Small group number 7 of order 32
G is the group 32gp7
The Hall-Senior number of this group is 47.
G has 2 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 3 maximal subgroups are:
D8xC2, 16gp6 (2x).
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 8 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- x1 in degree 2
- x2 in degree 2
- w1 in degree 3
- w2 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
There are 18 minimal relations:
- y1.y2 =
0
- y12 =
0
- y1.x2 =
0
- y1.x1 =
0
- x22 =
y22.x2
- x1.x2 =
y2.w2
- y1.w2 =
0
- y1.w1 =
0
- x2.w2 =
y22.w2
- x2.w1 =
y2.v1
+ y2.x12
- y1.v1 =
0
- w22 =
y2.x1.w2
- w1.w2 =
x1.v1
+ x13
- w12 =
x13
+ y2.x1.w1
+ y22.v1
+ y22.x12
+ y22.v2
- x2.v1 =
y2.x1.w2
+ y22.v1
+ y22.x12
- w2.v1 =
x12.w2
+ y2.x1.v1
+ y2.x13
- w1.v1 =
x12.w2
+ x12.w1
+ y2.x1.v1
+ y2.x13
+ y23.v1
+ y23.x12
+ y2.x2.v2
- v12 =
x14
+ y2.x12.w2
+ y22.x1.v1
+ y22.x13
+ y24.v1
+ y24.x12
+ y22.x2.v2
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
There is one minimal generator:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 8 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
v2
in degree 4
- h2 =
y22
in degree 2
- h3 =
x1
in degree 2
The first
term h1 forms
a regular sequence of maximum length.
The remaining
2 terms h2, h3 are all
annihilated by the class
y1.
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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y2.x2
in degree 3
-
v1
in degree 4
-
y2.w1
in degree 4
-
y2.v1
in degree 5
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 6.
-
y1
in degree 1
-
y2.x2
in degree 3
A basis for AnnR/(h1)(h2, h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 4.
A basis for AnnR/(h1)(h2)
/ h3 AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 6.
Restriction to maximal subgroup number 1, which is 16gp11
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- x1 restricts to
y2.y3
+ y1.y3
+ y32
- x2 restricts to
y12
- w1 restricts to
y13
+ y2.x
+ y1.x
+ y2.y32
+ y1.y32
+ y33
- w2 restricts to
y12.y3
+ y1.y32
- v1 restricts to
y14
+ y12.x
+ y22.y32
+ y1.y33
+ y34
- v2 restricts to
y12.x
+ y13.y3
+ x2
+ y2.y3.x
+ y1.y3.x
+ y32.x
+ y1.y33
Restriction to maximal subgroup number 2, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
y1.y2
- w1 restricts to
y23
- w2 restricts to
w
- v1 restricts to
y24
+ y2.w
- v2 restricts to
v
Restriction to maximal subgroup number 3, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
y1.y2
- w1 restricts to
y23
+ w
- w2 restricts to
w
- v1 restricts to
y24
+ y2.w
- v2 restricts to
v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- w1 restricts to
y33
+ y2.y32
+ y1.y22
+ y12.y2
- w2 restricts to
0
- v1 restricts to
y34
+ y22.y32
- v2 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
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y22
- w1 restricts to
y33
+ y2.y32
+ y23
+ y1.y22
+ y12.y2
- w2 restricts to
y2.y32
+ y22.y3
- v1 restricts to
y34
+ y2.y33
+ y24
+ y1.y23
+ y12.y22
- v2 restricts to
y2.y33
+ y23.y3
+ y1.y2.y32
+ y1.y22.y3
+ y1.y23
+ y12.y32
+ y12.y2.y3
+ y14
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
0
- v2 restricts to
y4
(1 + 2t + t2
+ t3 + 2t4 + t5) /
(1 - t2)2 (1 - t4)
Back to the groups of order 32