Small group number 2 of order 243
G is the group 243gp2
G has 2 minimal generators, rank 3 and exponent 9.
The centre has rank 3.
The 4 maximal subgroups are:
Ab(9,3,3) (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 7 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2, a nilpotent element
- x2 in degree 2, a nilpotent element
- x3 in degree 2, a regular element
- x4 in degree 2, a regular element
- x5 in degree 2, a regular element
There are 9 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x2 =
- y1.x2
- y2.x1 =
y1.x2
- y1.x1 =
0
- x22 =
0
- x1.x2 =
0
- x12 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There is one minimal generator:
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 6. 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 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x3
in degree 2
- h2 =
x4
in degree 2
- h3 =
x5
in degree 2
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
The first
3 terms h1, h2, h3 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, h3.
These free generators are:
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
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
y1.x2
in degree 3
Restriction to maximal subgroup number 1, which is 81gp11
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
x1
+ y1.y3
+ y1.y2
- x4 restricts to
x2
- x5 restricts to
x3
Restriction to maximal subgroup number 2, which is 81gp11
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
- y1.y3
- x2 restricts to
y1.y3
- x3 restricts to
x2
- y1.y3
- y1.y2
- x4 restricts to
x3
- x5 restricts to
x1
Restriction to maximal subgroup number 3, which is 81gp11
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
y1.y3
- x2 restricts to
0
- x3 restricts to
x2
- y1.y3
+ y1.y2
- x4 restricts to
- x3
- x5 restricts to
x3
+ x1
Restriction to maximal subgroup number 4, which is 81gp11
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
- y1.y3
- x2 restricts to
- y1.y3
- x3 restricts to
x2
- y1.y2
- x4 restricts to
x3
- x5 restricts to
x3
+ x1
Restriction to maximal elementary abelian number 1, which is V27
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x3
- x4 restricts to
x3
+ x1
- x5 restricts to
x2
Restriction to the greatest central elementary abelian, which is V27
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x3
- x2
- x4 restricts to
x2
- x1
- x5 restricts to
- x1
(1 + 2t + 2t2
+ t3) /
(1 - t2)3
Back to the groups of order 243