Small group number 13 of order 81
G = M27xC3 is Direct product M27 x C_3
G has 3 minimal generators, rank 3 and exponent 9.
The centre has rank 2.
The 13 maximal subgroups are:
Ab(9,3) (3x), M27 (9x), 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 8 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, a regular element
- w in degree 3, a nilpotent element
- u in degree 5, a nilpotent element
- t in degree 6, a regular element
There are 10 minimal relations:
- y32 =
0
- y22 =
0
- y12 =
0
- y1.x1 =
0
- y1.w =
0
- x1.w =
0
- w2 =
0
- y1.u =
0
- w.u =
0
- u2 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 4 minimal generators:
-
y1.y2.y3
-
y1.y2.x2
-
y2.y3.w
-
y2.x2.w
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 10 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x2
in degree 2
- h2 =
t
in degree 6
- h3 =
x1
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 4 as a module over the polynomial algebra
on h1, h2.
These free generators are:
- G1 =
y1.y2.y3
in degree 3
- G2 =
y1.y2.x2
in degree 4
- G3 =
y2.y3.w
in degree 5
- G4 =
y2.x2.w
in degree 6
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
-
y2.y3
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y1.y2.y3
in degree 3
-
y3.w
in degree 4
-
y2.w
in degree 4
-
u
in degree 5
-
y2.y3.w
in degree 5
-
y3.u
in degree 6
-
y2.u
in degree 6
-
y2.y3.u
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
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
w
in degree 3
-
y1.y2.y3
in degree 3
-
y3.w
in degree 4
-
y2.w
in degree 4
-
y2.y3.w
in degree 5
Restriction to maximal subgroup number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
- x2 restricts to
x1
- w restricts to
0
- u restricts to
y3.x22
- y2.x2.x3
- t restricts to
- x33
+ x22.x3
- y2.y3.x22
Restriction to maximal subgroup number 2, which is 27gp2
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
- y1.x22
- t restricts to
- x23
Restriction to maximal subgroup number 3, which is 27gp2
- y1 restricts to
- y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
y1.x22
- t restricts to
x23
Restriction to maximal subgroup number 4, which is 27gp2
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
x1
- w restricts to
y1.x2
- u restricts to
- y1.x22
- t restricts to
- x23
Restriction to maximal subgroup number 5, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
x
- x2 restricts to
0
- w restricts to
w
- u restricts to
u
- t restricts to
t
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
- w restricts to
w
- u restricts to
- u
- t restricts to
- t
Restriction to maximal subgroup number 7, which is 27gp4
- y1 restricts to
- y1
- y2 restricts to
y2
- y1
- y3 restricts to
- y1
- x1 restricts to
x
+ y1.y2
- x2 restricts to
y1.y2
- w restricts to
w
- u restricts to
- u
- t restricts to
- t
Restriction to maximal subgroup number 8, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- y2
- y1
- x1 restricts to
x
- y1.y2
- x2 restricts to
- x
+ y1.y2
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal subgroup number 9, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y1
- y3 restricts to
- y2
- x1 restricts to
x
+ y1.y2
- x2 restricts to
- x
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal subgroup number 10, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
- y2
- x1 restricts to
x
- y1.y2
- x2 restricts to
- x
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal subgroup number 11, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y1
- y3 restricts to
y2
- y1
- x1 restricts to
x
+ y1.y2
- x2 restricts to
x
+ y1.y2
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal subgroup number 12, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- y1
- x1 restricts to
x
- x2 restricts to
x
+ y1.y2
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal subgroup number 13, which is 27gp4
- y1 restricts to
y1
- y2 restricts to
y2
- y1
- y3 restricts to
y2
- x1 restricts to
x
+ y1.y2
- x2 restricts to
x
- w restricts to
w
- u restricts to
u
- t restricts to
t
Restriction to maximal elementary abelian number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
x3
- x2 restricts to
x2
- w restricts to
0
- u restricts to
- y3.x1.x3
+ y1.x32
- t restricts to
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
- x1
- w restricts to
0
- u restricts to
0
- t restricts to
- x23
(1 + 3t + 3t2
+ t3) /
(1 - t2)2 (1 - t6)
Back to the groups of order 81