Small group number 135 of order 64
G is the group 64gp135
The Hall-Senior number of this group is 263.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 8 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- w in degree 3
- u in degree 5
- r in degree 8, a regular element
There are 14 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y23 =
y1.x2
+ y1.y22
- y3.x1 =
0
- y2.w =
y22.x2
+ y12.x2
+ x1.x2
+ y1.y2.x1
+ x12
- y1.w =
y1.y2.x1
+ x12
- y1.y2.x2 =
y12.x2
+ y22.x1
+ y1.y2.x1
+ x12
- y1.x22 =
y1.x1.x2
+ y2.x12
- x1.w =
y2.x12
- w2 =
y3.u
+ x1.x22
+ x12.x2
+ y22.x12
- y2.u =
y22.x22
+ y14.y22
+ x1.x22
+ y12.x1.x2
+ y13.y2.x1
+ x12.x2
+ y22.x12
+ y12.x12
- y1.u =
y14.x2
+ y14.y22
+ y12.x1.x2
+ y12.y22.x1
- x1.u =
y13.x1.x2
+ y13.y22.x1
+ y1.x12.x2
+ y1.y22.x12
- u2 =
y3.x22.u
+ y32.w.u
+ y32.x24
+ y33.x22.w
+ y35.u
+ y35.x2.w
+ y18.y22
+ x1.x24
+ y16.x1.x2
+ y16.y22.x1
+ y17.y2.x1
+ y14.y22.x12
+ y15.y2.x12
+ y16.x12
+ y32.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y22.x1.x2 =
y12.x1.x2
+ x12.x2
+ y22.x12
+ y1.y2.x12
- x13 =
0
- y2.x12.x2 =
y1.x12.x2
- x12.x22 =
0
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 10 onwards, and
Benson's tests detect stability from degree 10
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
+ y12
in degree 2
- h3 =
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, 7, 8.
-
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
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
x1
in degree 2
-
w
in degree 3
-
y1.y22
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y2.x1
in degree 3
-
y1.x1
in degree 3
-
y12.y22
in degree 4
-
y14
in degree 4
-
y22.x1
in degree 4
-
y1.y2.x1
in degree 4
-
y12.x1
in degree 4
-
x12
in degree 4
-
u
in degree 5
-
y1.y22.x1
in degree 5
-
y12.y2.x1
in degree 5
-
y13.x1
in degree 5
-
y2.x12
in degree 5
-
y1.x12
in degree 5
-
y14.x1
in degree 6
-
y22.x12
in degree 6
-
y1.y2.x12
in degree 6
-
y12.x12
in degree 6
-
y1.y22.x12
in degree 7
-
y13.x12
in degree 7
-
w.u
in degree 8
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
x1
in degree 2
-
y1.y22
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y2.x1
in degree 3
-
y1.x1
in degree 3
-
y12.y22
in degree 4
-
y14
in degree 4
-
y22.x1
in degree 4
-
y1.y2.x1
in degree 4
-
y12.x1
in degree 4
-
x12
in degree 4
-
y1.y22.x1
in degree 5
-
y12.y2.x1
in degree 5
-
y13.x1
in degree 5
-
y2.x12
in degree 5
-
y1.x12
in degree 5
-
y14.x1
in degree 6
-
y22.x12
in degree 6
-
y1.y2.x12
in degree 6
-
y12.x12
in degree 6
-
y1.y22.x12
in degree 7
-
y13.x12
in degree 7