Small group number 25 of order 32
G is the group 32gp25
The Hall-Senior number of this group is 14.
G has 3 minimal generators, rank 3 and exponent 4.
The centre has rank 2.
The 7 maximal subgroups are:
Ab(4,2,2) (2x), D8xC2, Ab(4,4), 16gp3 (2x), 16gp4.
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 5 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2, a regular element
- x2 in degree 2, a regular element
There are 2 minimal relations:
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 12. The cohomology ring approximation
is stable from degree 2 onwards, and
Carlson's tests detect stability from degree 6
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x1
in degree 2
- h2 =
x2
in degree 2
- h3 =
y32
+ y22
in degree 2
The first
3 terms h1, h2, h3 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
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 6.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y1.y22
in degree 3
Restriction to maximal subgroup number 1, which is 16gp10
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
x
Restriction to maximal subgroup number 2, which is 16gp10
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
+ y1
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
x
Restriction to maximal subgroup number 3, which is 16gp2
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x2
+ x1
Restriction to maximal subgroup number 4, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x3
- x2 restricts to
y22
+ x2
Restriction to maximal subgroup number 5, which is 16gp11
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x
+ y2.y3
+ y1.y3
+ y32
- x2 restricts to
y32
Restriction to maximal subgroup number 6, which is 16gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x2
+ y1.y2
- x2 restricts to
x1
+ y1.y2
Restriction to maximal subgroup number 7, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
x1
+ x3
+ x2
- x2 restricts to
y22
+ x2
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y32
+ y12
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y32
+ y22
+ y12
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
y22
- x2 restricts to
y12
(1 + 3t + 3t2
+ t3) /
(1 - t2)3
Back to the groups of order 32