Small group number 32 of order 64
G is the group 64gp32
The Hall-Senior number of this group is 250.
G has 2 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
The 3 maximal subgroups are:
32gp27, 32gp6, 32gp7.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 11 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
- u in degree 5
There are 33 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x3 =
0
- y1.x2 =
0
- y1.x1 =
0
- x1.x3 =
0
- y2.w1 =
0
- y1.w3 =
0
- y1.w2 =
0
- y1.w1 =
0
- x3.w3 =
0
- x3.w2 =
0
- x2.w3 =
x1.w2
+ y2.v1
+ y2.x22
- x1.w1 =
0
- y1.v1 =
0
- w32 =
x13
+ y2.x1.w3
+ y22.x1.x2
+ y23.w2
- w2.w3 =
x12.x2
+ y2.u
+ y2.x2.w2
+ y2.x1.w2
+ y22.v1
+ y22.x22
- w22 =
x1.x22
+ y2.x2.w2
+ y22.v2
- w1.w3 =
0
- w1.w2 =
0
- w12 =
x22.x3
- x3.v1 =
x22.x3
- y1.u =
0
- w3.v1 =
x1.u
+ y2.x2.v1
+ y2.x23
+ y2.x1.x22
+ y22.x2.w2
- w2.v1 =
x2.u
+ y2.x2.v1
+ y2.x23
+ y2.x1.v2
- w1.v1 =
x22.w1
- x3.u =
0
- v12 =
x24
+ x1.x2.v1
+ y2.x22.w2
+ x12.v2
- w3.u =
x12.v1
+ y2.x2.u
+ y2.x22.w2
+ y2.x1.u
+ y2.x1.x2.w2
+ y22.x2.v1
+ y22.x23
+ y24.v2
- w2.u =
x1.x2.v1
+ y2.x22.w2
+ y2.w3.v2
+ y22.x2.v2
- w1.u =
0
- v1.u =
x23.w2
+ x1.x2.u
+ y2.x22.v1
+ y2.x24
+ x1.w3.v2
+ y2.x1.x2.v2
+ y23.x2.v2
- u2 =
x1.x24
+ x12.x2.v1
+ y2.x23.w2
+ y2.x1.x2.u
+ y2.x1.x22.w2
+ x13.v2
+ y2.x1.w3.v2
+ y22.x22.v2
+ y22.x1.x2.v2
+ y23.w2.v2
+ y24.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 10. The cohomology ring approximation
is stable from degree 10 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 4 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v2
in degree 4
- h2 =
x3
+ y22
in degree 2
- h3 =
x2
in degree 2
- h4 =
x1
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
2 terms h3, h4 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, h4) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
v1
in degree 4
-
y2.w3
in degree 4
-
y2.w2
in degree 4
-
u
in degree 5
-
y2.u
in degree 6
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.
-
y1
in degree 1
-
y22
in degree 2
-
w1
in degree 3
-
y2.w2
in degree 4
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 6.
A basis for AnnR/(h1, h2)(h3)
/ h4 AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
Restriction to maximal subgroup number 1, which is 32gp27
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
y1.y2
+ y12
- x2 restricts to
x1
+ x3
- x3 restricts to
y32
- w1 restricts to
y3.x3
- w2 restricts to
y1.x1
+ y2.x2
+ y1.x3
- w3 restricts to
y2.x1
+ y1.y22
+ y13
- v1 restricts to
y1.y2.x1
+ y12.x1
+ x1.x3
+ y1.y2.x3
+ y1.y2.x2
+ y12.x3
+ y12.x2
+ x32
- v2 restricts to
x1.x2
+ x2.x3
+ x22
- u restricts to
y13.x1
+ y2.x1.x3
+ y23.x2
+ y1.x1.x3
+ y1.y22.x3
+ y1.y22.x2
+ y12.y2.x3
+ y13.x3
+ y13.x2
+ y2.x2.x3
+ y1.x32
Restriction to maximal subgroup number 2, which is 32gp6
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y22
- x2 restricts to
x2
+ x1
- x3 restricts to
x3
- w1 restricts to
w2
- w2 restricts to
y2.x2
+ y1.x1
- w3 restricts to
y23
- v1 restricts to
x22
+ x12
+ y2.w1
- v2 restricts to
x1.x3
+ x1.x2
+ v
- u restricts to
y2.x22
+ y22.w1
+ y1.x12
Restriction to maximal subgroup number 3, which is 32gp7
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x2
+ y22
- x2 restricts to
x1
- x3 restricts to
x2
- w1 restricts to
w2
- w2 restricts to
w2
+ y2.x1
- w3 restricts to
y2.x2
+ y23
- v1 restricts to
v1
+ y2.w2
+ y2.w1
+ y22.x1
- v2 restricts to
v1
+ x12
+ y2.w2
+ y22.x2
+ v2
- u restricts to
x1.w2
+ y2.v1
+ y22.w2
+ y22.w1
+ y23.x1
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- x3 restricts to
y22
- w1 restricts to
y2.y32
+ y22.y3
- w2 restricts to
0
- w3 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
- u restricts to
0
Restriction to maximal elementary abelian number 2, which is V16
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y42
+ y3.y4
+ y2.y4
+ y1.y2
- x3 restricts to
0
- w1 restricts to
0
- w2 restricts to
y3.y42
+ y32.y4
+ y2.y3.y4
+ y12.y2
- w3 restricts to
y33
+ y2.y3.y4
+ y22.y3
+ y1.y22
- v1 restricts to
y44
+ y3.y43
+ y32.y42
+ y33.y4
+ y22.y42
+ y22.y3.y4
+ y1.y33
+ y1.y2.y42
+ y1.y22.y4
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
- v2 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
- u restricts to
y3.y44
+ y32.y43
+ y33.y42
+ y34.y4
+ y2.y3.y43
+ y2.y33.y4
+ y22.y3.y42
+ y22.y32.y4
+ y23.y3.y4
+ y1.y34
+ y1.y2.y33
+ y1.y22.y42
+ y1.y22.y32
+ y1.y23.y4
+ y1.y23.y3
+ y12.y33
+ y12.y2.y42
+ y12.y22.y4
+ y12.y22.y3
+ y12.y23
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
- x3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- w3 restricts to
0
- v1 restricts to
0
- v2 restricts to
y4
- u restricts to
0
(1 + 2t + t2
+ t3 + 2t4 + t5) /
(1 - t2)3 (1 - t4)
Back to the groups of order 64