Small group number 6 of order 32
G is the group 32gp6
The Hall-Senior number of this group is 46.
G has 2 minimal generators, rank 3 and exponent 4.
The centre has rank 1.
The 3 maximal subgroups are:
D8xC2, 16gp3 (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
- 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 =
y1.x1
- x2.x3 =
x12
- y2.w2 =
0
- y1.w2 =
0
- y1.w1 =
0
- x3.w1 =
x1.w2
+ y1.x12
- x2.w2 =
x1.w1
+ y1.x12
- w22 =
x12.x3
+ x12.x2
- w1.w2 =
x1.x22
+ x13
- w12 =
x23
+ x12.x2
+ y2.x2.w1
+ y22.v
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
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 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
+ x2
+ x1
+ y22
in degree 2
- h3 =
x3
+ 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.x1.
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
-
y22
in degree 2
-
w2
in degree 3
-
w1
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.
Restriction to maximal subgroup number 1, which is 16gp11
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
y12
+ y1.y3
- x2 restricts to
y12
+ y2.y3
+ y32
- x3 restricts to
y12
- w1 restricts to
y2.x
+ y12.y3
+ y2.y32
+ y33
- w2 restricts to
y12.y3
+ y1.y32
- v restricts to
x2
+ y2.y3.x
+ y1.y3.x
+ y12.y32
+ y32.x
+ y1.y33
Restriction to maximal subgroup number 2, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
x1
- x2 restricts to
y22
- x3 restricts to
x2
- w1 restricts to
y2.x1
+ y23
- w2 restricts to
y2.x1
+ y2.x2
+ y1.x3
- v restricts to
y22.x1
+ x1.x3
+ x1.x2
+ x32
+ x2.x3
Restriction to maximal subgroup number 3, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x1
+ x2
- x2 restricts to
x2
- x3 restricts to
y22
+ x2
- w1 restricts to
y2.x2
+ y1.x3
+ y1.x2
- w2 restricts to
y2.x1
+ y2.x2
+ y1.x3
- v restricts to
y22.x1
+ x1.x3
+ y22.x2
+ x32
+ x2.x3
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y32
- x3 restricts to
y32
+ y22
- w1 restricts to
y2.y32
- w2 restricts to
y2.y32
+ y22.y3
- v restricts to
y22.y32
+ y23.y3
+ 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
0
- x2 restricts to
y32
+ y2.y3
- x3 restricts to
0
- w1 restricts to
y33
+ 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 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 32