Small group number 7 of order 81
G = Syl3(A9) is Sylow 3-subgroup of A_9
G has 2 minimal generators, rank 3 and exponent 9.
The centre has rank 1.
The 4 maximal subgroups are:
E27, M27 (2x), V27.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 16 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
- x3 in degree 2
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- w3 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, a nilpotent element
- t2 in degree 6
- t3 in degree 6, a regular element
- s in degree 7, a nilpotent element
There are 88 minimal relations:
- y22 =
0
- y1.y2 =
0
- y12 =
0
- y2.x3 =
0
- y1.x2 =
0
- y2.x1 =
0
- y1.x1 =
0
- x2.x3 =
0
- x1.x3 =
0
- x1.x2 =
- y2.w1
- x12 =
0
- y2.w2 =
0
- y1.w3 =
0
- y1.w1 =
0
- x3.w3 =
0
- x3.w1 =
0
- x2.w2 =
- y2.v1
- y1.v2 =
0
- x1.w3 =
y2.v1
- x1.w2 =
0
- x1.w1 =
0
- y1.v1 =
0
- x3.v2 =
0
- x3.v1 =
0
- x2.v1 =
- w1.w3
- y2.u1
- x1.v2 =
- y2.u1
- w32 =
0
- w2.w3 =
0
- w22 =
0
- w1.w2 =
0
- w12 =
0
- x1.v1 =
0
- y1.u2 =
0
- y1.u1 =
0
- w2.v2 =
- y2.t1
- w1.v2 =
x2.u1
- y2.t2
- x3.u2 =
0
- x3.u1 =
0
- y1.t2 =
0
- w3.v1 =
y2.t1
- w2.v1 =
0
- w1.v1 =
0
- x1.u2 =
y2.t1
- x1.u1 =
0
- y1.t1 =
0
- x3.t2 =
0
- v1.v2 =
- w1.u2
- x3.t1 =
0
- x2.t1 =
- w1.u2
- y2.s
- x1.t2 =
y2.w3.v2
- y2.x2.u2
- y2.x2.u1
- v12 =
0
- w3.u1 =
- w1.u2
- y2.s
- y2.w3.v2
+ y2.x2.u2
+ y2.x2.u1
- w2.u2 =
0
- w2.u1 =
0
- w1.u1 =
y2.w3.v2
- y2.x2.u2
- y2.x2.u1
- x1.t1 =
0
- y1.s =
0
- v2.u1 =
w3.t2
- x2.s
- x2.w3.v2
+ x22.u2
+ x22.u1
+ y2.v22
- y2.x2.t2
- w2.t2 =
- y2.w3.u2
+ y2.w1.u2
- w1.t2 =
- x2.w3.v2
+ x22.u2
+ x22.u1
+ y2.v22
- y2.x2.t2
- x3.s =
0
- v1.u2 =
0
- v1.u1 =
y2.w3.u2
- y2.w1.u2
- w3.t1 =
y2.w3.u2
- y2.w1.u2
- w2.t1 =
0
- w1.t1 =
- y2.w3.u2
+ y2.w1.u2
- x1.s =
y2.w3.u2
- y2.w1.u2
- v2.t1 =
- w3.s
+ x2.w3.u2
- x2.w1.u2
- y2.v2.u2
- v1.t2 =
x2.w3.u2
- x2.w1.u2
- y2.v2.u2
- u22 =
0
- u1.u2 =
w3.s
- x2.w3.u2
+ x2.w1.u2
+ y2.v2.u2
- y2.w1.t3
- u12 =
0
- v1.t1 =
0
- w2.s =
0
- w1.s =
- x2.w3.u2
+ x2.w1.u2
+ y2.v2.u2
- u2.t2 =
v2.s
- x2.w1.t3
- u1.t2 =
- w3.v22
+ x2.v2.u2
+ x2.w3.t2
- x22.s
- x22.w3.v2
+ x23.u2
+ x23.u1
+ y2.v2.t2
+ y2.x2.v22
- y2.x22.t2
- y2.x22.t3
- u2.t1 =
y2.v1.t3
- u1.t1 =
0
- v1.s =
0
- t22 =
- v23
- x2.v2.t2
- x23.t3
- t1.t2 =
w3.v2.u2
- x2.w3.s
+ x22.w3.u2
- x22.w1.u2
- y2.v2.s
- y2.x2.v2.u2
+ y2.x2.w3.t3
+ y2.x2.w1.t3
- t12 =
0
- u2.s =
- w1.w3.t3
- y2.u1.t3
- u1.s =
- w3.v2.u2
+ x2.w3.s
- x22.w3.u2
+ x22.w1.u2
+ y2.v2.s
+ y2.x2.v2.u2
- y2.x2.w3.t3
- y2.x2.w1.t3
- t2.s =
- v22.u2
- x2.v2.s
- x22.w3.t3
- x22.w1.t3
+ y2.x2.v2.t3
- t1.s =
0
- s2 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
- w1.w3.u2 =
- y2.w3.s
+ y2.x2.w3.u2
- y2.x2.w1.u2
Essential ideal:
Zero ideal
Nilradical:
There are 11 minimal generators:
-
y2
-
y1
-
x1
-
w3
-
w2
-
w1
-
v1
-
u2
-
u1
-
t1
-
s
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 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
t3
in degree 6
- h2 =
x3
+ x2
in degree 2
- h3 =
v2
in degree 4
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
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
the zero ideal.
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 12.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
v1
in degree 4
-
y2.w3
in degree 4
-
y1.w2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y2.v1
in degree 5
-
t2
in degree 6
-
t1
in degree 6
-
y2.u2
in degree 6
-
s
in degree 7
-
w3.u2
in degree 8
-
y2.w3.u2
in degree 9
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
-
x2
in degree 2
-
y1.w2
in degree 4
-
y2.v1
in degree 5
Restriction to maximal subgroup number 1, which is V27
- y1 restricts to
0
- y2 restricts to
y1
- x1 restricts to
- y1.y2
- x2 restricts to
x1
- x3 restricts to
0
- w1 restricts to
y2.x1
- y1.x2
- w2 restricts to
y1.y2.y3
- w3 restricts to
y3.x1
- y2.x2
+ y2.x1
+ y1.x3
+ y1.x2
- v1 restricts to
- y2.y3.x1
+ y1.y3.x2
- y1.y2.x3
- v2 restricts to
- x22
- x1.x3
- x1.x2
- u1 restricts to
- y2.x22
- y2.x1.x3
- y2.x1.x2
+ y1.x2.x3
- y1.x1.x3
- u2 restricts to
- y3.x22
- y3.x1.x3
- y3.x1.x2
+ y2.x2.x3
- y2.x1.x3
+ y1.x32
- y1.x2.x3
- t1 restricts to
y2.y3.x22
+ y2.y3.x1.x3
+ y2.y3.x1.x2
- y1.y3.x2.x3
+ y1.y3.x1.x3
- y1.y2.x32
- y1.y2.x2.x3
- y1.y2.x1.x3
- t2 restricts to
- x23
- x1.x22
- x12.x3
- t3 restricts to
x33
- x22.x3
+ x1.x32
- x1.x2.x3
- s restricts to
- y3.x23
- y3.x1.x22
- y3.x12.x3
+ y2.x22.x3
- y2.x1.x32
- y2.x1.x2.x3
+ y2.x12.x3
+ y1.x2.x32
- y1.x22.x3
+ y1.x1.x32
- y1.x1.x2.x3
Restriction to maximal subgroup number 2, which is 27gp3
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
x4
- w1 restricts to
0
- w2 restricts to
w1
+ y1.x2
- w3 restricts to
- y2.x3
+ y1.x2
+ y1.x1
- v1 restricts to
0
- v2 restricts to
- x32
+ x12
- y1.w2
+ y1.w1
- u1 restricts to
y2.x32
- y1.x1.x2
- y1.x12
- u2 restricts to
x3.w2
- x1.w2
- t1 restricts to
y2.x3.w2
- y1.x1.w2
- t2 restricts to
x33
- x13
- t3 restricts to
x12.x2
- t
- y1.x4.w1
- y1.x1.w2
+ y1.x1.w1
- s restricts to
- x32.w2
+ x12.w2
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
y1.y2
- x3 restricts to
- y1.y2
- w1 restricts to
0
- w2 restricts to
- w
- w3 restricts to
w
- y2.x
- v1 restricts to
- y2.w
- v2 restricts to
- x2
- y2.w
- u1 restricts to
- y2.x2
- u2 restricts to
u
- t1 restricts to
- y2.u
- t2 restricts to
- x3
- t3 restricts to
t
+ y2.u
- s restricts to
x.u
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
- y1.y2
- x3 restricts to
- y1.y2
- w1 restricts to
0
- w2 restricts to
w
- w3 restricts to
w
- y2.x
- v1 restricts to
y2.w
- v2 restricts to
- x2
- y2.w
- u1 restricts to
y2.x2
- u2 restricts to
- u
- t1 restricts to
- y2.u
- t2 restricts to
x3
- t3 restricts to
- t
- y2.u
- s restricts to
x.u
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
- w1 restricts to
0
- w2 restricts to
- y2.x1
+ y1.x2
- w3 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
0
- t3 restricts to
x1.x22
- x13
- s restricts to
0
Restriction to maximal elementary abelian number 2, which is V27
- y1 restricts to
0
- y2 restricts to
y3
- x1 restricts to
- y2.y3
- y1.y3
- x2 restricts to
x3
- x3 restricts to
0
- w1 restricts to
y3.x2
+ y3.x1
- y2.x3
- y1.x3
- w2 restricts to
y1.y2.y3
- w3 restricts to
y3.x1
- y2.x2
- y2.x1
+ y1.x3
- y1.x2
- y1.x1
- v1 restricts to
- y2.y3.x1
+ y1.y3.x2
- y1.y2.x3
- v2 restricts to
- x22
- x1.x3
+ x1.x2
- x12
- u1 restricts to
- y3.x2.x3
- y3.x22
+ y3.x1.x3
+ y3.x12
+ y2.x22
+ y2.x1.x3
- y2.x1.x2
+ y2.x12
+ y1.x22
+ y1.x1.x3
- y1.x1.x2
+ y1.x12
- u2 restricts to
- y3.x22
+ y3.x1.x2
+ y2.x2.x3
+ y2.x1.x3
+ y2.x1.x2
+ y2.x12
+ y1.x2.x3
- y1.x22
- y1.x1.x2
- t1 restricts to
y2.y3.x2.x3
- y2.y3.x22
- y2.y3.x1.x3
- y2.y3.x1.x2
- y2.y3.x12
+ y1.y3.x22
- y1.y3.x1.x2
+ y1.y2.x22
+ y1.y2.x1.x3
- y1.y2.x1.x2
+ y1.y2.x12
- t2 restricts to
- x2.x32
- x22.x3
+ x23
+ x1.x32
+ x1.x2.x3
- x12.x3
+ x13
- t3 restricts to
- x22.x3
+ x1.x2.x3
- x1.x22
+ x12.x2
- s restricts to
- y3.x22.x3
+ y3.x23
+ y3.x1.x2.x3
- y3.x12.x2
+ y2.x2.x32
- y2.x22.x3
- y2.x1.x32
- y2.x1.x2.x3
- y2.x1.x22
+ y2.x12.x3
+ y2.x12.x2
- y2.x13
+ y1.x22.x3
+ y1.x23
- y1.x1.x22
+ y1.x12.x2
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
- w1 restricts to
0
- w2 restricts to
0
- w3 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
0
- t3 restricts to
x3
- s restricts to
0
(1 + 2t + 2t2
+ 3t3 + 3t4 + 2t5
+ 2t6 + t7) /
(1 - t2) (1 - t4) (1 - t6)
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