Small group number 91 of order 64
G is the group 64gp91
The Hall-Senior number of this group is 126.
G has 3 minimal generators, rank 3 and exponent 4.
The centre has rank 1.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 10 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 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 27 minimal relations:
- y1.y3 =
0
- y12 =
0
- y3.x2 =
y22.y3
+ y1.x1
- y1.y22 =
0
- y3.w =
y22.x1
- y22.x2 =
y24
+ y1.w
- y22.y32 =
y24
- y22.w =
y22.y3.x1
- y1.x1.x2 =
y1.x12
- w2 =
x1.x22
+ x12.x2
+ y24.x1
+ y1.x2.w
+ y1.x1.w
- y3.u2 =
y24.x1
- y1.u2 =
y1.x2.w
+ y1.x1.w
- y1.u1 =
0
- x2.u2 =
x2.u1
+ x22.w
+ x1.u2
+ x1.x2.w
+ y22.u1
+ y1.x13
- y3.t =
y3.x13
+ y32.u1
+ y22.u1
+ y24.y3.x1
+ y26.y3
+ y1.x13
- y22.u2 =
y24.y3.x1
- y1.t =
0
- w.u2 =
x2.t
+ x1.x23
+ x13.x2
+ y24.x12
+ y28
- w.u1 =
x2.t
+ x1.t
+ x12.x22
+ x14
+ y3.x1.u1
+ y22.x13
+ y28
+ y1.x22.w
+ y1.x12.w
- y22.t =
y22.x13
+ y26.x1
+ y28
+ y1.x12.w
- w.t =
x1.x2.u1
+ x12.x2.w
+ x13.w
+ y22.x1.u1
+ y22.y3.x13
+ y24.y3.x12
+ y26.y3.x1
+ y1.x14
- u22 =
x1.x24
+ x13.x22
+ y24.x13
+ y26.x12
+ y28.x1
+ y1.x23.w
+ y1.x13.w
- u1.u2 =
x22.t
+ x1.x2.t
+ y22.x14
+ y22.y3.x1.u1
+ y24.x13
+ y26.x12
+ y210
+ y1.x23.w
- u12 =
x12.x23
+ x13.x22
+ x14.x2
+ x15
+ y3.x12.u1
+ y34.x13
+ y35.u1
+ y22.x14
+ y24.y3.u1
+ y28.x1
+ y210
+ y1.x13.w
+ y32.r
- u2.t =
x1.x22.u1
+ x12.x2.u1
+ y22.x12.u1
+ y22.y3.x14
+ y24.x1.u1
+ y26.y3.x12
+ y28.y3.x1
- u1.t =
x12.x2.u1
+ x12.x22.w
+ x13.u1
+ x14.w
+ y3.x15
+ y32.x12.u1
+ y35.x13
+ y36.u1
+ y24.x1.u1
+ y1.x15
+ y33.r
+ y22.y3.r
- t2 =
x13.x23
+ x14.x22
+ x15.x2
+ x16
+ y32.x15
+ y33.x12.u1
+ y36.x13
+ y37.u1
+ y22.y3.x12.u1
+ y24.x14
+ y26.y3.u1
+ y28.x12
+ y212
+ y34.r
+ y24.r
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 12 onwards, and
Benson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x2
+ y32
+ y2.y3
in degree 2
- h3 =
x1
in degree 2
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, 7, 9.
-
Filter degree type:
-1, -2, -3, -3.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3) is as follows.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y2.w
in degree 4
-
y24
in degree 4
-
y1.w
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y1.y2.w
in degree 5
-
t
in degree 6
-
y3.u1
in degree 6
-
y2.u2
in degree 6
-
y2.u1
in degree 6
-
y2.t
in degree 7
-
y2.y3.u1
in degree 7
-
y22.u1
in degree 7
-
y22.y3.u1
in degree 8
-
y23.u1
in degree 8
-
y24.u1
in degree 9
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1.h
in degree 3
-
y1.y2.h
in degree 4
-
y1.w.h
in degree 6
-
y1.y2.w.h
in degree 7