Small group number 26 of order 32
G is the group 32gp26
The Hall-Senior number of this group is 15.
G has 3 minimal generators, rank 2 and exponent 4.
The centre has rank 2.
The 7 maximal subgroups are:
Q8xC2, Ab(4,4) (3x), 16gp4 (3x).
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 5 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a nilpotent element
- x in degree 2, a regular element
- v in degree 4, a regular element
There are 3 minimal relations:
- y32 =
y12
- y22 =
y1.y2
+ y12
- y13 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 2 minimal generators:
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 4 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 =
x
in degree 2
- h2 =
v
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 3 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y2.y3
+ y12.y2
in degree 3
- G2 =
y12.y3
in degree 3
- G3 =
y12.y2.y3
+ y13.y2
in degree 4
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
-
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
-
y1.y2.y3
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y12.y2.y3
in degree 4
Restriction to maximal subgroup number 1, which is 16gp2
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
x1
- v restricts to
x22
Restriction to maximal subgroup number 2, which is 16gp2
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- x restricts to
x2
+ x1
- v restricts to
x22
Restriction to maximal subgroup number 3, which is 16gp2
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
x2
+ x1
- v restricts to
x22
Restriction to maximal subgroup number 4, which is 16gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
0
- x restricts to
x1
+ y1.y2
- v restricts to
x22
+ x12
Restriction to maximal subgroup number 5, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- x restricts to
y32
+ y1.y2
- v restricts to
v
+ y34
Restriction to maximal subgroup number 6, which is 16gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y2
- x restricts to
x1
- v restricts to
x22
+ x12
Restriction to maximal subgroup number 7, which is 16gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
- x restricts to
x1
- v restricts to
x22
+ x12
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x restricts to
y22
+ y12
- v restricts to
y14
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x restricts to
y12
- v restricts to
y24
(1 + 3t + 4t2
+ 3t3 + t4) /
(1 - t2) (1 - t4)
Back to the groups of order 32