Small group number 937 of order 128
G is the group 128gp937
G has 3 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
64gp137 (3x), Q8xQ8, 64gp37 (3x).
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 9 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1
- x1 in degree 2, a nilpotent element
- x2 in degree 2, a nilpotent element
- v in degree 4
- u1 in degree 5, a nilpotent element
- u2 in degree 5, a nilpotent element
- r in degree 8, a regular element
There are 21 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y2.x2 =
0
- y2.x1 =
y1.x2
+ y1.y22
+ y12.y2
+ y13
- y1.x1 =
y1.y22
+ y12.y2
+ y13
- x22 =
x1.x2
+ x12
+ y12.y22
+ y13.y2
- y24 =
0
- y14 =
0
- y3.v =
0
- y3.u2 =
y34.x2
+ y34.x1
+ y32.x12
- y3.u1 =
y34.x2
+ y32.x1.x2
- y2.u1 =
y1.u2
+ y1.y2.v
+ y12.v
- x13 =
0
- x2.u2 =
y33.x12
+ y3.x12.x2
+ y23.v
- x2.u1 =
x1.u2
+ y3.x12.x2
+ y23.v
+ y1.y2.u2
+ y1.y22.v
+ y12.u2
+ y12.u1
+ y13.v
- x1.u1 =
y33.x1.x2
+ y1.x2.v
+ y3.x12.x2
+ y23.v
+ y1.y2.u2
+ y1.y22.v
+ y12.u2
+ y12.u1
+ y13.v
- x1.x2.v =
y23.u2
+ y13.u1
- x12.v =
y12.y22.v
+ y13.u1
+ y13.y2.v
- x2.v2 =
u1.u2
+ u12
+ y36.x1.x2
+ y1.y2.v2
+ y12.v2
+ y1.y2.r
+ y12.r
+ y34.x12.x2
+ y13.y22.u2
- x1.v2 =
u12
+ y36.x1.x2
+ y36.x12
+ y1.v.u1
+ y12.v2
+ y12.r
- u22 =
u1.u2
+ u12
+ y2.v.u2
+ y1.y2.v2
+ y12.v2
+ y22.r
+ y1.y2.r
+ y12.r
+ y34.x12.x2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y12.x2 =
y1.y23
+ y14
- y13.y23 =
0
- x12.u2 =
y33.x12.x2
+ y12.y22.u2
+ y12.y23.v
+ y13.y2.u2
+ y13.y22.v
- y1.y23.u2 =
y12.y23.v
- y1.u12 =
y1.y22.v2
+ y12.v.u1
+ y12.y2.v2
+ y13.r
- y1.y23.v2 =
y12.u1.u2
+ y12.y22.v2
+ y13.v.u1
+ y13.y2.r
- u13 =
y23.v3
+ y1.v.u1.u2
+ y1.y2.v2.u2
+ y1.y22.v3
+ y12.v2.u2
+ y12.y2.v3
+ y12.u1.r
+ y12.y2.v.r
+ y13.v.r
Essential ideal:
Zero ideal
Nilradical:
There are 6 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 12
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
v
+ y34
in degree 4
The first
2 terms h1, h2 form
a regular sequence of maximum length.
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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 12.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y33
in degree 3
-
y3.x2
in degree 3
-
y3.x1
in degree 3
-
y1.x2
in degree 3
-
y23
in degree 3
-
y1.y22
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y34
in degree 4
-
y32.x2
in degree 4
-
y32.x1
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
y1.y23
in degree 4
-
y12.y22
in degree 4
-
y13.y2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y33.x2
in degree 5
-
y33.x1
in degree 5
-
y3.x1.x2
in degree 5
-
y3.x12
in degree 5
-
y12.y23
in degree 5
-
y13.y22
in degree 5
-
y34.x2
in degree 6
-
y34.x1
in degree 6
-
y32.x1.x2
in degree 6
-
y32.x12
in degree 6
-
y2.u2
in degree 6
-
y1.u2
in degree 6
-
y1.u1
in degree 6
-
x12.x2
in degree 6
-
x1.u2
in degree 7
-
y33.x1.x2
in degree 7
-
y33.x12
in degree 7
-
y3.x12.x2
in degree 7
-
y22.u2
in degree 7
-
y1.y2.u2
in degree 7
-
y12.u2
in degree 7
-
y12.u1
in degree 7
-
y32.x12.x2
in degree 8
-
y23.u2
in degree 8
-
y1.y22.u2
in degree 8
-
y12.y2.u2
in degree 8
-
y13.u2
in degree 8
-
y13.u1
in degree 8
-
y33.x12.x2
in degree 9
-
y12.y22.u2
in degree 9
-
y13.y2.u2
in degree 9
-
y13.y22.u2
in degree 10
Restriction to maximal subgroup number 1, which is 64gp239
- y1 restricts to
y4
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
0
- x1 restricts to
y3.y4
+ y2.y4
+ y22
+ y1.y3
+ y12
- x2 restricts to
y2.y3
+ y22
+ y1.y3
+ y1.y2
- v restricts to
v2
+ v1
+ y12.y22
- u1 restricts to
y4.v2
+ y3.v2
+ y3.v1
+ y2.v2
+ y2.v1
+ y12.y2.y3.y4
- u2 restricts to
y3.v1
+ y2.v2
+ y1.v1
- r restricts to
v22
+ v1.v2
+ v12
+ y1.y2.y3.y4.v2
+ y1.y22.y4.v1
+ y1.y22.y3.v2
+ y12.y3.y4.v2
+ y12.y2.y3.v2
+ y12.y22.v1
Restriction to maximal subgroup number 2, which is 64gp137
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y3
- x1 restricts to
y2.y3
+ y22
- x2 restricts to
y2.y3
+ y22
+ x1
+ y12
- v restricts to
x22
+ y22.x2
+ y23.y3
+ y24
+ y12.x2
- u1 restricts to
y2.x22
+ y2.y34
+ y24.y3
+ y33.x1
+ y2.y32.x1
+ y23.x1
+ y2.x12
+ y13.x2
- u2 restricts to
y2.x22
+ y24.y3
+ y25
+ u
+ y33.x1
+ y2.x1.x2
+ y22.y3.x1
+ y3.x12
+ y2.x12
+ y13.x2
- r restricts to
x24
+ r
+ y2.y35.x1
+ y23.u
+ y26.x1
+ y24.x12
+ y12.x23
Restriction to maximal subgroup number 3, which is 64gp37
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
+ x1
- v restricts to
v
+ y2.w2
+ y2.w1
+ y1.w2
- u1 restricts to
u1
+ y22.w1
- u2 restricts to
u2
+ u1
+ x2.w2
+ y22.w1
- r restricts to
v2
+ r
+ w2.u2
+ w2.u1
+ y22.t
Restriction to maximal subgroup number 4, which is 64gp137
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
y2.y3
+ y22
+ x1
- x2 restricts to
y2.y3
+ y22
+ y12
- v restricts to
x22
+ y22.x2
+ y23.y3
+ y24
+ y12.x2
- u1 restricts to
y2.x22
+ y2.y34
+ y25
+ u
+ y2.x1.x2
+ y22.y3.x1
+ y2.x12
+ y13.x2
- u2 restricts to
y2.x22
+ y33.x1
+ y2.y32.x1
+ y23.x1
+ y1.x22
+ y2.x12
- r restricts to
x24
+ r
+ y2.y35.x1
+ y23.u
+ y26.x1
+ y34.x12
+ y24.x12
+ y12.x23
+ y13.u
Restriction to maximal subgroup number 5, which is 64gp37
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y1
- x1 restricts to
x2
+ x1
+ y22
- x2 restricts to
x1
+ y22
- v restricts to
v
+ y2.w2
+ y2.w1
+ y1.w2
- u1 restricts to
u2
+ u1
+ x2.w2
+ y22.w1
- u2 restricts to
u1
+ y2.v
- r restricts to
v2
+ r
+ w2.u2
+ w2.u1
+ y22.t
+ y23.u2
Restriction to maximal subgroup number 6, which is 64gp137
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y3
- x1 restricts to
x1
- x2 restricts to
y2.y3
+ y22
+ x1
+ y12
- v restricts to
x22
+ y22.x2
+ y23.y3
+ y24
+ y12.x2
- u1 restricts to
y2.y34
+ y25
+ u
+ y33.x1
+ y2.x1.x2
+ y2.y32.x1
+ y22.y3.x1
+ y23.x1
- u2 restricts to
y2.x22
+ y2.y34
+ y25
+ u
+ y2.x1.x2
+ y22.y3.x1
+ y2.x12
+ y13.x2
- r restricts to
x24
+ r
+ y2.y35.x1
+ y23.u
+ y26.x1
+ y24.x12
+ y12.x23
Restriction to maximal subgroup number 7, which is 64gp37
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
x2
+ x1
+ y22
- x2 restricts to
x2
- v restricts to
v
+ y2.w2
+ y2.w1
+ y1.w2
- u1 restricts to
u2
+ u1
+ y2.v
+ x2.w2
- u2 restricts to
u2
+ y2.v
+ x2.w2
+ y22.w1
- r restricts to
v2
+ r
+ w2.u2
+ w2.u1
+ y22.t
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- v restricts to
y24
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y28
+ y14.y24
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y8
(1 + 3t + 6t2
+ 8t3 + 8t4 + 8t5
+ 8t6 + 8t7 + 6t8
+ 3t9 + t10) /
(1 - t4) (1 - t8)
Back to the groups of order 128