Small group number 11 of order 32
G is the group 32gp11
The Hall-Senior number of this group is 31.
G has 2 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 3 maximal subgroups are:
16gp13, Ab(4,4), 16gp6.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 6 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- w in degree 3, a nilpotent element
- v in degree 4, a regular element
There are 9 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x2 =
y1.x2
- y1.x1 =
0
- x1.x2 =
y1.w
- y2.w =
y1.w
- x12 =
0
- x1.w =
0
- w2 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
There are 3 minimal generators:
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 6
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
x2
+ y22
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 6.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
x1
in degree 2
-
w
in degree 3
-
y2.x1
in degree 3
-
y1.w
in degree 4
Restriction to maximal subgroup number 1, which is 16gp13
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
y1.y3
+ y1.y2
- x2 restricts to
y32
+ y1.y3
- w restricts to
y2.y32
+ y1.y2.y3
- v restricts to
y13.y3
+ v
Restriction to maximal subgroup number 2, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ y1.y2
- w restricts to
w
- v restricts to
v
+ y2.w
Restriction to maximal subgroup number 3, which is 16gp2
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
x1
- w restricts to
y2.x1
+ y1.x2
- v restricts to
x22
+ x1.x2
+ y1.y2.x1
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
y22
- w restricts to
0
- v restricts to
y12.y22
+ y14
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- w restricts to
0
- v restricts to
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
- w restricts to
0
- v restricts to
y4
(1 + 2t + 2t2
+ 2t3 + t4) /
(1 - t2) (1 - t4)
Back to the groups of order 32