Small group number 23 of order 64
G is the group 64gp23
The Hall-Senior number of this group is 128.
G has 2 minimal generators, rank 4 and exponent 4.
The centre has rank 2.
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 13 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- x5 in degree 2, a regular element
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- w4 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
There are 44 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x4 =
y1.x2
- y2.x3 =
y1.x2
- y1.x3 =
y1.x2
- y2.x1 =
0
- y1.x1 =
0
- x32 =
x2.x4
- x1.x4 =
y1.w3
+ y1.w2
- x1.x3 =
y1.w3
- x1.x2 =
y2.w3
- y2.w4 =
y1.w3
- y2.w2 =
0
- y2.w1 =
0
- y1.w4 =
y1.w3
+ y1.w2
- y1.w1 =
0
- x12 =
0
- x4.w4 =
x4.w2
+ x3.w2
+ x3.w1
+ x2.w4
+ x2.w1
- x4.w3 =
x3.w2
+ x2.w4
+ x2.w1
+ y1.x22
- x3.w4 =
x3.w2
+ x2.w4
+ x2.w1
+ y1.x22
- x3.w3 =
x2.w4
- x1.w4 =
y1.x22
+ y1.x4.x5
- x1.w3 =
y2.x22
+ y1.x2.x5
- x1.w2 =
y1.x4.x5
+ y1.x2.x5
- x1.w1 =
0
- y2.v1 =
y2.x22
+ y1.x22
- y1.v1 =
0
- w42 =
x22.x4
+ x42.x5
- w3.w4 =
x22.x3
+ x3.x4.x5
- w32 =
x23
+ y2.x2.w3
+ y1.x2.w3
+ x2.x4.x5
- w2.w4 =
x3.v1
+ x2.x3.x4
+ x22.x3
+ x42.x5
+ x3.x4.x5
- w2.w3 =
x2.v1
+ x22.x4
+ x23
+ x3.x4.x5
+ x2.x4.x5
- w22 =
x42.x5
+ x2.x4.x5
- w1.w4 =
x4.v1
+ x2.x42
+ x2.x3.x4
+ y1.x2.w3
- w1.w3 =
x3.v1
+ x2.x3.x4
+ x22.x4
+ y1.x2.w3
- w1.w2 =
x4.v1
+ x3.v1
+ x2.x42
+ x2.x3.x4
+ x22.x4
+ x22.x3
- w12 =
x2.x42
+ x22.x4
- x1.v1 =
y2.x2.w3
+ y1.x2.w3
- w4.v1 =
x2.x4.w2
+ x2.x3.w1
+ x22.w1
+ x4.x5.w1
+ y1.x22.x5
- w3.v1 =
x2.x3.w2
+ x22.w4
+ x22.w3
+ x22.w2
+ x22.w1
+ y1.x23
+ x3.x5.w1
+ y1.x22.x5
- w2.v1 =
x2.x4.w2
+ x22.w2
+ x4.x5.w1
+ x3.x5.w1
- w1.v1 =
x2.x4.w2
+ x2.x4.w1
+ x2.x3.w2
+ x22.w1
- v12 =
x22.x42
+ x24
+ x2.x42.x5
+ x22.x4.x5
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.x2.x4 =
y1.x22
- y1.x2.w2 =
0
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 8 onwards, and
Benson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 4 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x5
in degree 2
- h2 =
v2
in degree 4
- h3 =
x4
+ x3
+ x2
in degree 2
- h4 =
x2
in degree 2
The first
3 terms h1, h2, h3 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, -1, 4, 6.
-
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
-
y2
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x1
in degree 2
-
w4
in degree 3
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
v
in degree 4
-
y1.w3
in degree 4
-
x3.v
in degree 6
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y2
in degree 1
-
y1.h
in degree 3
-
y1.w3
in degree 4
-
x1.h
in degree 4