ + y₃².x₂.x₃³ + y₃².x₂³.x₃ + y₃².x₂⁴ + y₃².x₁.x₂².x₃ + y₃².x₁.x₂³ + y₃³.x₃.u₂ + y₃³.x₃².w + y₃³.x₂.u₂ + y₃³.x₂.u₁ + y₃³.x₂.x₃.w + y₃³.x₁.x₃.w + y₃³.x₁.x₂.w + y₃⁴.t + y₃⁴.x₃³ + y₃⁴.x₂.x₃² + y₃⁴.x₂².x₃ + y₃⁴.x₁.x₂.x₃ + y₃⁵.u₂ + y₃⁵.u₁ + y₃⁵.x₂.w + y₃⁶.x₂.x₃ + y₃⁶.x₂² + y₃⁶.x₁.x₃ + y₃⁶.x₁.x₂ + y₃⁷.w + y₂².x₃⁴ + y₂³.x₃².w + y₂⁴.x₃³ + y₂⁶.x₃² + y₃².r
u₁.u₂ = x₃.w.u₂ + x₃.w.u₁ + x₃².t + x₃⁵ + x₂.x₃.t + x₂.x₃⁴ + y₃.w.t + y₃.x₃².u₂ + y₃.x₃².u₁ + y₃.x₂.x₃.u₂ + y₃.x₂.x₃².w + y₃.x₂².u₂ + y₃.x₂².u₁ + y₃.x₂².x₃.w + y₃².x₃⁴ + y₃².x₂.t + y₃².x₂.x₃³ + y₃².x₂⁴ + y₃³.x₃.u₁ + y₃³.x₃².w + y₃⁴.x₃³ + y₃⁴.x₂².x₃ + y₃⁴.x₁.x₂.x₃ + y₃⁵.x₁.w + y₃⁶.x₁.x₃ + y₃⁷.w + y₃⁸.x₁ + y₃¹⁰ + y₂.x₃².u₁ + y₂².x₃.t + y₂².x₃⁴ + y₂³.x₃².w + y₂⁵.x₃.w + y₁.x₂².x₃.w + y₁.y₂⁶.w + y₁.y₂⁷.x₃
u₁² = x₂².x₃³ + x₂³.x₃² + y₃.x₃².u₂ + y₃.x₃².u₁ + y₃.x₂.x₃.u₁ + y₃².w.u₁ + y₃².x₂⁴ + y₃³.x₃.u₂ + y₃³.x₃².w + y₃³.x₂.u₂ + y₃³.x₂.u₁ + y₃⁴.x₃³ + y₃⁴.x₂².x₃ + y₃⁴.x₂³ + y₃⁴.x₁.x₂.x₃ + y₃⁵.u₁ + y₃⁵.x₂.w + y₃⁵.x₁.w + y₃⁶.x₂² + y₃⁶.x₁.x₃ + y₂.x₃³.w + y₂².x₃⁴ + y₂³.x₃².w + y₂⁴.x₃³ + y₂⁶.x₃² + y₂⁸.x₁ + y₁.x₂².x₃.w + y₁.y₂⁶.w + y₁.y₂⁷.x₁
u₂.t = x₃.w.t + x₃³.u₂ + x₃⁴.w + x₂.x₃².u₂ + x₂.x₃².u₁ + x₂².x₃.u₁ + x₂².x₃².w + y₃.x₃.w.u₂ + y₃.x₂.w.u₂ + y₃.x₂.w.u₁ + y₃.x₂.x₃.t + y₃.x₂².t + y₃.x₂³.x₃² + y₃.x₂⁴.x₃ + y₃.x₁.x₂³.x₃ + y₃².x₂.x₃².w + y₃².x₂².u₁ + y₃².x₂².x₃.w + y₃².x₁.x₂.x₃.w + y₃³.w.u₂ + y₃³.x₁.x₂².x₃ + y₃³.x₁.x₂³ + y₃⁴.x₃.u₂ + y₃⁴.x₃².w + y₃⁴.x₂.x₃.w + y₃⁴.x₁.x₃.w + y₃⁵.x₂.x₃² + y₃⁵.x₂³ + y₃⁵.x₁.x₂² + y₃⁶.x₃.w + y₃⁷.x₂² + y₃⁷.x₁.x₃ + y₃⁷.x₁.x₂ + y₃⁹.x₃ + y₃⁹.x₂ + y₃⁹.x₁ + y₃¹¹ + y₂.x₃².t + y₂.x₃⁵ + y₂².x₃².u₁ + y₂³.x₃⁴ + y₂⁴.x₃.u₁ + y₂⁶.x₃.w + y₁.x₂⁴.x₃ + y₁.y₂⁷.w + y₁.y₂⁸.x₃ + y₃.x₁.r + y₃³.r + y₁.x₁.r
u₁.t = x₃.w.t + x₃³.u₁ + x₃⁴.w + x₂.x₃².u₂ + x₂.x₃².u₁ + x₂².x₃.u₂ + x₂².x₃².w + y₃.x₃.w.u₁ + y₃.x₃².t + y₃.x₃⁵ + y₃.x₂.w.u₂ + y₃.x₂.w.u₁ + y₃.x₂.x₃⁴ + y₃.x₂².t + y₃.x₂².x₃³ + y₃.x₂⁴.x₃ + y₃².w.t + y₃².x₃².u₂ + y₃².x₃².u₁ + y₃².x₂.x₃.u₁ + y₃².x₂.x₃².w + y₃².x₂².u₂ + y₃³.w.u₁ + y₃³.x₃.t + y₃³.x₂.x₃³ + y₃³.x₂².x₃² + y₃³.x₂⁴ + y₃⁴.x₃².w + y₃⁴.x₂.u₂ + y₃⁵.t + y₃⁵.x₂².x₃ + y₃⁵.x₂³ + y₃⁵.x₁.x₂.x₃ + y₃⁶.u₂ + y₃⁶.u₁ + y₃⁶.x₂.w + y₃⁶.x₁.w + y₃⁷.x₃² + y₃⁷.x₂.x₃ + y₃⁷.x₂² + y₃⁹.x₃ + y₃⁹.x₂ + y₃⁹.x₁ + y₂.x₃².t + y₂.x₃⁵ + y₂².x₃².u₁ + y₂².x₃³.w + y₂³.x₃.t + y₂⁵.x₃³ + y₁.x₂⁴.x₃ + y₁.y₂⁷.w + y₁.y₂⁸.x₃ + y₁.y₂⁸.x₁
t² = x₃⁶ + x₂⁴.x₃² + y₃.x₂.x₃².u₂ + y₃.x₂.x₃².u₁ + y₃.x₂².x₃.u₂ + y₃.x₂².x₃.u₁ + y₃².x₃².t + y₃².x₃⁵ + y₃².x₂.x₃⁴ + y₃².x₂².x₃³ + y₃².x₂³.x₃² + y₃².x₂⁴.x₃ + y₃².x₁.x₂⁴ + y₃³.x₃².u₂ + y₃³.x₂.x₃.u₂ + y₃³.x₂.x₃².w + y₃³.x₂².x₃.w + y₃³.x₁.x₂².w + y₃⁴.x₃.t + y₃⁴.x₃⁴ + y₃⁴.x₂.t + y₃⁴.x₂³.x₃ + y₃⁴.x₂⁴ + y₃⁴.x₁.x₂².x₃ + y₃⁴.x₁.x₂³ + y₃⁵.x₂.u₂ + y₃⁵.x₁.x₃.w + y₃⁵.x₁.x₂.w + y₃⁶.t + y₃⁶.x₂.x₃² + y₃⁶.x₂².x₃ + y₃⁶.x₂³ + y₃⁷.u₂ + y₃⁸.x₃² + y₃⁸.x₂.x₃ + y₃⁸.x₂² + y₃⁸.x₁.x₂ + y₃⁹.w + y₃¹⁰.x₂ + y₂³.x₃³.w + y₂⁷.x₃.w + y₂⁸.x₃² + y₁.y₂⁸.w + y₁.y₂⁹.x₃ + y₃².x₂.r + y₃².x₁.r + y₃⁴.r

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

y₂.x₁.x₃ = y₂².w + y₂³.x₃ + y₁.y₂.w + y₁.y₂².x₃
y₁.x₁.x₃ = y₁.y₂.w + y₁.y₂².x₃
y₁.x₁.x₂ = y₁.y₂².x₁
y₂.x₁.w = 0
y₁.x₁.w = 0
y₁.y₂.x₃.w = 0
y₂.w.u₁ = y₂.x₃.t + y₂.x₃⁴ + y₂².x₃².w + y₂⁴.x₃.w + y₁.x₂³.x₃
y₂.w.t = y₂.x₃³.w + y₂².x₃.t + y₂².x₃⁴ + y₁.x₂².x₃.w + y₁.y₂⁶.w + y₁.y₂⁷.x₃

Completion information

This cohomology ring was obtained from a calculation out to degree 13. The cohomology ring approximation is stable from degree 12 onwards, and Benson's tests detect stability from degree 13 onwards.

This cohomology ring has dimension 4 and depth 2. Here is a homogeneous system of parameters:

h₁ = r in degree 8
h₂ = x₃² + x₂.x₃ + x₂² + y₃⁴ + y₂.w + y₂².x₃ + y₂².x₁ + y₂⁴ + y₁.y₂.x₃ + y₁.y₂.x₁ in degree 4
h₃ = x₃ + y₃² in degree 2
h₄ = y₃ in degree 1

The first 2 terms h₁, h₂ form 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, -1, 8, 10, 11.
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
y₂ in degree 1
y₁ in degree 1
x₄ in degree 2
x₂ in degree 2
x₁ in degree 2
y₂² in degree 2
w in degree 3
y₂.x₁ in degree 3
y₂³ in degree 3
y₁.x₂ in degree 3
x₁.x₂ in degree 4
y₂².x₁ in degree 4
y₂⁴ in degree 4
y₁.w in degree 4
u₂ in degree 5
u₁ in degree 5
x₂.w in degree 5
x₁.w in degree 5
y₂³.x₁ in degree 5
t in degree 6
y₂.u₁ in degree 6
y₁.x₂.w in degree 6
x₂.u₂ in degree 7
x₂.u₁ in degree 7
x₁.x₂.w in degree 7
y₂.t in degree 7
y₂².u₁ in degree 7
w.u₂ in degree 8
w.u₁ in degree 8
x₂.t in degree 8
y₂².t in degree 8
y₂³.u₁ in degree 8
w.t in degree 9
y₂³.t in degree 9
y₂⁴.u₁ in degree 9
x₂.w.u₂ in degree 10
x₂.w.u₁ in degree 10
y₂⁴.t in degree 10
x₂.w.t in degree 11

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

y₁ in degree 1
y₂.h in degree 2
y₁.x₂ in degree 3
x₄.h + x₁.h in degree 3
y₂².h in degree 3
y₁.w in degree 4
y₂.x₁.h in degree 4
y₂³.h in degree 4
u₁ + x₁.w + y₂³.x₁ + x₂.h³ + x₁.h³ + h⁵ in degree 5
y₂².x₁.h in degree 5
y₂⁴.h in degree 5
y₂.u₁ + y₂.x₁.w + y₂⁴.x₁ + y₂.x₂.h³ + y₂.x₁.h³ + y₂.h⁵ in degree 6
y₁.x₂.w in degree 6
y₂³.x₁.h in degree 6
x₂.u₁ + x₁.x₂.w + y₂³.x₁.x₂ + x₂².h³ + x₁.x₂.h³ + x₂.h⁵ in degree 7
y₂.t in degree 7
y₂².u₁ + y₂².x₁.w + y₂⁵.x₁ + y₂².x₂.h³ + y₂².x₁.h³ + y₂².h⁵ in degree 7
w.u₁ + x₁.w² + y₂³.x₁.w + x₂.w.h³ + x₁.w.h³ + w.h⁵ in degree 8
y₂².t in degree 8
y₂³.u₁ + y₂³.x₁.w + y₂⁶.x₁ + y₂³.x₂.h³ + y₂³.x₁.h³ + y₂³.h⁵ in degree 8
y₂³.t in degree 9
y₂⁴.u₁ + y₂⁴.x₁.w + y₂⁷.x₁ + y₂⁴.x₂.h³ + y₂⁴.x₁.h³ + y₂⁴.h⁵ in degree 9
x₂.w.u₁ + x₁.x₂.w² + y₂³.x₁.x₂.w + x₂².w.h³ + x₁.x₂.w.h³ + x₂.w.h⁵ in degree 10
y₂⁴.t in degree 10

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

y₁.x₂ + y₁.x₁ + y₁.h in degree 3
y₁.y₂.h in degree 4
y₁.y₂⁴ + y₁.h² in degree 5
y₁.x₁.h in degree 5
y₁.y₂².h in degree 5
y₁.x₂.w + y₁.x₁.w + y₁.w.h in degree 6
y₁.y₂.x₁.h in degree 6
y₁.y₂³.h in degree 6
y₁.y₂².x₁.h in degree 7
y₁.h³ in degree 7
y₁.y₂³.x₁.h in degree 8
y₁.w.h² in degree 8

Restriction information

Restriction to special subgroup number 1, which is 2gp1

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to 0
x₃ restricts to 0
x₄ restricts to 0
w restricts to 0
u₁ restricts to 0
u₂ restricts to 0
t restricts to 0
r restricts to y⁸

Restriction to special subgroup number 2, which is 8gp5

y₁ restricts to 0
y₂ restricts to y₂
y₃ restricts to 0
x₁ restricts to 0
x₂ restricts to 0
x₃ restricts to y₃² + y₂.y₃
x₄ restricts to 0
w restricts to y₂.y₃² + y₂².y₃
u₁ restricts to y₂³.y₃² + y₂⁴.y₃
u₂ restricts to y₂.y₃⁴ + y₂⁴.y₃
t restricts to y₃⁶ + y₂.y₃⁵ + y₂³.y₃³ + y₂⁴.y₃²
r restricts to y₂².y₃⁶ + y₂³.y₃⁵ + y₂⁵.y₃³ + y₂⁷.y₃ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 3, which is 8gp5

y₁ restricts to 0
y₂ restricts to y₃
y₃ restricts to 0
x₁ restricts to y₃²
x₂ restricts to y₃²
x₃ restricts to y₂.y₃ + y₂²
x₄ restricts to 0
w restricts to 0
u₁ restricts to y₃⁵ + y₂².y₃³ + y₂⁴.y₃
u₂ restricts to y₂².y₃³ + y₂⁴.y₃
t restricts to y₂³.y₃³ + y₂⁴.y₃² + y₂⁵.y₃ + y₂⁶
r restricts to y₃⁸ + y₂.y₃⁷ + y₂⁴.y₃⁴ + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₃² + y₁⁴.y₃⁴ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Restriction to special subgroup number 4, which is 16gp14

y₁ restricts to 0
y₂ restricts to 0
y₃ restricts to y₄
x₁ restricts to y₃.y₄
x₂ restricts to y₃.y₄ + y₃²
x₃ restricts to y₂.y₄ + y₂²
x₄ restricts to y₃.y₄
w restricts to y₃².y₄ + y₂.y₃² + y₂².y₃ + y₁.y₄² + y₁².y₄
u₁ restricts to y₄⁵ + y₃².y₄³ + y₃⁴.y₄ + y₂.y₄⁴ + y₂.y₃².y₄² + y₂².y₃³ + y₂³.y₃.y₄ + y₂³.y₃² + y₂⁴.y₄ + y₁.y₄⁴ + y₁.y₃.y₄³ + y₁.y₂.y₄³ + y₁.y₂².y₄² + y₁².y₄³ + y₁².y₃.y₄² + y₁².y₂.y₄² + y₁².y₂².y₄
u₂ restricts to y₄⁵ + y₃.y₄⁴ + y₃².y₄³ + y₃³.y₄² + y₃⁴.y₄ + y₂.y₃³.y₄ + y₂².y₃.y₄² + y₂².y₃².y₄ + y₂².y₃³ + y₂³.y₃.y₄ + y₂³.y₃² + y₁.y₃.y₄³ + y₁.y₃².y₄² + y₁.y₂.y₄³ + y₁.y₂².y₄² + y₁².y₄³ + y₁².y₃.y₄² + y₁².y₃².y₄ + y₁².y₂.y₄² + y₁².y₂².y₄ + y₁⁴.y₄
t restricts to y₃.y₄⁵ + y₂.y₄⁵ + y₂.y₃³.y₄² + y₂².y₄⁴ + y₂².y₃².y₄² + y₂².y₃³.y₄ + y₂².y₃⁴ + y₂³.y₄³ + y₂³.y₃.y₄² + y₂³.y₃².y₄ + y₂⁴.y₄² + y₂⁵.y₄ + y₂⁶ + y₁.y₄⁵ + y₁.y₃².y₄³ + y₁.y₃³.y₄² + y₁.y₂.y₄⁴ + y₁.y₂².y₄³ + y₁².y₃.y₄³ + y₁².y₃².y₄² + y₁².y₃³.y₄ + y₁².y₂.y₄³ + y₁².y₂².y₄² + y₁⁴.y₄² + y₁⁴.y₃.y₄
r restricts to y₄⁸ + y₃⁵.y₄³ + y₃⁶.y₄² + y₃⁷.y₄ + y₂.y₄⁷ + y₂.y₃².y₄⁵ + y₂.y₃⁶.y₄ + y₂².y₄⁶ + y₂².y₃.y₄⁵ + y₂².y₃².y₄⁴ + y₂².y₃³.y₄³ + y₂².y₃⁴.y₄² + y₂³.y₃².y₄³ + y₂³.y₃³.y₄² + y₂³.y₃⁴.y₄ + y₂³.y₃⁵ + y₂⁴.y₃.y₄³ + y₂⁴.y₃².y₄² + y₂⁴.y₃³.y₄ + y₂⁴.y₃⁴ + y₂⁵.y₃.y₄² + y₂⁵.y₃³ + y₂⁶.y₃.y₄ + y₂⁶.y₃² + y₁.y₃².y₄⁵ + y₁.y₂.y₄⁶ + y₁.y₂.y₃.y₄⁵ + y₁.y₂.y₃².y₄⁴ + y₁.y₂.y₃⁴.y₄² + y₁.y₂².y₄⁵ + y₁.y₂².y₃.y₄⁴ + y₁.y₂².y₃⁴.y₄ + y₁.y₂⁴.y₃².y₄ + y₁².y₄⁶ + y₁².y₃.y₄⁵ + y₁².y₃².y₄⁴ + y₁².y₃⁴.y₄² + y₁².y₂.y₃.y₄⁴ + y₁².y₂.y₃⁴.y₄ + y₁².y₂².y₄⁴ + y₁².y₂².y₃.y₄³ + y₁².y₂².y₃².y₄² + y₁².y₂².y₃⁴ + y₁².y₂⁴.y₄² + y₁².y₂⁴.y₃² + y₁⁴.y₃.y₄³ + y₁⁴.y₃⁴ + y₁⁴.y₂.y₄³ + y₁⁴.y₂.y₃².y₄ + y₁⁴.y₂².y₃² + y₁⁴.y₂⁴ + y₁⁸

Poincaré series

(1 + 2t + 3t² + 3t³ + t⁴ + t⁵ + 2t⁸ + t⁹ + t¹⁰ + t¹¹) / (1 - t) (1 - t²) (1 - t⁴) (1 - t⁸)

Back to the groups of order 128