Small group number 102 of order 64
G is the group 64gp102
The Hall-Senior number of this group is 120.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
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, a nilpotent element
- x2 in degree 2
- w in degree 3
- u1 in degree 5, a nilpotent element
- u2 in degree 5
- t in degree 6, a nilpotent element
- r in degree 8, a regular element
There are 27 minimal relations:
- y1.y3 =
0
- y12 =
0
- y3.x2 =
y22.y3
+ y1.x2
- y1.x1 =
0
- y3.w =
y23.y3
+ y22.x1
+ y1.w
- x1.x2 =
y22.x1
+ y1.w
+ y1.y2.x2
- x12 =
0
- x1.w =
y23.x1
+ y1.y2.w
- y1.y22.x2 =
0
- w2 =
y24.x2
+ y1.y2.x22
+ y1.y22.w
- y3.u1 =
y1.u1
- y1.u2 =
y1.y2.x22
+ y1.y22.w
+ y1.u1
- y1.x2.w =
y1.u1
- x2.u2 =
x22.w
+ y2.x23
+ y22.u2
+ y24.w
+ y25.x2
+ y22.u1
+ y1.x23
+ y1.y2.u1
- x1.u2 =
y3.t
+ y24.y3.x1
- x1.u1 =
y1.y2.u1
- y1.t =
0
- w.u2 =
y2.x22.w
+ y23.u2
+ y25.w
+ y22.t
+ y26.x1
+ y1.y2.x23
+ y1.y24.w
+ y1.x2.u1
- w.u1 =
x2.t
+ y22.t
+ y23.u1
+ y1.y24.w
- x1.t =
0
- w.t =
y23.t
+ y24.u1
+ y1.y25.w
+ y1.y2.x2.u1
- u22 =
y2.y34.u2
+ y22.x24
+ y22.y33.u2
+ y24.x23
+ y24.y3.u2
+ y24.y36
+ y26.x22
+ y26.y34
+ y28.x2
+ y28.y32
+ y2.y33.t
+ y22.y32.t
+ y23.y35.x1
+ y25.y33.x1
+ y26.y32.x1
+ y28.x1
+ u12
+ y32.r
- u1.u2 =
x22.t
+ y2.x22.u1
+ y24.t
+ y25.u1
+ y1.y26.w
+ y1.x22.u1
- y1.y2.x24 =
u12
- u2.t =
y2.x22.t
+ y2.y34.t
+ y22.y33.t
+ y24.x2.u1
+ y24.y35.x1
+ y25.t
+ y25.y34.x1
+ y26.u1
+ y28.y3.x1
+ y1.y27.w
+ y1.y2.x22.u1
+ y3.x1.r
- u1.t =
y1.y2.x22.u1
- t2 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y22.u1 =
0
- y2.u12 =
0
- y1.u12 =
0
- u14 =
0
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 =
x22
+ y34
+ y22.x2
+ y22.y32
+ y24
in degree 4
- h3 =
y3
+ 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, 9, 10.
-
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
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
y22
in degree 2
-
x1
in degree 2
-
w
in degree 3
-
y23
in degree 3
-
y2.x1
in degree 3
-
y1.x2
in degree 3
-
y24
in degree 4
-
y22.x1
in degree 4
-
y1.w
in degree 4
-
u2
in degree 5
-
x2.w
in degree 5
-
u1
in degree 5
-
y23.x1
in degree 5
-
y2.u2
in degree 6
-
t
in degree 6
-
y1.u1
in degree 6
-
y22.u2
in degree 7
-
x2.u1
in degree 7
-
y2.t
in degree 7
-
y23.u2
in degree 8
-
x2.t
in degree 8
-
y22.t
in degree 8
-
y23.t
in degree 9
-
y24.t
in degree 10
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1.x2.h
in degree 4
-
y1.h5
in degree 6
-
y1.u1.h
in degree 7
-
y1.w.h5
in degree 9