Small group number 753 of order 128
G is the group 128gp753
G has 3 minimal generators, rank 5 and exponent 4.
The centre has rank 2.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 4, 5.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 13 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- x5 in degree 2, a regular element
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- v1 in degree 4
- v2 in degree 4, a regular element
There are 34 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y1.y2 =
0
- y3.x4 =
0
- y3.x3 =
y2.x1
- y3.x1 =
y2.x1
- y2.x4 =
0
- y2.x2 =
y2.x1
- x42 =
y12.x5
- x3.x4 =
x1.x4
+ y1.w2
- x2.x4 =
x1.x4
+ y1.w3
- x2.x3 =
x12
+ y1.w1
- y3.w2 =
0
- y3.w1 =
0
- y2.w3 =
0
- y2.w1 =
0
- x4.w3 =
y1.x2.x5
+ y1.x1.x5
- x4.w2 =
y1.x3.x5
+ y1.x1.x5
- x4.w1 =
y1.v1
- x3.w3 =
x1.w2
+ y1.v1
- x2.w2 =
x1.w3
+ y1.v1
- y3.v1 =
0
- y2.v1 =
0
- w32 =
y3.x2.w3
+ x22.x5
+ x12.x5
+ y32.v2
- w2.w3 =
x1.x3.x5
+ x1.x2.x5
+ y1.x5.w1
- w22 =
y2.x3.w2
+ x32.x5
+ x12.x5
+ y22.v2
- w1.w3 =
x2.v1
+ x1.v1
- w1.w2 =
x3.v1
+ x1.v1
- w12 =
x12.x3
+ x12.x2
+ y1.x1.w1
+ y12.x1.x2
+ y12.x12
+ y12.v2
- x4.v1 =
y1.x5.w1
- w3.v1 =
x2.x5.w1
+ x1.x5.w1
- w2.v1 =
x3.x5.w1
+ x1.x5.w1
- w1.v1 =
x12.w3
+ x12.w2
+ y1.x1.v1
+ y12.x1.w3
+ y1.x4.v2
- v12 =
x12.x3.x5
+ x12.x2.x5
+ y1.x1.x5.w1
+ y12.x1.x2.x5
+ y12.x12.x5
+ y12.x5.v2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y22.x1 =
0
- y2.x1.x3 =
y2.x12
- y2.x1.w2 =
0
This cohomology ring was obtained from a calculation
out to degree 18. 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 =
x5
in degree 2
- h2 =
v2
in degree 4
- h3 =
x32
+ x22
+ x12
+ y34
+ y24
+ y1.w1
+ y14
in degree 4
- h4 =
x12.x3
+ x12.x2
+ y32.x22
+ y22.x32
+ y1.x3.w1
+ y1.x2.w1
+ y12.x32
+ y12.x22
+ y12.x12
+ y13.w1
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
-
y3
in degree 1
-
y2
in degree 1
-
x4
in degree 2
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y32
in degree 2
-
y22
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
y3.x2
in degree 3
-
y33
in degree 3
-
y2.x3
in degree 3
-
y2.x1
in degree 3
-
y23
in degree 3
-
v
in degree 4
-
x22
in degree 4
-
x1.x4
in degree 4
-
x1.x3
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
y3.w3
in degree 4
-
y32.x2
in degree 4
-
y34
in degree 4
-
y2.w2
in degree 4
-
y22.x3
in degree 4
-
y24
in degree 4
-
x3.w2
in degree 5
-
x3.w1
in degree 5
-
x2.w3
in degree 5
-
x2.w1
in degree 5
-
x1.w3
in degree 5
-
x1.w2
in degree 5
-
x1.w1
in degree 5
-
y32.w3
in degree 5
-
y33.x2
in degree 5
-
y35
in degree 5
-
y2.x12
in degree 5
-
y22.w2
in degree 5
-
y23.x3
in degree 5
-
y25
in degree 5
-
x3.v
in degree 6
-
x2.v
in degree 6
-
x1.v
in degree 6
-
x1.x22
in degree 6
-
x12.x2
in degree 6
-
x13
in degree 6
-
y3.x2.w3
in degree 6
-
y33.w3
in degree 6
-
y34.x2
in degree 6
-
y36
in degree 6
-
y2.x3.w2
in degree 6
-
y23.w2
in degree 6
-
y24.x3
in degree 6
-
y26
in degree 6
-
x22.w1
in degree 7
-
x1.x3.w2
in degree 7
-
x1.x3.w1
in degree 7
-
x1.x2.w3
in degree 7
-
x1.x2.w1
in degree 7
-
x12.w3
in degree 7
-
x12.w2
in degree 7
-
x12.w1
in degree 7
-
y32.x2.w3
in degree 7
-
y34.w3
in degree 7
-
y35.x2
in degree 7
-
y22.x3.w2
in degree 7
-
y24.w2
in degree 7
-
y25.x3
in degree 7
-
x22.v
in degree 8
-
x1.x3.v
in degree 8
-
x1.x2.v
in degree 8
-
x12.v
in degree 8
-
x13.x2
in degree 8
-
y33.x2.w3
in degree 8
-
y35.w3
in degree 8
-
y36.x2
in degree 8
-
y23.x3.w2
in degree 8
-
y25.w2
in degree 8
-
y26.x3
in degree 8
-
x1.x22.w1
in degree 9
-
x12.x2.w1
in degree 9
-
x13.w3
in degree 9
-
x13.w2
in degree 9
-
x13.w1
in degree 9
-
y34.x2.w3
in degree 9
-
y36.w3
in degree 9
-
y24.x3.w2
in degree 9
-
y26.w2
in degree 9
-
x1.x22.v
in degree 10
-
x12.x2.v
in degree 10
-
x13.v
in degree 10
-
y35.x2.w3
in degree 10
-
y25.x3.w2
in degree 10
-
x13.x2.w1
in degree 11
-
y36.x2.w3
in degree 11
-
y26.x3.w2
in degree 11
-
x13.x2.v
in degree 12
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
-
y3
in degree 1
-
y2
in degree 1
-
y32
in degree 2
-
y22
in degree 2
-
y3.x2
in degree 3
-
y33
in degree 3
-
y2.x3
in degree 3
-
y2.x1
in degree 3
-
y23
in degree 3
-
y3.w3
in degree 4
-
y32.x2
in degree 4
-
y34
in degree 4
-
y2.w2
in degree 4
-
y22.x3
in degree 4
-
y24
in degree 4
-
y32.w3
in degree 5
-
y33.x2
in degree 5
-
y35
in degree 5
-
y2.x12
in degree 5
-
y22.w2
in degree 5
-
y23.x3
in degree 5
-
y25
in degree 5
-
y3.x2.w3
in degree 6
-
y33.w3
in degree 6
-
y34.x2
in degree 6
-
y36
in degree 6
-
y2.x3.w2
in degree 6
-
y23.w2
in degree 6
-
y24.x3
in degree 6
-
y26
in degree 6
-
x12.w3
+ x12.w2
+ x3.v.h
+ x2.v.h
+ x4.h5
in degree 7
-
y32.x2.w3
in degree 7
-
y34.w3
in degree 7
-
y35.x2
in degree 7
-
y22.x3.w2
in degree 7
-
y24.w2
in degree 7
-
y25.x3
in degree 7
-
y33.x2.w3
in degree 8
-
y35.w3
in degree 8
-
y36.x2
in degree 8
-
y23.x3.w2
in degree 8
-
y25.w2
in degree 8
-
y26.x3
in degree 8
-
x13.w3
+ x13.w2
+ x1.x3.v.h
+ x1.x2.v.h
+ x1.x4.h5
in degree 9
-
y34.x2.w3
in degree 9
-
y36.w3
in degree 9
-
y24.x3.w2
in degree 9
-
y26.w2
in degree 9
-
y35.x2.w3
in degree 10
-
y25.x3.w2
in degree 10
-
y36.x2.w3
in degree 11
-
y26.x3.w2
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y2.x1
in degree 3
-
y2.x12
in degree 5
Restriction to special subgroup number 1, which is 4gp2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
y12
- w1 restricts to
0
- w2 restricts to
0
- w3 restricts to
0
- v1 restricts to
0
- v2 restricts to
y24
Restriction to special subgroup number 2, which is 16gp14
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
y42
+ y3.y4
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
y1.y3
+ y12
- w1 restricts to
0
- w2 restricts to
0
- w3 restricts to
y2.y32
+ y22.y3
+ y1.y42
+ y1.y3.y4
- v1 restricts to
0
- v2 restricts to
y2.y3.y42
+ y2.y32.y4
+ y22.y42
+ y22.y3.y4
+ y22.y32
+ y24
Restriction to special subgroup number 3, which is 16gp14
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
y42
+ y3.y4
- x4 restricts to
0
- x5 restricts to
y1.y3
+ y12
- w1 restricts to
0
- w2 restricts to
y2.y32
+ y22.y3
+ y1.y42
+ y1.y3.y4
- w3 restricts to
0
- v1 restricts to
0
- v2 restricts to
y2.y3.y42
+ y2.y32.y4
+ y22.y42
+ y22.y3.y4
+ y22.y32
+ y24
Restriction to special subgroup number 4, which is 32gp51
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
y4.y5
+ y2.y3
- x2 restricts to
y42
+ y3.y4
- x3 restricts to
y52
+ y3.y5
- x4 restricts to
y1.y3
- x5 restricts to
y12
- w1 restricts to
y4.y52
+ y42.y5
+ y3.y4.y5
+ y22.y3
- w2 restricts to
y1.y52
+ y1.y4.y5
+ y1.y3.y5
+ y1.y2.y3
- w3 restricts to
y1.y4.y5
+ y1.y42
+ y1.y3.y4
+ y1.y2.y3
- v1 restricts to
y1.y4.y52
+ y1.y42.y5
+ y1.y3.y4.y5
+ y1.y22.y3
- v2 restricts to
y42.y52
+ y43.y5
+ y3.y42.y5
+ y2.y4.y52
+ y2.y42.y5
+ y2.y3.y4.y5
+ y2.y3.y42
+ y2.y32.y4
+ y22.y52
+ y22.y4.y5
+ y22.y42
+ y22.y3.y5
+ y22.y3.y4
+ y22.y32
+ y23.y3
+ y24
(1 + 2t + 4t2
+ 6t3 + 7t4 + 8t5
+ 7t6 + 6t7 + 4t8
+ 2t9 + t10) /
(1 - t) (1 - t2) (1 - t4)2 (1 - t6\
)
Back to the groups of order 128