Small group number 37 of order 32
G is the group 32gp37
The Hall-Senior number of this group is 13.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 2.
The 7 maximal subgroups are:
Ab(4,2,2), Ab(8,2) (2x), 16gp6 (4x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 5 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1, a regular element
- w in degree 3, a nilpotent element
- v in degree 4, a regular element
There are 4 minimal relations:
- y12 =
0
- y1.y22 =
0
- y1.w =
0
- w2 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There is one minimal generator:
Nilradical:
There are 2 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 8
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
y3
in degree 1
- h2 =
v
in degree 4
- h3 =
y22
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
2 terms h1, h2 form
a complete Duflot regular sequence.
That is, their restrictions to the greatest central elementary abelian
subgroup form a regular sequence of maximal length.
The ideal of essential classes is
free of rank 1 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y2.y3
in degree 3
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 7.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y1.y2
in degree 2
-
w
in degree 3
-
y2.w
in degree 4
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 5.
-
y1
in degree 1
-
y1.y2
in degree 2
Restriction to maximal subgroup number 1, which is 16gp10
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- w restricts to
y1.y32
- v restricts to
x2
+ y32.x
+ y1.y33
Restriction to maximal subgroup number 2, which is 16gp5
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- w restricts to
y1.x
- v restricts to
x2
Restriction to maximal subgroup number 3, which is 16gp5
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- w restricts to
y1.x
- v restricts to
x2
Restriction to maximal subgroup number 4, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- w restricts to
w
- v restricts to
v
Restriction to maximal subgroup number 5, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- w restricts to
w
- v restricts to
v
Restriction to maximal subgroup number 6, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- w restricts to
w
- v restricts to
v
Restriction to maximal subgroup number 7, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- w restricts to
w
- v restricts to
v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- w restricts to
0
- v restricts to
y12.y32
+ y14
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- w restricts to
0
- v restricts to
y14
(1 + 2t + t2) /
(1 - t) (1 - t2) (1 - t4)
Back to the groups of order 32