Small group number 8 of order 81
G is the group 81gp8
G has 2 minimal generators, rank 2 and exponent 9.
The centre has rank 1.
The 4 maximal subgroups are:
Ab(9,3), E27, M27 (2x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 13 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
- w in degree 3, a nilpotent element
- v1 in degree 4, a nilpotent element
- v2 in degree 4
- u1 in degree 5, a nilpotent element
- u2 in degree 5, a nilpotent element
- t1 in degree 6
- t2 in degree 6, a regular element
- s in degree 7, a nilpotent element
There are 65 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x3 =
0
- y2.x2 =
0
- y2.x1 =
y1.x2
- y1.x1 =
0
- x2.x3 =
0
- x1.x3 =
0
- x22 =
0
- x1.x2 =
0
- x12 =
0
- y2.w =
0
- y1.w =
0
- x3.w =
0
- y1.v2 =
0
- x2.w =
0
- x1.w =
0
- y2.v1 =
0
- y1.v1 =
0
- x3.v2 =
y1.u1
- x3.v1 =
- y1.u1
- x2.v2 =
y2.u2
- x1.v2 =
0
- w2 =
0
- x2.v1 =
0
- x1.v1 =
0
- y2.u1 =
0
- y1.u2 =
y1.u1
- w.v2 =
- y2.t1
- x3.u2 =
x3.u1
- y1.t1 =
0
- w.v1 =
0
- x2.u2 =
0
- x2.u1 =
0
- x1.u2 =
0
- x1.u1 =
0
- x3.t1 =
- y1.x3.u1
- v1.v2 =
y2.s
- x2.t1 =
y2.s
- x1.t1 =
0
- v12 =
0
- w.u2 =
- y2.s
- w.u1 =
0
- y1.s =
y1.x3.u1
- v2.u1 =
y2.v22
- w.t1 =
y2.v22
- x3.s =
x32.u1
- y1.x3.t2
- v1.u2 =
0
- v1.u1 =
0
- x2.s =
- y1.x2.t2
- x1.s =
y1.x2.t2
- v1.t1 =
- y2.v2.u2
- u22 =
0
- u1.u2 =
y2.v2.u2
- u12 =
0
- w.s =
y2.v2.u2
- u2.t1 =
v2.s
- y2.v2.t2
- u1.t1 =
y2.v2.t1
- v1.s =
0
- t12 =
- v23
- u2.s =
- y2.u2.t2
+ y1.u1.t2
- u1.s =
y2.v2.s
+ y1.u1.t2
- t1.s =
- v22.u2
+ y2.t1.t2
- s2 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There is one minimal generator:
Nilradical:
There are 9 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 14. The cohomology ring approximation
is stable from degree 14 onwards, and
Carlson's tests detect stability from degree 14
onwards.
This cohomology ring has dimension 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
t2
in degree 6
- h2 =
v2
+ x32
in degree 4
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
x1.
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:
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
-
y2
in degree 1
-
y1
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
w
in degree 3
-
y1.x3
in degree 3
-
y1.x2
in degree 3
-
x32
in degree 4
-
v1
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
t1
in degree 6
-
y1.u1
in degree 6
-
s
in degree 7
-
x3.u1
in degree 7
-
y1.x3.u1
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.
-
x1
in degree 2
-
y1.x2
in degree 3
Restriction to maximal subgroup number 1, which is 27gp2
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
0
- w restricts to
- y1.x1
- v1 restricts to
y1.y2.x1
- v2 restricts to
- x12
- y1.y2.x1
- u1 restricts to
- y1.x12
- u2 restricts to
- y2.x12
+ y1.x1.x2
+ y1.x12
- t1 restricts to
- x13
- t2 restricts to
x23
- x12.x2
+ x13
- s restricts to
- y2.x13
+ y1.x23
- y1.x13
Restriction to maximal subgroup number 2, which is 27gp3
- y1 restricts to
y2
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x3
- w restricts to
0
- v1 restricts to
y2.w2
- y1.w2
+ y1.w1
- v2 restricts to
- x42
+ x12
- y2.w2
- u1 restricts to
- x3.w2
+ x1.w1
- y1.x1.x2
- u2 restricts to
- x3.w2
+ x1.w1
+ y1.x42
- y1.x1.x2
- y1.x12
- t1 restricts to
x43
- x12.x2
- x13
+ y2.x3.w2
- y1.x1.w2
- t2 restricts to
- x43
+ x13
+ t
+ y1.x4.w1
+ y1.x1.w2
- s restricts to
- x32.w2
+ x12.w1
- y1.x43
- y1.x12.x2
+ y1.x13
- y2.t
Restriction to maximal subgroup number 3, which is 27gp4
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
0
- x3 restricts to
y1.y2
- w restricts to
0
- v1 restricts to
0
- v2 restricts to
- x2
- y2.w
- u1 restricts to
0
- u2 restricts to
y2.x2
- t1 restricts to
x3
- t2 restricts to
- x3
+ t
+ y2.u
- s restricts to
- y2.x3
- y1.t
Restriction to maximal subgroup number 4, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
0
- x3 restricts to
- y1.y2
- w restricts to
0
- v1 restricts to
0
- v2 restricts to
- x2
- y2.w
- u1 restricts to
0
- u2 restricts to
- y2.x2
- t1 restricts to
- x3
- t2 restricts to
x3
+ t
+ y2.u
- s restricts to
- y2.x3
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
x2
- w restricts to
0
- v1 restricts to
y1.y2.x2
- v2 restricts to
- y1.y2.x2
- u1 restricts to
- y2.x1.x2
+ y1.x22
- u2 restricts to
- y2.x1.x2
+ y1.x22
- t1 restricts to
y1.y2.x22
- t2 restricts to
x1.x22
- x13
- s restricts to
y2.x1.x22
+ y2.x13
+ y1.x23
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
0
- w restricts to
0
- v1 restricts to
0
- v2 restricts to
- x12
- u1 restricts to
0
- u2 restricts to
y1.x12
- t1 restricts to
x13
- t2 restricts to
- x23
+ x12.x2
- x13
- s restricts to
- y1.x13
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
- w restricts to
0
- v1 restricts to
0
- v2 restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- t1 restricts to
0
- t2 restricts to
x3
- s restricts to
0
(1 + 2t + 3t2
+ 3t3 + 2t4 + 2t5
+ t6 + t7 + t8) /
(1 - t4) (1 - t6)
Back to the groups of order 81