Small group number 32 of order 64
G is the group 64gp32
The Hall-Senior number of this group is 250.
G has 2 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 11 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
- u in degree 5
There are 33 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x3 =
0
- y1.x2 =
0
- y1.x1 =
0
- x1.x3 =
0
- y2.w1 =
0
- y1.w3 =
0
- y1.w2 =
0
- y1.w1 =
0
- x3.w3 =
0
- x3.w2 =
0
- x2.w3 =
x1.w2
+ y2.v1
+ y2.x22
- x1.w1 =
0
- y1.v1 =
0
- w32 =
x13
+ y2.x1.w3
+ y22.x1.x2
+ y23.w2
- w2.w3 =
x12.x2
+ y2.u
+ y2.x2.w2
+ y2.x1.w2
+ y22.v1
+ y22.x22
- w22 =
x1.x22
+ y2.x2.w2
+ y22.v2
- w1.w3 =
0
- w1.w2 =
0
- w12 =
x22.x3
- x3.v1 =
x22.x3
- y1.u =
0
- w3.v1 =
x1.u
+ y2.x2.v1
+ y2.x23
+ y2.x1.x22
+ y22.x2.w2
- w2.v1 =
x2.u
+ y2.x2.v1
+ y2.x23
+ y2.x1.v2
- w1.v1 =
x22.w1
- x3.u =
0
- v12 =
x24
+ x1.x2.v1
+ y2.x22.w2
+ x12.v2
- w3.u =
x12.v1
+ y2.x2.u
+ y2.x22.w2
+ y2.x1.u
+ y2.x1.x2.w2
+ y22.x2.v1
+ y22.x23
+ y24.v2
- w2.u =
x1.x2.v1
+ y2.x22.w2
+ y2.w3.v2
+ y22.x2.v2
- w1.u =
0
- v1.u =
x23.w2
+ x1.x2.u
+ y2.x22.v1
+ y2.x24
+ x1.w3.v2
+ y2.x1.x2.v2
+ y23.x2.v2
- u2 =
x1.x24
+ x12.x2.v1
+ y2.x23.w2
+ y2.x1.x2.u
+ y2.x1.x22.w2
+ x13.v2
+ y2.x1.w3.v2
+ y22.x22.v2
+ y22.x1.x2.v2
+ y23.w2.v2
+ y24.x2.v2
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
This cohomology ring was obtained from a calculation
out to degree 13. The cohomology ring approximation
is stable from degree 10 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 =
v2
in degree 4
- h2 =
x32
+ x22
+ x12
+ y2.w2
+ y24
in degree 4
- h3 =
x22.x3
+ x1.x22
+ y2.x1.w2
+ y22.x22
+ y22.x12
+ y23.w2
in degree 6
- h4 =
y2
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, 3, 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.
A basis for R/(h1, h2, h3, h4) is as follows.
-
1
in degree 0
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
y1.x3
in degree 3
-
v
in degree 4
-
x2.x3
in degree 4
-
x22
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
u
in degree 5
-
x3.w1
in degree 5
-
x2.w2
in degree 5
-
x2.w1
in degree 5
-
x1.w3
in degree 5
-
x1.w2
in degree 5
-
x2.v
in degree 6
-
x23
in degree 6
-
x1.v
in degree 6
-
x12.x2
in degree 6
-
x13
in degree 6
-
x2.u
in degree 7
-
x2.x3.w1
in degree 7
-
x22.w1
in degree 7
-
x1.u
in degree 7
-
x12.w3
in degree 7
-
x12.w2
in degree 7
-
x1.x2.v
in degree 8
-
x12.v
in degree 8
-
x13.x2
in degree 8
-
x23.w1
in degree 9
-
x1.x2.u
in degree 9
-
x12.u
in degree 9
-
x12.x2.v
in degree 10
-
x12.x2.u
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y1
in degree 1
-
x3
in degree 2
-
w1
in degree 3
-
y1.x3
in degree 3
-
x2.x3
in degree 4
-
x22
+ x12
+ w2.h
+ h4
in degree 4
-
x3.w1
in degree 5
-
x2.w1
in degree 5
-
x23
+ x12.x2
+ x2.w2.h
+ x2.h4
in degree 6
-
x13
+ x1.h4
+ h6
in degree 6
-
x2.x3.w1
in degree 7
-
x22.w1
in degree 7
-
x13.x2
+ x1.x2.h4
+ x2.h6
in degree 8
-
x23.w1
in degree 9
-
x12.x2.v
+ x12.u.h
+ x12.v.h2
+ x12.w3.h3
+ x12.w2.h3
+ x2.v.h4
+ x12.x2.h4
+ u.h5
+ x2.w2.h5
+ x1.w3.h5
+ v.h6
+ x12.h6
+ w3.h7
+ x2.h8
+ x1.h8
+ h10
in degree 10
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1
in degree 1
-
y1.x3
in degree 3