Small group number 928 of order 128
G = Syl2(S8) is Sylow 2-subgroup of Symmetric Group S_8
G has 3 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
There are 5 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 4, 4, 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 9 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- w1 in degree 3
- w2 in degree 3
- v in degree 4, a regular element
There are 14 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y3.x2 =
0
- y2.x3 =
0
- y1.x1 =
0
- x1.x3 =
0
- y3.w2 =
0
- y3.w1 =
0
- y2.w2 =
y1.w1
- x3.w1 =
0
- x1.w2 =
0
- w22 =
x22.x3
+ y1.x2.w2
+ y12.v
- w1.w2 =
y1.x2.w1
+ y1.y2.v
- w12 =
x1.x22
+ y2.x2.w1
+ y22.v
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 6 onwards, and
Benson's tests detect stability from degree 13
onwards.
This cohomology ring has dimension 4 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
x32
+ x22
+ x12
+ y34
+ y2.w1
+ y22.x2
+ y24
+ y1.w2
+ y1.y2.x2
+ y12.y22
+ y14
in degree 4
- h3 =
x22.x3
+ x1.x22
+ y32.x32
+ y32.x12
+ y2.x1.w1
+ y22.x22
+ y22.x1.x2
+ y22.x12
+ y23.w1
+ y24.x2
+ y1.x3.w2
+ y1.y2.x22
+ y1.y22.w1
+ y12.x32
+ y12.x22
+ y12.y24
+ y13.w2
+ y13.w1
+ y13.y2.x2
+ y14.y22
in degree 6
- h4 =
y2
+ y1
in degree 1
The first
3 terms h1, h2, h3 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, -1, 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
-
y3
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y32
in degree 2
-
y12
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y3.x3
in degree 3
-
y3.x1
in degree 3
-
y33
in degree 3
-
y1.x2
in degree 3
-
y13
in degree 3
-
x2.x3
in degree 4
-
x22
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
y32.x3
in degree 4
-
y32.x1
in degree 4
-
y34
in degree 4
-
y1.w1
in degree 4
-
y12.x2
in degree 4
-
y14
in degree 4
-
x3.w2
in degree 5
-
x2.w2
in degree 5
-
x2.w1
in degree 5
-
x1.w1
in degree 5
-
y3.x12
in degree 5
-
y33.x3
in degree 5
-
y33.x1
in degree 5
-
y35
in degree 5
-
y12.w1
in degree 5
-
y13.x2
in degree 5
-
y15
in degree 5
-
x23
in degree 6
-
x12.x2
in degree 6
-
x13
in degree 6
-
y32.x12
in degree 6
-
y34.x3
in degree 6
-
y34.x1
in degree 6
-
y36
in degree 6
-
y1.x2.w1
in degree 6
-
y13.w1
in degree 6
-
y14.x2
in degree 6
-
y16
in degree 6
-
x2.x3.w2
in degree 7
-
x22.w2
in degree 7
-
x1.x2.w1
in degree 7
-
x12.w1
in degree 7
-
y33.x12
in degree 7
-
y35.x3
in degree 7
-
y35.x1
in degree 7
-
y12.x2.w1
in degree 7
-
y14.w1
in degree 7
-
y15.x2
in degree 7
-
x13.x2
in degree 8
-
y34.x12
in degree 8
-
y36.x3
in degree 8
-
y36.x1
in degree 8
-
y13.x2.w1
in degree 8
-
y15.w1
in degree 8
-
y16.x2
in degree 8
-
x23.w2
in degree 9
-
x12.x2.w1
in degree 9
-
y35.x12
in degree 9
-
y14.x2.w1
in degree 9
-
y16.w1
in degree 9
-
y36.x12
in degree 10
-
y15.x2.w1
in degree 10
-
y16.x2.w1
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y3
in degree 1
-
y32
in degree 2
-
y3.x3
in degree 3
-
y3.x1
in degree 3
-
y33
in degree 3
-
y32.x3
in degree 4
-
y32.x1
in degree 4
-
y34
in degree 4
-
y3.x12
in degree 5
-
y33.x3
in degree 5
-
y33.x1
in degree 5
-
y35
in degree 5
-
y32.x12
in degree 6
-
y34.x3
in degree 6
-
y34.x1
in degree 6
-
y36
in degree 6
-
y33.x12
in degree 7
-
y35.x3
in degree 7
-
y35.x1
in degree 7
-
y34.x12
in degree 8
-
y36.x3
in degree 8
-
y36.x1
in degree 8
-
y35.x12
in degree 9
-
y36.x12
in degree 10