Small group number 14 of order 81
G = E27*C9 is Central product E27 * C_9
G has 3 minimal generators, rank 2 and exponent 9.
The centre has rank 1.
The 13 maximal subgroups are:
Ab(9,3) (4x), E27, M27 (8x).
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 7 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
- v in degree 4
- t in degree 6, a regular element
There are 9 minimal relations:
- y32 =
0
- y22 =
0
- y12 =
0
- y2.x2 =
y1.x1
- y2.v =
y1.x1.x2
- y1.v =
y1.x12
- x2.v =
x12.x2
- x1.v =
x1.x22
- v2 =
x12.x22
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y2.x1 =
0
- y1.x1.x22 =
y1.x13
- x1.x23 =
x13.x2
Essential ideal:
There is one minimal generator:
Nilradical:
There are 3 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 8 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
t
in degree 6
- h2 =
x22
+ x12
in degree 4
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y1.y2.
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
free of rank 1 as a module over the polynomial algebra
on h1.
These free generators are:
- G1 =
y1.y2.y3
in degree 3
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 10.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y3.x2
in degree 3
-
y3.x1
in degree 3
-
y2.x1
in degree 3
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y1.y2.y3
in degree 3
-
v
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
y2.y3.x1
in degree 4
-
y1.y3.x2
in degree 4
-
y1.y3.x1
in degree 4
-
y3.v
in degree 5
-
y3.x1.x2
in degree 5
-
y3.x12
in degree 5
-
y1.x1.x2
in degree 5
-
y1.x12
in degree 5
-
x12.x2
in degree 6
-
x13
in degree 6
-
y1.y3.x1.x2
in degree 6
-
y1.y3.x12
in degree 6
-
y3.x12.x2
in degree 7
-
y3.x13
in degree 7
-
y1.x12.x2
in degree 7
-
y1.y3.x12.x2
in degree 8
A basis for AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 6.
-
y1.y2
in degree 2
-
y1.y2.y3
in degree 3
Restriction to maximal subgroup number 1, which is 27gp2
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
0
- v restricts to
0
- t restricts to
x23
- x12.x2
Restriction to maximal subgroup number 2, which is 27gp2
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- x1 restricts to
0
- x2 restricts to
x1
- v restricts to
0
- t restricts to
x23
- x12.x2
Restriction to maximal subgroup number 3, which is 27gp2
- y1 restricts to
- y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
- x1
- v restricts to
x12
- t restricts to
x23
- x12.x2
Restriction to maximal subgroup number 4, which is 27gp2
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x1
- v restricts to
x12
- t restricts to
x23
- x12.x2
Restriction to maximal subgroup number 5, which is 27gp3
- y1 restricts to
- y2
- y2 restricts to
y2
+ y1
- y3 restricts to
0
- x1 restricts to
x4
+ x3
- x2 restricts to
- x3
- v restricts to
x32
- x1.x2
- y1.w2
+ y1.w1
- t restricts to
x12.x2
- t
- y1.x4.w1
- y1.x1.w2
+ y1.x1.w1
Restriction to maximal subgroup number 6, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
- y2
- y3 restricts to
- y1
- x1 restricts to
- x
- x2 restricts to
- y1.y2
- v restricts to
- y2.w
- t restricts to
t
+ y2.u
Restriction to maximal subgroup number 7, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x
- x2 restricts to
- y1.y2
- v restricts to
- y2.w
- t restricts to
- t
- y2.u
Restriction to maximal subgroup number 8, which is 27gp4
- y1 restricts to
y2
- y1
- y2 restricts to
y1
- y3 restricts to
- y1
- x1 restricts to
- y1.y2
- x2 restricts to
x
+ y1.y2
- v restricts to
- y2.w
- t restricts to
t
+ y2.u
Restriction to maximal subgroup number 9, which is 27gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
- y2
+ y1
- y3 restricts to
y1
- x1 restricts to
- x
- y1.y2
- x2 restricts to
x
- y1.y2
- v restricts to
x2
- y2.w
- t restricts to
- t
- y2.u
Restriction to maximal subgroup number 10, which is 27gp4
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
- y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
x
- v restricts to
x2
- y2.w
- t restricts to
t
+ y2.u
Restriction to maximal subgroup number 11, which is 27gp4
- y1 restricts to
- y2
- y1
- y2 restricts to
y1
- y3 restricts to
y1
- x1 restricts to
- y1.y2
- x2 restricts to
- x
+ y1.y2
- v restricts to
- y2.w
- t restricts to
- t
- y2.u
Restriction to maximal subgroup number 12, which is 27gp4
- y1 restricts to
- y2
+ y1
- y2 restricts to
- y2
- y1
- y3 restricts to
y1
- x1 restricts to
- x
+ y1.y2
- x2 restricts to
- x
- y1.y2
- v restricts to
x2
- y2.w
- t restricts to
- t
- y2.u
Restriction to maximal subgroup number 13, which is 27gp4
- y1 restricts to
- y2
- y1
- y2 restricts to
y2
- y3 restricts to
- y1
- x1 restricts to
x
- x2 restricts to
- x
+ y1.y2
- v restricts to
x2
- y2.w
- t restricts to
t
+ y2.u
Restriction to maximal elementary abelian number 1, which is V9
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
0
- v restricts to
0
- t restricts to
x1.x22
- x13
Restriction to maximal elementary abelian number 2, which is V9
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
x2
+ x1
- v 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
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
x2
- v restricts to
x22
- t restricts to
x1.x22
- x13
Restriction to maximal elementary abelian number 4, which is V9
- y1 restricts to
- y2
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x2
- x2 restricts to
- x2
- v restricts to
x22
- t restricts to
x1.x22
- x13
Restriction to the greatest central elementary abelian, which is C3
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- v restricts to
0
- t restricts to
- x3
(1 + 3t + 5t2
+ 6t3 + 6t4 + 5t5
+ 3t6 + 2t7 + t8) /
(1 - t4) (1 - t6)
Back to the groups of order 81