Small group number 3 of order 81
G is the group 81gp3
G has 2 minimal generators, rank 3 and exponent 9.
The centre has rank 2.
The 4 maximal subgroups are:
Ab(9,3) (3x), V27.
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 12 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
- x4 in degree 2, a regular element
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- v in degree 4, a nilpotent element
- u in degree 5, a nilpotent element
- t1 in degree 6, a nilpotent element
- t2 in degree 6, a regular element
There are 44 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y1.x3 =
0
- y2.x2 =
- y1.x2
- y2.x1 =
y1.x2
- y1.x1 =
0
- x2.x3 =
y2.w2
- y1.w2
- x1.x3 =
y1.w2
- x22 =
y1.w2
- x1.x2 =
0
- x12 =
0
- y2.w1 =
y1.w2
- y1.w1 =
0
- x3.w1 =
0
- x2.w2 =
- y2.v
- x2.w1 =
y2.v
- x1.w2 =
y2.v
- x1.w1 =
0
- y1.v =
0
- x3.v =
y2.u
+ y2.x3.w2
- y1.x4.w2
- w22 =
0
- w1.w2 =
0
- w12 =
0
- x2.v =
0
- x1.v =
0
- y1.u =
0
- w2.v =
- y2.t1
+ y2.x4.v
- w1.v =
0
- x2.u =
y2.t1
- x1.u =
0
- y1.t1 =
0
- x3.t1 =
w2.u
+ y2.x32.w2
+ y2.x4.u
+ y2.x3.x4.w2
- y1.x42.w2
- v2 =
0
- w1.u =
0
- x2.t1 =
0
- x1.t1 =
0
- v.u =
y2.w2.u
- w2.t1 =
- y2.x4.t1
+ y1.x2.t2
+ y2.x42.v
- w1.t1 =
0
- u2 =
0
- v.t1 =
0
- u.t1 =
y2.x3.w2.u
+ y2.x4.w2.u
- t12 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 3 minimal generators:
Nilradical:
There are 9 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 12 onwards, and
Carlson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x4
in degree 2
- h2 =
t2
in degree 6
- h3 =
x3
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y1.
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
free of rank 3 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.x2
in degree 3
- G2 =
y1.w2
in degree 4
- G3 =
y2.v
in degree 5
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
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.x2
in degree 3
-
v
in degree 4
-
u
in degree 5
-
y2.v
in degree 5
-
t1
in degree 6
-
y2.t1
in degree 7
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y1
in degree 1
-
w1
in degree 3
-
y1.x2
in degree 3
-
y1.w2
in degree 4
-
y2.v
in degree 5
Restriction to maximal subgroup number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y1.y2
- x3 restricts to
x1
- x4 restricts to
x3
- w1 restricts to
0
- w2 restricts to
y2.x1
- y1.x2
- v restricts to
y1.y3.x1
- y1.y2.x1
- u restricts to
y3.x12
+ y2.x12
- y1.x1.x2
- t1 restricts to
y2.y3.x12
+ y1.y3.x1.x3
- y1.y3.x1.x2
- y1.y2.x1.x3
+ y1.y2.x12
- t2 restricts to
x23
- x1.x32
+ x12.x3
- x12.x2
+ y1.y3.x1.x3
+ y1.y2.x12
Restriction to maximal subgroup number 2, which is 27gp2
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
- y1.y2
- x2 restricts to
y1.y2
- x3 restricts to
0
- x4 restricts to
x2
- w1 restricts to
- y1.x1
- w2 restricts to
0
- v restricts to
y1.y2.x1
- u restricts to
- y1.x1.x2
- y1.x12
- t1 restricts to
y1.y2.x1.x2
+ y1.y2.x12
- t2 restricts to
x13
- y1.y2.x22
+ y1.y2.x1.x2
Restriction to maximal subgroup number 3, which is 27gp2
- y1 restricts to
- y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
- x2
- w1 restricts to
y1.x1
- w2 restricts to
- y1.x1
- v restricts to
- y1.y2.x1
- u restricts to
- y1.x1.x2
+ y1.x12
- t1 restricts to
y1.y2.x1.x2
- y1.y2.x12
- t2 restricts to
x13
+ y1.y2.x22
+ y1.y2.x1.x2
Restriction to maximal subgroup number 4, which is 27gp2
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
- y1.y2
- x2 restricts to
- y1.y2
- x3 restricts to
0
- x4 restricts to
x2
- w1 restricts to
- y1.x1
- w2 restricts to
- y1.x1
- v restricts to
y1.y2.x1
- u restricts to
- y1.x1.x2
- y1.x12
- t1 restricts to
y1.y2.x1.x2
+ y1.y2.x12
- t2 restricts to
x13
- y1.y2.x22
+ y1.y2.x1.x2
Restriction to maximal elementary abelian number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y2.y3
- x3 restricts to
x2
- x4 restricts to
x3
+ x1
- w1 restricts to
0
- w2 restricts to
y3.x2
- y2.x3
- v restricts to
- y1.y2.x2
- u restricts to
- y3.x22
- y2.x2.x3
+ y1.x22
- t1 restricts to
- y2.y3.x2.x3
+ y2.y3.x22
- y1.y3.x22
- y1.y2.x1.x2
- t2 restricts to
x33
- x2.x32
+ x1.x2.x3
+ x1.x22
- x12.x2
+ y2.y3.x2.x3
+ y2.y3.x22
+ y2.y3.x1.x2
- y1.y2.x2.x3
- y1.y2.x1.x2
Restriction to the greatest central elementary abelian, which is V9
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
- x1
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
0
- u restricts to
0
- t1 restricts to
0
- t2 restricts to
x23
(1 + 2t + 2t2
+ 2t3 + t4) /
(1 - t2)2 (1 - t6)
Back to the groups of order 81