Small group number 32 of order 32
G is the group 32gp32
The Hall-Senior number of this group is 40.
G has 3 minimal generators, rank 2 and exponent 4.
The centre has rank 2.
The 7 maximal subgroups are:
Ab(4,4), 16gp4 (6x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 7 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a nilpotent element
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- v1 in degree 4, a regular element
- v2 in degree 4, a regular element
There are 10 minimal relations:
- y32 =
y1.y2
+ y12
- y22 =
y1.y3
- y12.y3 =
y12.y2
- y13 =
0
- y3.w2 =
y2.w2
+ y1.w1
- y3.w1 =
y2.w1
+ y1.w2
+ y1.w1
- y12.w1 =
0
- w22 =
y1.y3.v2
+ y1.y2.v2
+ y12.v1
+ y12.y2.w2
- w1.w2 =
y1.y3.v2
+ y1.y3.v1
+ y1.y2.v2
+ y1.y2.v1
+ y12.v2
+ y12.y2.w2
- w12 =
y1.y3.v1
+ y1.y2.v1
+ y12.v2
+ y12.v1
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 5 minimal generators:
-
y1.y2.y3
-
y12.y2
-
y1.y2.w2
-
y1.y2.w1
-
y12.w2
Nilradical:
There are 5 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 22. 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 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v1
in degree 4
- h2 =
v2
in degree 4
The first
2 terms h1, h2 form
a regular sequence of maximum length.
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 6 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y2.y3
in degree 3
- G2 =
y12.y2
in degree 3
- G3 =
y1.y2.w2
in degree 5
- G4 =
y1.y2.w1
in degree 5
- G5 =
y12.w2
in degree 5
- G6 =
y12.y2.w2
in degree 6
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 8.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.y2.y3
in degree 3
-
y12.y2
in degree 3
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
y1.y2.w2
in degree 5
-
y1.y2.w1
in degree 5
-
y12.w2
in degree 5
-
y12.y2.w2
in degree 6
Restriction to maximal subgroup number 1, which is 16gp2
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- w1 restricts to
y2.x2
+ y2.x1
+ y1.x2
+ y1.x1
- w2 restricts to
y2.x1
+ y1.x1
- v1 restricts to
x22
+ x12
+ y1.y2.x2
- v2 restricts to
x12
Restriction to maximal subgroup number 2, which is 16gp4
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y2
+ y1
- w1 restricts to
y2.x2
+ y1.x2
+ y1.x1
- w2 restricts to
y2.x2
+ y2.x1
+ y1.x1
- v1 restricts to
x22
+ x12
+ y1.y2.x1
- v2 restricts to
x12
+ y1.y2.x2
Restriction to maximal subgroup number 3, which is 16gp4
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- w1 restricts to
y2.x2
+ y1.x2
+ y1.x1
- w2 restricts to
y2.x2
+ y2.x1
+ y1.x1
- v1 restricts to
x22
+ x12
+ y1.y2.x2
- v2 restricts to
x12
+ y1.y2.x2
Restriction to maximal subgroup number 4, which is 16gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
0
- w1 restricts to
y2.x2
+ y1.x1
- w2 restricts to
y2.x1
+ y1.x2
+ y1.x1
- v1 restricts to
x22
+ x12
+ y1.y2.x2
- v2 restricts to
x22
+ y1.y2.x2
Restriction to maximal subgroup number 5, which is 16gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- w1 restricts to
y2.x2
+ y1.x1
- w2 restricts to
y2.x1
+ y1.x2
+ y1.x1
- v1 restricts to
x22
+ x12
- v2 restricts to
x22
+ y1.y2.x2
Restriction to maximal subgroup number 6, which is 16gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- w1 restricts to
y1.x2
+ y1.x1
- w2 restricts to
y1.x2
- v1 restricts to
x12
+ y1.y2.x1
- v2 restricts to
x22
+ x12
Restriction to maximal subgroup number 7, which is 16gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- w1 restricts to
y1.x2
+ y1.x1
- w2 restricts to
y1.x1
- v1 restricts to
x12
+ y1.y2.x2
- v2 restricts to
x22
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
y24
+ y14
- v2 restricts to
y24
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
y14
- v2 restricts to
y24
+ y14
(1 + 3t + 4t2
+ 4t3 + 4t4 + 3t5
+ t6) /
(1 - t4)2
Back to the groups of order 32