Small group number 928 of order 128
G is the group 128gp928
G has 3 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
Syl2(M12), 64gp138 (2x), 64gp226, 64gp32 (2x), 64gp34.
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.
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 6 onwards, and
Carlson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 4 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
x2
+ y32
in degree 2
- h3 =
x3
+ x1
+ y12
in degree 2
- h4 =
y24
+ y12.x3
in degree 4
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
The remaining
term h4 is
annihilated by the class
y3.
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.
The ideal of essential classes is
the zero ideal.
The essential ideal squares to zero.
A basis for R/(h1, h2, h3, h4) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 12.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x1
in degree 2
-
y32
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y3.x1
in degree 3
-
y23
in degree 3
-
y1.y22
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y32.x1
in degree 4
-
y2.w1
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
y1.y23
in degree 4
-
y12.y22
in degree 4
-
y14
in degree 4
-
y22.w1
in degree 5
-
y1.y2.w1
in degree 5
-
y12.w2
in degree 5
-
y12.w1
in degree 5
-
y12.y23
in degree 5
-
y15
in degree 5
-
y23.w1
in degree 6
-
y1.y22.w1
in degree 6
-
y12.y2.w1
in degree 6
-
y13.w2
in degree 6
-
y12.y22.w1
in degree 7
-
y14.w2
in degree 7
-
y15.w2
in degree 8
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y3
in degree 1
-
y32
in degree 2
-
y3.x1
in degree 3
-
y32.x1
in degree 4
Restriction to maximal subgroup number 1, which is 64gp226
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y42
+ y3.y4
- x2 restricts to
y3.y4
+ y1.y3
+ x2
+ x1
- x3 restricts to
y1.y2
- w1 restricts to
y3.y42
+ y32.y4
+ y4.x2
+ y4.x1
+ y3.x2
- w2 restricts to
y2.x2
+ y1.x1
- v restricts to
y3.y4.x1
+ y1.y3.x1
+ x1.x2
Restriction to maximal subgroup number 2, which is 64gp138
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
+ y1
- x1 restricts to
x3
+ x2
- x2 restricts to
x1
+ y32
- x3 restricts to
y1.y2
- w1 restricts to
w
+ y3.x1
- w2 restricts to
y1.x1
- v restricts to
y3.w
+ y32.x3
+ y32.x2
+ y32.x1
+ v
Restriction to maximal subgroup number 3, which is 64gp32
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
+ y22
- x3 restricts to
x3
- w1 restricts to
w2
+ y2.x2
- w2 restricts to
w1
- v restricts to
y2.w2
+ y22.x2
+ y22.x1
+ v2
Restriction to maximal subgroup number 4, which is 64gp138
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y2
+ y1
- x1 restricts to
y1.y2
- x2 restricts to
x1
+ y32
- x3 restricts to
x3
+ x2
- w1 restricts to
y1.x1
- w2 restricts to
w
+ y33
- v restricts to
y3.w
+ y32.x3
+ y32.x2
+ y32.x1
+ v
Restriction to maximal subgroup number 5, which is 64gp32
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
- x1 restricts to
x3
- x2 restricts to
x2
- x3 restricts to
x1
- w1 restricts to
w1
- w2 restricts to
w2
- v restricts to
v2
Restriction to maximal subgroup number 6, which is 64gp134
- y1 restricts to
y3
- y2 restricts to
y3
- y3 restricts to
y1
- x1 restricts to
x2
+ y22
+ y1.y2
- x2 restricts to
x1
- x3 restricts to
x2
- w1 restricts to
w
+ y2.x1
- w2 restricts to
w
- v restricts to
y3.w
+ y22.x2
+ v
Restriction to maximal subgroup number 7, which is 64gp34
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
+ x1
- w1 restricts to
w1
+ y2.x2
- w2 restricts to
w2
+ w1
+ y2.x2
- v restricts to
v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
y32
+ y2.y3
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- x3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 3, which is V16
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y42
+ y3.y4
+ y1.y3
+ y1.y2
- x3 restricts to
0
- w1 restricts to
y3.y42
+ y32.y4
+ y1.y32
+ y12.y3
- w2 restricts to
y2.y42
+ y2.y3.y4
+ y1.y2.y3
+ y12.y2
- v restricts to
y1.y2.y42
+ y1.y2.y3.y4
+ y12.y42
+ y12.y3.y4
+ y12.y2.y3
+ y13.y3
+ y13.y2
+ y14
Restriction to maximal elementary abelian number 4, which is V16
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y42
+ y3.y4
+ y1.y2
- x3 restricts to
y32
+ y2.y3
- w1 restricts to
0
- w2 restricts to
y3.y42
+ y32.y4
+ y2.y42
+ y2.y3.y4
+ y12.y2
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y1.y2.y42
+ y1.y2.y3.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y12.y2.y3
+ y13.y2
+ y14
Restriction to maximal elementary abelian number 5, which is V16
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y42
+ y3.y4
+ y2.y4
+ y1.y2
- x3 restricts to
0
- w1 restricts to
y3.y42
+ y32.y4
+ y2.y42
+ y22.y4
+ y1.y22
+ y12.y2
- w2 restricts to
0
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y1.y2.y3.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y12.y2.y4
+ y12.y2.y3
+ y13.y2
+ y14
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y4
(1 + 3t + 5t2
+ 7t3 + 7t4 + 5t5
+ 3t6 + t7) /
(1 - t2)2 (1 - t4)2
Back to the groups of order 128