Small group number 6661 of order 256
G = Syl2(S4(7)) is Sylow 2-group of Symplectic group S4(7)
G has 3 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
There are 6 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3, 3, 4, 4.
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
- x2 in degree 2
- x3 in degree 2
- w in degree 3
- v in degree 4, a regular element
There are 9 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y2.x3 =
0
- y2.x2 =
y1.x2
- y1.x1 =
0
- x1.x3 =
0
- y2.w =
0
- y1.w =
0
- w2 =
x22.x3
+ x1.x22
+ y3.x2.w
+ y32.v
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
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
+ y3.w
+ y32.x2
+ y34
+ y24
+ y12.y22
+ y14
in degree 4
- h3 =
x22.x3
+ x1.x22
+ y3.x3.w
+ y3.x1.w
+ y32.x32
+ y32.x2.x3
+ y32.x22
+ y32.x1.x2
+ y32.x12
+ y33.w
+ y34.x2
+ y22.x12
+ y12.x32
+ y12.x22
+ y12.y24
+ y14.y22
in degree 6
- h4 =
y3
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
-
y2
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w
in degree 3
-
y2.x1
in degree 3
-
y23
in degree 3
-
y1.x3
in degree 3
-
y1.x2
in degree 3
-
y1.y22
in degree 3
-
y12.y2
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
-
y22.x1
in degree 4
-
y24
in degree 4
-
y1.y23
in degree 4
-
y12.x3
in degree 4
-
y12.x2
in degree 4
-
y12.y22
in degree 4
-
y13.y2
in degree 4
-
y14
in degree 4
-
x3.w
in degree 5
-
x2.w
in degree 5
-
x1.w
in degree 5
-
y23.x1
in degree 5
-
y25
in degree 5
-
y1.x22
in degree 5
-
y1.y24
in degree 5
-
y12.y23
in degree 5
-
y13.x3
in degree 5
-
y13.x2
in degree 5
-
y13.y22
in degree 5
-
y14.y2
in degree 5
-
y15
in degree 5
-
x23
in degree 6
-
x12.x2
in degree 6
-
x13
in degree 6
-
y24.x1
in degree 6
-
y26
in degree 6
-
y12.x22
in degree 6
-
y12.y24
in degree 6
-
y13.y23
in degree 6
-
y14.x3
in degree 6
-
y14.x2
in degree 6
-
y14.y22
in degree 6
-
y15.y2
in degree 6
-
y16
in degree 6
-
x2.x3.w
in degree 7
-
x22.w
in degree 7
-
x1.x2.w
in degree 7
-
x12.w
in degree 7
-
y25.x1
in degree 7
-
y13.x22
in degree 7
-
y13.y24
in degree 7
-
y14.y23
in degree 7
-
y15.x3
in degree 7
-
y15.x2
in degree 7
-
y15.y22
in degree 7
-
y16.y2
in degree 7
-
x13.x2
in degree 8
-
y26.x1
in degree 8
-
y14.x22
in degree 8
-
y14.y24
in degree 8
-
y15.y23
in degree 8
-
y16.x3
in degree 8
-
y16.x2
in degree 8
-
y16.y22
in degree 8
-
x23.w
in degree 9
-
x12.x2.w
in degree 9
-
x13.w
in degree 9
-
y15.x22
in degree 9
-
y15.y24
in degree 9
-
y16.y23
in degree 9
-
y16.x22
in degree 10
-
y16.y24
in degree 10
-
x13.x2.w
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y2.x1
in degree 3
-
y23
in degree 3
-
y1.x3
in degree 3
-
y1.x2
in degree 3
-
y1.y22
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y22.x1
in degree 4
-
y24
in degree 4
-
y1.y23
in degree 4
-
y12.x3
in degree 4
-
y12.x2
in degree 4
-
y12.y22
in degree 4
-
y13.y2
in degree 4
-
y14
in degree 4
-
y23.x1
in degree 5
-
y25
in degree 5
-
y1.x22
in degree 5
-
y1.y24
in degree 5
-
y12.y23
in degree 5
-
y13.x3
in degree 5
-
y13.x2
in degree 5
-
y13.y22
in degree 5
-
y14.y2
in degree 5
-
y15
in degree 5
-
y24.x1
in degree 6
-
y26
in degree 6
-
y12.x22
in degree 6
-
y12.y24
in degree 6
-
y13.y23
in degree 6
-
y14.x3
in degree 6
-
y14.x2
in degree 6
-
y14.y22
in degree 6
-
y15.y2
in degree 6
-
y16
in degree 6
-
y25.x1
in degree 7
-
y13.x22
in degree 7
-
y13.y24
in degree 7
-
y14.y23
in degree 7
-
y15.x3
in degree 7
-
y15.x2
in degree 7
-
y15.y22
in degree 7
-
y16.y2
in degree 7
-
y26.x1
in degree 8
-
y14.x22
in degree 8
-
y14.y24
in degree 8
-
y15.y23
in degree 8
-
y16.x3
in degree 8
-
y16.x2
in degree 8
-
y16.y22
in degree 8
-
y15.x22
in degree 9
-
y15.y24
in degree 9
-
y16.y23
in degree 9
-
y16.x22
in degree 10
-
y16.y24
in degree 10