Small group number 1735 of order 128
G is the group 128gp1735
G has 4 minimal generators, rank 5 and exponent 8.
The centre has rank 2.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 5.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 10 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- x1 in degree 2
- x2 in degree 2, a regular element
- w1 in degree 3
- w2 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
There are 18 minimal relations:
- y2.y4 =
0
- y1.y4 =
0
- y4.x1 =
0
- y32.y4 =
0
- x12 =
y1.y2.x1
+ y12.x2
- y4.w2 =
0
- y4.w1 =
0
- y32.x1 =
y2.w2
+ y1.w1
- x1.w2 =
y1.v1
- x1.w1 =
y2.v1
+ y22.w2
+ y1.y2.w1
+ y1.y32.x2
- y4.v1 =
0
- w22 =
y1.y32.w2
+ y12.v2
- w1.w2 =
y32.v1
+ y2.y32.w2
+ y1.y2.v2
- w12 =
y2.y32.w1
+ y34.x2
+ y22.v2
- x1.v1 =
y1.y2.v1
+ y1.x2.w2
- w2.v1 =
y1.y32.v1
+ y1.x1.v2
- w1.v1 =
y32.x2.w2
+ y2.x1.v2
- v12 =
y1.y2.y32.v1
+ y1.y32.x2.w2
+ y1.y2.x1.v2
+ y12.x2.v2
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
This cohomology ring was obtained from a calculation
out to degree 13. The cohomology ring approximation
is stable from degree 8 onwards, and
Benson's tests detect stability from degree 13
onwards.
This cohomology ring has dimension 5 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x2
in degree 2
- h2 =
v2
in degree 4
- h3 =
y44
+ y34
+ y22.y32
+ y24
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
in degree 4
- h4 =
y22.y34
+ y24.y32
+ y1.y2.y34
+ y1.y24.y3
+ y12.y34
+ y12.y22.y32
+ y12.y24
+ y14.y32
+ y14.y2.y3
+ y14.y22
in degree 6
- h5 =
y1
in degree 1
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
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.
Data for Benson's test:
-
Raw filter degree type:
-1, -1, -1, 5, 11, 12.
-
Filter degree type:
-1, -2, -3, -4, -5, -5.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3, h4, h5) is as follows.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
x
in degree 2
-
y42
in degree 2
-
y3.y4
in degree 2
-
y32
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y43
in degree 3
-
y3.x
in degree 3
-
y3.y42
in degree 3
-
y33
in degree 3
-
y2.x
in degree 3
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
v
in degree 4
-
y3.w2
in degree 4
-
y3.w1
in degree 4
-
y3.y43
in degree 4
-
y34
in degree 4
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y2.y3.x
in degree 4
-
y2.y33
in degree 4
-
y22.x
in degree 4
-
y22.y32
in degree 4
-
y23.y3
in degree 4
-
y24
in degree 4
-
y3.v
in degree 5
-
y32.w2
in degree 5
-
y32.w1
in degree 5
-
y35
in degree 5
-
y2.v
in degree 5
-
y2.y3.w2
in degree 5
-
y2.y3.w1
in degree 5
-
y22.w2
in degree 5
-
y22.w1
in degree 5
-
y22.y3.x
in degree 5
-
y22.y33
in degree 5
-
y23.x
in degree 5
-
y23.y32
in degree 5
-
y24.y3
in degree 5
-
y25
in degree 5
-
y32.v
in degree 6
-
y33.w2
in degree 6
-
y33.w1
in degree 6
-
y2.y3.v
in degree 6
-
y2.y32.w1
in degree 6
-
y22.v
in degree 6
-
y22.y3.w2
in degree 6
-
y22.y3.w1
in degree 6
-
y23.w2
in degree 6
-
y23.w1
in degree 6
-
y23.y3.x
in degree 6
-
y23.y33
in degree 6
-
y24.x
in degree 6
-
y24.y32
in degree 6
-
y25.y3
in degree 6
-
y33.v
in degree 7
-
y2.y32.v
in degree 7
-
y2.y33.w1
in degree 7
-
y22.y3.v
in degree 7
-
y22.y32.w1
in degree 7
-
y23.v
in degree 7
-
y23.y3.w2
in degree 7
-
y23.y3.w1
in degree 7
-
y24.w2
in degree 7
-
y24.w1
in degree 7
-
y24.y3.x
in degree 7
-
y24.y33
in degree 7
-
y25.x
in degree 7
-
y25.y32
in degree 7
-
y2.y33.v
in degree 8
-
y22.y32.v
in degree 8
-
y22.y33.w1
in degree 8
-
y23.y3.v
in degree 8
-
y23.y32.w1
in degree 8
-
y24.v
in degree 8
-
y24.y3.w2
in degree 8
-
y24.y3.w1
in degree 8
-
y25.w2
in degree 8
-
y25.w1
in degree 8
-
y25.y3.x
in degree 8
-
y25.y33
in degree 8
-
y22.y33.v
in degree 9
-
y23.y32.v
in degree 9
-
y23.y33.w1
in degree 9
-
y24.y3.v
in degree 9
-
y24.y32.w1
in degree 9
-
y25.v
in degree 9
-
y25.y3.w2
in degree 9
-
y25.y3.w1
in degree 9
-
y23.y33.v
in degree 10
-
y24.y32.v
in degree 10
-
y24.y33.w1
in degree 10
-
y25.y3.v
in degree 10
-
y25.y32.w1
in degree 10
-
y24.y33.v
in degree 11
-
y25.y32.v
in degree 11
-
y25.y33.w1
in degree 11
-
y25.y33.v
in degree 12
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
-
y4
in degree 1
-
y42
in degree 2
-
y3.y4
in degree 2
-
y43
in degree 3
-
y3.y42
in degree 3
-
y3.y43
in degree 4
-
y34
+ y22.y32
+ y24
+ y2.y32.h
+ y22.y3.h
+ y32.h2
+ y2.y3.h2
+ y22.h2
+ h4
in degree 4
-
y35
+ y22.y33
+ y24.y3
+ y2.y33.h
+ y22.y32.h
+ y33.h2
+ y2.y32.h2
+ y22.y3.h2
+ y3.h4
in degree 5
-
y25.y32.w1
+ y24.y32.w1.h
+ y23.y32.w1.h2
+ y25.w1.h2
+ y22.y32.w1.h3
+ y24.w1.h3
+ y25.x.h3
+ y2.y32.w1.h4
+ y23.w1.h4
+ y24.x.h4
+ y32.w2.h5
+ y32.w1.h5
+ y22.w2.h5
+ y22.w1.h5
+ y2.w2.h6
+ y2.y3.x.h6
+ y22.x.h6
+ w2.h7
+ y3.x.h7
+ x.h8
in degree 10
-
y25.y33.w1
+ y24.y33.w1.h
+ y23.y33.w1.h2
+ y25.y3.w1.h2
+ y22.y33.w1.h3
+ y24.y3.w1.h3
+ y25.y3.x.h3
+ y2.y33.w1.h4
+ y23.y3.w1.h4
+ y24.y3.x.h4
+ y33.w2.h5
+ y33.w1.h5
+ y22.y3.w2.h5
+ y22.y3.w1.h5
+ y2.y3.w2.h6
+ y2.y32.x.h6
+ y22.y3.x.h6
+ y3.w2.h7
+ y32.x.h7
+ y3.x.h8
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y4
in degree 1
-
y42
in degree 2
-
y3.y4
in degree 2
-
y43
in degree 3
-
y3.y42
in degree 3
-
y3.y43
in degree 4
-
y34
+ y22.y32
+ y24
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
in degree 4
-
y35
+ y22.y33
+ y24.y3
+ y1.y2.y33
+ y1.y22.y32
+ y12.y33
+ y12.y2.y32
+ y12.y22.y3
+ y14.y3
in degree 5
Restriction to special subgroup number 1, which is 4gp2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- x1 restricts to
0
- x2 restricts to
y12
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
0
- v2 restricts to
y24
Restriction to special subgroup number 2, which is 8gp5
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y3
- x1 restricts to
0
- x2 restricts to
y1.y3
+ y12
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
0
- v2 restricts to
y22.y32
+ y24
Restriction to special subgroup number 3, which is 32gp51
- y1 restricts to
y3
- y2 restricts to
y4
- y3 restricts to
y5
- y4 restricts to
0
- x1 restricts to
y1.y3
- x2 restricts to
y1.y4
+ y12
- w1 restricts to
y2.y3.y4
+ y22.y4
+ y1.y52
- w2 restricts to
y2.y32
+ y22.y3
- v1 restricts to
y1.y2.y32
+ y1.y22.y3
- v2 restricts to
y2.y3.y52
+ y22.y52
+ y22.y32
+ y24
(1 + 3t + 5t2
+ 8t3 + 11t4 + 13t5
+ 14t6 + 13t7 + 11t8
+ 8t9 + 5t10 + 3t11
+ t12) /
(1 - t) (1 - t2) (1 - t4)2 (1 - t6\
)
Back to the groups of order 128