Small group number 12 of order 81
G = E27xC3 is Direct product E27 x C_3
G has 3 minimal generators, rank 3 and exponent 3.
The centre has rank 2.
The 13 maximal subgroups are:
E27 (9x), V27 (4x).
There are 4 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 11 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a nilpotent element
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- x5 in degree 2, a regular element
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- t in degree 6, a regular element
There are 22 minimal relations:
- y32 =
0
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x4 =
y1.x2
- y2.x3 =
y1.x2
+ y1.x1
- y2.x1 =
- y1.x1
- y1.x3 =
y1.x2
- x3.x4 =
x2.x3
+ x12
+ y1.w2
+ y1.w1
- x32 =
x2.x3
- y1.w2
+ y1.w1
- x2.x4 =
x2.x3
+ x12
+ y1.w2
- y1.w1
- x1.x4 =
- x12
+ y1.w2
- x1.x3 =
x12
- y1.w2
+ y1.w1
- x1.x2 =
x12
+ y2.w2
- y2.w1 =
y1.w2
- y1.w1
- x4.w2 =
x4.w1
+ x2.w1
- y1.x12
- x3.w2 =
- x2.w1
+ x1.w1
+ y1.x12
- x3.w1 =
x2.w1
- x1.w2 =
y1.x12
- w22 =
0
- w1.w2 =
- y1.x1.w1
- w12 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
- y1.x2.w2 =
- y1.x2.w1
+ y1.x1.w1
Essential ideal:
Zero ideal
Nilradical:
There are 5 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 6 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x5
in degree 2
- h2 =
t
in degree 6
- h3 =
x4
- x3
- x2
- x1
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 10.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y3.x3
in degree 3
-
y3.x2
in degree 3
-
y3.x1
in degree 3
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y3.w2
in degree 4
-
y3.w1
in degree 4
-
y2.w2
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
y1.y3.x2
in degree 4
-
y1.y3.x1
in degree 4
-
x2.w1
in degree 5
-
x1.w1
in degree 5
-
y2.y3.w2
in degree 5
-
y1.y3.w2
in degree 5
-
y1.y3.w1
in degree 5
-
y3.x2.w1
in degree 6
-
y3.x1.w1
in degree 6
-
y1.x1.w1
in degree 6
-
y1.y3.x1.w1
in degree 7
Restriction to maximal subgroup number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
y2.y3
- x2 restricts to
x2
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
x1
- w1 restricts to
0
- w2 restricts to
y3.x2
- y2.x3
- t restricts to
x33
- x22.x3
Restriction to maximal subgroup number 2, which is V27
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- x1 restricts to
y2.y3
- x2 restricts to
0
- x3 restricts to
- y2.y3
- x4 restricts to
x2
- x5 restricts to
x1
- w1 restricts to
y3.x2
- y2.x3
- w2 restricts to
y3.x2
- y2.x3
- t restricts to
x33
- x22.x3
- y2.y3.x22
Restriction to maximal subgroup number 3, which is V27
- y1 restricts to
- y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x2
- x3 restricts to
x2
+ y2.y3
- x4 restricts to
- x2
- x5 restricts to
x1
- w1 restricts to
- y3.x2
+ y2.x3
+ y2.x2
- w2 restricts to
- y2.x2
- t restricts to
x33
- x22.x3
+ x23
+ y2.y3.x22
Restriction to maximal subgroup number 4, which is V27
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
- y2.y3
- x2 restricts to
x2
- x3 restricts to
x2
- y2.y3
- x4 restricts to
x2
- x5 restricts to
x1
- w1 restricts to
y3.x2
- y2.x3
- w2 restricts to
- y3.x2
+ y2.x3
- t restricts to
x33
- x22.x3
Restriction to maximal subgroup number 5, which is 27gp3
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
0
- x1 restricts to
- x2
+ x1
- x2 restricts to
x4
- x3 restricts to
x4
+ x2
- x4 restricts to
x4
+ x3
- x5 restricts to
0
- w1 restricts to
- w2
+ y1.x1
- w2 restricts to
- w2
- w1
+ y1.x2
- y1.x1
- t restricts to
x13
- t
+ y2.x3.w2
- y1.x4.w1
- y1.x1.w2
Restriction to maximal subgroup number 6, which is 27gp3
- y1 restricts to
- y2
+ y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y1
- x1 restricts to
x3
- x1
- x2 restricts to
x4
+ x3
- x3 restricts to
x4
+ x3
- x2
+ x1
- x4 restricts to
x4
- x3
- x5 restricts to
- x4
+ x3
- w1 restricts to
w2
+ 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 7, which is 27gp3
- y1 restricts to
- y2
+ y1
- y2 restricts to
- y2
- y3 restricts to
- y2
+ y1
- x1 restricts to
- x2
- x2 restricts to
- x3
- x3 restricts to
- x3
- x2
+ x1
- x4 restricts to
x4
- x3
- x5 restricts to
x4
- x3
- w1 restricts to
w2
+ w1
- y1.x2
- w2 restricts to
- w2
- y1.x2
- t restricts to
- t
- y1.x1.w2
- y1.x1.w1
Restriction to maximal subgroup number 8, which is 27gp3
- y1 restricts to
- y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- y2
- y1
- x1 restricts to
x4
- x2
- x1
- x2 restricts to
x4
+ x3
- x3 restricts to
x4
+ x1
- x4 restricts to
- x4
- x5 restricts to
- x4
- x3
- w1 restricts to
w1
+ y1.x4
- y1.x1
- w2 restricts to
- w2
+ w1
- y1.x4
+ y1.x1
- t restricts to
x43
- x13
- t
+ y1.x4.w1
+ y1.x1.w2
Restriction to maximal subgroup number 9, which is 27gp3
- y1 restricts to
- y2
+ y1
- y2 restricts to
- y2
- y1
- y3 restricts to
- y2
- x1 restricts to
- x4
+ x2
+ x1
- x2 restricts to
- x4
- x3
- x3 restricts to
- x4
- x3
+ x2
- x1
- x4 restricts to
x4
- x3
- x5 restricts to
- x3
- w1 restricts to
- w2
- w1
+ y1.x4
- y1.x2
- y1.x1
- w2 restricts to
w2
- w1
- y1.x4
- y1.x2
+ y1.x1
- t restricts to
- x43
+ x13
+ t
+ y1.x1.w2
+ y1.x1.w1
Restriction to maximal subgroup number 10, which is 27gp3
- y1 restricts to
y2
- y2 restricts to
- y2
+ y1
- y3 restricts to
- y2
- y1
- x1 restricts to
- x3
+ x1
- x2 restricts to
x4
- x3
- x3 restricts to
- x3
+ x2
+ x1
- x4 restricts to
x3
- x5 restricts to
- x4
- x3
- w1 restricts to
- w2
+ w1
+ y2.x3
+ y1.x2
- y1.x1
- w2 restricts to
- w1
- y2.x3
+ y1.x1
- t restricts to
- x33
+ x12.x2
+ x13
- t
+ y2.x3.w2
- y1.x4.w1
+ y1.x1.w1
Restriction to maximal subgroup number 11, which is 27gp3
- y1 restricts to
- y2
- y1
- y2 restricts to
- y2
+ y1
- y3 restricts to
- y2
+ y1
- x1 restricts to
x4
- x2
- x1
- x2 restricts to
x4
- x3
- x3 restricts to
x4
- x3
- x2
- x4 restricts to
- x4
- x3
- x5 restricts to
x4
- x3
- w1 restricts to
w2
+ y1.x4
+ y1.x2
- y1.x1
- w2 restricts to
- w2
+ w1
- y1.x4
+ y1.x1
- t restricts to
x43
- x13
- t
+ y1.x4.w1
- y1.x1.w2
- y1.x1.w1
Restriction to maximal subgroup number 12, which is 27gp3
- y1 restricts to
y2
+ y1
- y2 restricts to
- y1
- y3 restricts to
- y2
+ y1
- x1 restricts to
- x4
+ x2
+ x1
- x2 restricts to
- x4
- x3 restricts to
- x4
- x2
- x4 restricts to
x4
+ x3
- x5 restricts to
x4
- x3
- w1 restricts to
w2
+ y1.x4
+ y1.x2
- y1.x1
- w2 restricts to
w2
- w1
- y1.x4
- y1.x2
+ y1.x1
- t restricts to
- x43
+ x13
+ t
- y2.x3.w2
- y1.x1.w1
Restriction to maximal subgroup number 13, which is 27gp3
- y1 restricts to
- y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x3
- x1
- x2 restricts to
x4
+ x3
- x3 restricts to
x3
+ x2
+ x1
- x4 restricts to
- x3
- x5 restricts to
x4
- w1 restricts to
- w2
+ 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
Restriction to maximal elementary abelian number 1, which is V27
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
- y1.y2
- x2 restricts to
0
- x3 restricts to
y1.y2
- x4 restricts to
x2
+ x1
- x5 restricts to
x3
- w1 restricts to
- y2.x1
+ y1.x2
- w2 restricts to
- y2.x1
+ y1.x2
- t restricts to
- x1.x22
+ x12.x2
+ y1.y2.x22
- y1.y2.x1.x2
+ y1.y2.x12
Restriction to maximal elementary abelian number 2, which is V27
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
- y1.y3
- x2 restricts to
x3
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
x2
- w1 restricts to
0
- w2 restricts to
- y3.x1
+ y1.x3
- t restricts to
- x1.x32
+ x13
Restriction to maximal elementary abelian number 3, which is V27
- y1 restricts to
y3
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
y1.y3
- x2 restricts to
x3
- x3 restricts to
x3
+ y1.y3
- x4 restricts to
x3
- x5 restricts to
x2
- w1 restricts to
- y3.x1
+ y1.x3
- w2 restricts to
y3.x1
- y1.x3
- t restricts to
- x1.x32
+ x13
Restriction to maximal elementary abelian number 4, which is V27
- y1 restricts to
- y3
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
x3
- x2 restricts to
x3
- x3 restricts to
x3
- y1.y3
- x4 restricts to
- x3
- x5 restricts to
x2
- w1 restricts to
y3.x3
+ y3.x1
- y1.x3
- w2 restricts to
- y3.x3
- t restricts to
x33
- x1.x32
+ x13
- y1.y3.x32
Restriction to the greatest central elementary abelian, which is V9
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
- y1
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
- x1
- w1 restricts to
0
- w2 restricts to
0
- t restricts to
x23
(1 + 3t + 5t2
+ 7t3 + 7t4 + 5t5
+ 3t6 + t7) /
(1 - t2)2 (1 - t6)
Back to the groups of order 81