Small group number 26 of order 243
G is the group 243gp26
G has 2 minimal generators, rank 2 and exponent 9.
The centre has rank 1.
The 4 maximal subgroups are:
Ab(9,9), Syl3(U3(8)) (3x).
There are 4 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2, 2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 9 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- t in degree 6, a regular element
There are 21 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x4 =
y1.x1
- y2.x2 =
y1.x1
- y2.x1 =
y1.x2
- y1.x3 =
y1.x1
- x3.x4 =
x1.x2
+ y1.w1
- x2.x4 =
x12
- x2.x3 =
x1.x2
+ y2.w2
- y1.w2
+ y1.w1
- x22 =
x12
- x1.x4 =
x1.x2
- x1.x3 =
x12
+ y1.w2
- y1.w1
- y2.w1 =
y1.w2
- y1.w1
- x4.w2 =
x4.w1
+ x1.w1
+ y1.x1.x2
- y1.x12
- x3.w1 =
x1.w1
- x2.w2 =
x1.w2
- y1.x1.x2
+ y1.x12
- x2.w1 =
x1.w2
- x1.w1
+ y1.x1.x2
- y1.x12
- w22 =
0
- w1.w2 =
y1.x1.w2
+ y1.x1.w1
- w12 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 8. 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 =
t
in degree 6
- h2 =
x4
+ x3
- x2
+ x1
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The first
term h1 forms
a complete Duflot regular sequence.
That is, its restriction to the greatest central elementary abelian
subgroup forms 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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y2.w2
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
x1.w2
in degree 5
-
x1.w1
in degree 5
-
y1.x1.w1
in degree 6
Restriction to maximal subgroup number 1, which is 81gp2
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
x1
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
y2.x1
- y1.x2
- t restricts to
- x23
+ x12.x2
+ y1.y2.x12
Restriction to maximal subgroup number 2, which is 81gp9
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x3
- x2
- x1
- x4 restricts to
x4
- w1 restricts to
w1
- y2.x3
- y1.x2
- y1.x1
- w2 restricts to
w1
+ y2.x3
+ y1.x2
+ y1.x1
- t restricts to
- x33
- x12.x2
- x13
- t
+ y2.x3.w2
+ y1.x4.w1
- y1.x1.w2
+ y1.x1.w1
Restriction to maximal subgroup number 3, which is 81gp9
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
x4
- x2 restricts to
- x4
- x3 restricts to
x4
- x3
+ x2
+ x1
- x4 restricts to
- x4
- w1 restricts to
- w1
+ y2.x3
+ y1.x4
+ y1.x2
+ y1.x1
- w2 restricts to
y2.x3
+ y1.x4
+ y1.x2
+ y1.x1
- t restricts to
x43
- x33
- x12.x2
- x13
- t
+ y2.x3.w2
- y1.x4.w1
- y1.x1.w2
+ y1.x1.w1
Restriction to maximal subgroup number 4, which is 81gp9
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x4
- x2 restricts to
x4
- x3 restricts to
x4
- x3
+ x2
+ x1
- x4 restricts to
x4
- w1 restricts to
- w1
+ y2.x3
+ y1.x2
+ y1.x1
- w2 restricts to
w1
+ y2.x3
+ y1.x2
+ y1.x1
- t restricts to
- x43
+ x33
+ x12.x2
+ x13
+ t
- y2.x3.w2
+ y1.x1.w2
- y1.x1.w1
Restriction to maximal elementary abelian number 1, which is V9
- y1 restricts to
y2
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
y1.y2
- x4 restricts to
x2
- w1 restricts to
y2.x1
- y1.x2
- w2 restricts to
y2.x1
- y1.x2
- t restricts to
- x1.x22
+ x13
Restriction to maximal elementary abelian number 2, which is V9
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x2
+ x1
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- t restricts to
- x1.x22
+ x12.x2
Restriction to maximal elementary abelian number 3, which is V9
- y1 restricts to
y2
- y2 restricts to
y2
- x1 restricts to
x2
- x2 restricts to
x2
- x3 restricts to
x2
+ y1.y2
- x4 restricts to
x2
- w1 restricts to
y2.x1
- y1.x2
- w2 restricts to
- y2.x1
+ y1.x2
- t restricts to
- x23
- x1.x22
+ x13
- y1.y2.x22
Restriction to maximal elementary abelian number 4, which is V9
- y1 restricts to
- y2
- y2 restricts to
y2
- x1 restricts to
x2
- x2 restricts to
- x2
- x3 restricts to
x2
- y1.y2
- x4 restricts to
- x2
- w1 restricts to
y2.x2
- y2.x1
+ y1.x2
- w2 restricts to
y2.x2
- t restricts to
x23
- x1.x22
+ x13
+ y1.y2.x22
Restriction to the greatest central elementary abelian, which is C3
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- t restricts to
x3
(1 + 2t + 3t2
+ 4t3 + 3t4 + 2t5
+ t6) /
(1 - t2) (1 - t6)
Back to the groups of order 243