Small group number 34 of order 64
G is the group 64gp34
The Hall-Senior number of this group is 252.
G has 2 minimal generators, rank 3 and exponent 4.
The centre has rank 1.
The 3 maximal subgroups are:
32gp34, 32gp6 (2x).
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 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
- x3 in degree 2
- w1 in degree 3
- w2 in degree 3
- v in degree 4, a regular element
There are 14 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x3 =
y1.x1
- y2.x1 =
y1.x1
- y1.x2 =
0
- x1.x3 =
x12
- y2.w2 =
0
- y1.w2 =
0
- y1.w1 =
0
- x3.w1 =
x1.w1
- x1.w2 =
x1.w1
- w22 =
x22.x3
- w1.w2 =
x1.x22
- 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:
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 6 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
x3
+ y22
in degree 2
- h3 =
x2
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
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
-
x1
in degree 2
-
y22
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.x1
in degree 3
-
y2.w1
in degree 4
-
y22.w1
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
-
y1.x1
in degree 3
Restriction to maximal subgroup number 1, which is 32gp34
- y1 restricts to
0
- y2 restricts to
y3
+ y2
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ x2
- x3 restricts to
y22
- w1 restricts to
y13
+ y3.x1
+ y2.x1
+ y1.x2
- w2 restricts to
y23
+ y2.x2
- v restricts to
y13.y2
+ y14
+ y22.x1
+ y12.x2
+ x1.x2
+ x12
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
+ y22
- w1 restricts to
y2.x2
+ y1.x1
- w2 restricts to
w2
+ y2.x2
+ y1.x1
- v restricts to
x1.x3
+ x1.x2
+ v
Restriction to maximal subgroup number 3, which is 32gp6
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x3
- x2 restricts to
x2
+ x1
- x3 restricts to
x3
+ y22
- w1 restricts to
w2
- w2 restricts to
w2
+ y2.x2
+ y1.x1
- v restricts to
x1.x3
+ x1.x2
+ v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- x3 restricts to
0
- w1 restricts to
y2.y32
+ y22.y3
+ y1.y22
+ y12.y2
- 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
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- x3 restricts to
y22
- w1 restricts to
0
- w2 restricts to
y2.y32
+ y22.y3
- 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 V8
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
y32
- x2 restricts to
y2.y3
+ y22
- x3 restricts to
y32
- w1 restricts to
y2.y32
+ y22.y3
- w2 restricts to
y2.y32
+ y22.y3
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ 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
- x3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y4
(1 + 2t + 2t2
+ 2t3 + t4) /
(1 - t2)2 (1 - t4)
Back to the groups of order 64