Small group number 4 of order 27
G = M27 is Extraspecial 3-group of order 27 and exponent 9
G has 2 minimal generators, rank 2 and exponent 9.
The centre has rank 1.
The 4 maximal subgroups are:
C9 (3x), V9.
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 6 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x in degree 2
- w in degree 3, a nilpotent element
- u in degree 5, a nilpotent element
- t in degree 6, a regular element
There are 9 minimal relations:
- y22 =
0
- y12 =
0
- y1.x =
0
- y1.w =
0
- x.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 2 minimal generators:
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 12. 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 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
t
in degree 6
- h2 =
x
in degree 2
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y1.
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 2 as a module over the polynomial algebra
on h1.
These free generators are:
- G1 =
y1.y2
in degree 2
- G2 =
y2.w
in degree 4
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 8.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y1.y2
in degree 2
-
w
in degree 3
-
y2.w
in degree 4
-
u
in degree 5
-
y2.u
in degree 6
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
in degree 1
-
y1.y2
in degree 2
-
w
in degree 3
-
y2.w
in degree 4
Restriction to maximal subgroup number 1, which is V9
- y1 restricts to
0
- y2 restricts to
y1
- x restricts to
x1
- w restricts to
0
- u restricts to
y2.x12
- y1.x1.x2
- t restricts to
- x23
+ x12.x2
- y1.y2.x12
Restriction to maximal subgroup number 2, which is C9
- y1 restricts to
y
- y2 restricts to
0
- x restricts to
0
- w restricts to
y.x
- u restricts to
- y.x2
- t restricts to
- x3
Restriction to maximal subgroup number 3, which is C9
- y1 restricts to
- y
- y2 restricts to
y
- x restricts to
0
- w restricts to
y.x
- u restricts to
y.x2
- t restricts to
x3
Restriction to maximal subgroup number 4, which is C9
- y1 restricts to
y
- y2 restricts to
y
- x restricts to
0
- w restricts to
y.x
- u restricts to
- y.x2
- t restricts to
- x3
Restriction to maximal elementary abelian number 1, which is V9
- y1 restricts to
0
- y2 restricts to
y2
- x restricts to
x2
- w restricts to
0
- u restricts to
- y2.x1.x2
+ y1.x22
- t restricts to
x1.x22
- x13
+ y1.y2.x22
Restriction to the greatest central elementary abelian, which is C3
- y1 restricts to
0
- y2 restricts to
0
- x restricts to
0
- w restricts to
0
- u restricts to
0
- t restricts to
- x3
(1 + 2t + t2) /
(1 - t2) (1 - t6)
Back to the groups of order 27