Small group number 170 of order 128
G is the group 128gp170
G has 3 minimal generators, rank 5 and exponent 4.
The centre has rank 4.
The 7 maximal subgroups are:
64gp193 (4x), 64gp261, 64gp56 (2x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
5, 5.
This cohomology ring calculation is only complete
out to degree 6.
Ring structure
| Restriction information
The cohomology ring has 12 generators:
- y1 in degree 1, a nilpotent element
- 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
- x6 in degree 2, a regular element
- x7 in degree 2, a regular element
- x8 in degree 2, a regular element
- w in degree 3
There are 27 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y1.y2 =
0
- y12 =
0
- y3.x3 =
y1.x2
- y3.x2 =
0
- y2.x4 =
y1.x2
- y2.x2 =
y2.x1
- y1.x4 =
0
- y1.x3 =
0
- y1.x1 =
0
- x42 =
y32.x5
- x3.x4 =
0
- x32 =
y22.x5
- x2.x4 =
0
- x2.x3 =
y2.w
+ y22.x6
- x22 =
y22.x6
- x1.x4 =
y3.w
- x1.x3 =
y2.w
+ y22.x6
- x1.x2 =
y22.x6
- x12 =
y32.x7
+ y22.x6
- y1.w =
0
- x4.w =
y3.x1.x5
+ y1.x2.x6
- x3.w =
y2.x3.x6
+ y2.x1.x5
- x2.w =
y2.x3.x6
+ y2.x1.x6
- x1.w =
y3.x4.x7
+ y2.x3.x6
+ y2.x1.x6
- w2 =
y32.x5.x7
+ y22.x62
+ y22.x5.x6
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
There is one minimal generator:
Restriction to maximal subgroup number 1, which is 64gp261
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y2
- x1 restricts to
y22
+ y2.y3
+ y1.y5
+ y1.y3
- x2 restricts to
y1.y5
+ y1.y3
- x3 restricts to
y12
+ y1.y5
+ y1.y4
+ y1.y3
- x4 restricts to
y22
+ y2.y5
+ y2.y4
+ y2.y3
- x5 restricts to
y22
+ y12
+ y52
+ y42
+ y32
- x6 restricts to
y2.y5
+ y2.y3
+ y52
+ y32
- x7 restricts to
y22
+ y1.y3
+ y32
- x8 restricts to
x
+ y2.y5
+ y2.y4
+ y1.y5
+ y1.y4
+ y52
+ y42
- w restricts to
y23
+ y22.y5
+ y22.y4
+ y12.y5
+ y12.y3
+ y2.y3.y5
+ y2.y3.y4
+ y2.y32
+ y1.y4.y5
+ y1.y3.y4
Restriction to maximal subgroup number 2, which is 64gp193
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
0
- x1 restricts to
y22
+ y2.y4
+ y1.y4
- x2 restricts to
y22
+ y2.y4
+ y1.y4
+ y1.y3
- x3 restricts to
x1
+ y22
+ y2.y3
+ y1.y3
- x4 restricts to
y1.y4
- x5 restricts to
y22
+ x2
+ y32
- x6 restricts to
y22
+ y42
+ y1.y4
- x7 restricts to
x1
+ x3
+ x2
- x8 restricts to
y2.y3
+ y32
+ y1.y3
- w restricts to
y2.x1
+ y4.x1
+ y22.y4
+ y22.y3
+ y2.y42
+ y2.y3.y4
+ y1.y42
+ y1.y3.y4
Restriction to maximal subgroup number 3, which is 64gp193
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x1
+ y22
+ y1.y4
- x2 restricts to
x1
+ y22
- x3 restricts to
x1
+ y22
+ y2.y3
+ y1.y4
+ y1.y3
- x4 restricts to
y1.y3
- x5 restricts to
y22
+ x2
+ y32
- x6 restricts to
y22
+ x2
- x7 restricts to
x1
+ x3
+ x2
+ y2.y4
+ y42
- x8 restricts to
x1
+ x3
+ x2
- w restricts to
y3.x1
+ y22.y3
+ y1.y3.y4
Restriction to maximal subgroup number 4, which is 64gp193
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
y22
+ y2.y4
- x2 restricts to
y1.y4
+ y1.y3
- x3 restricts to
y1.y4
- x4 restricts to
x1
+ y2.y3
- x5 restricts to
x2
+ y32
- x6 restricts to
x3
- x7 restricts to
y22
+ y42
+ y1.y4
- x8 restricts to
y2.y4
+ y2.y3
+ y42
+ y32
- w restricts to
y2.x1
+ y4.x1
+ y22.y3
+ y2.y3.y4
Restriction to maximal subgroup number 5, which is 64gp193
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x1
+ y2.y4
+ y2.y3
- x2 restricts to
y1.y4
+ y1.y3
- x3 restricts to
y1.y3
- x4 restricts to
x1
+ y22
+ y2.y4
+ y1.y3
- x5 restricts to
y22
+ x2
+ y42
- x6 restricts to
x1
+ x3
+ x2
+ y2.y4
+ y2.y3
+ y42
+ y32
- x7 restricts to
x2
+ y42
+ y32
- x8 restricts to
x1
+ x3
+ x2
- w restricts to
y2.x1
+ y3.x1
+ y2.x2
+ y22.y4
+ y22.y3
+ y2.y42
+ y2.y3.y4
+ y1.x3
Restriction to maximal subgroup number 6, which is 64gp56
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- x1 restricts to
y2.y3
+ y1.y3
- x2 restricts to
x2
+ x1
+ y2.y3
- x3 restricts to
x2
+ y2.y3
- x4 restricts to
x2
- x5 restricts to
x5
- x6 restricts to
x5
+ x4
+ x2
- x7 restricts to
x5
+ x4
+ y32
+ x2
+ y1.y3
- x8 restricts to
x5
+ x3
+ y32
- w restricts to
y3.x2
+ y1.x5
+ y1.x4
+ y1.x2
Restriction to maximal subgroup number 7, which is 64gp56
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
+ y1.y3
- x3 restricts to
x1
+ y2.y3
+ y1.y3
- x4 restricts to
x1
+ y2.y3
- x5 restricts to
x3
+ y32
- x6 restricts to
x4
- x7 restricts to
x4
+ x3
+ x1
- x8 restricts to
x5
+ x4
+ y32
+ x2
+ y1.y3
- w restricts to
y3.x1
+ y2.x4
+ y2.x3
Restriction to maximal elementary abelian number 1, which is V32
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y5
- x1 restricts to
y2.y5
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
y4.y5
- x5 restricts to
y42
- x6 restricts to
y1.y5
+ y12
- x7 restricts to
y22
- x8 restricts to
y3.y5
+ y32
- w restricts to
y2.y4.y5
Restriction to maximal elementary abelian number 2, which is V32
- y1 restricts to
0
- y2 restricts to
y5
- y3 restricts to
0
- x1 restricts to
y1.y5
- x2 restricts to
y1.y5
- x3 restricts to
y4.y5
- x4 restricts to
0
- x5 restricts to
y42
- x6 restricts to
y12
- x7 restricts to
y2.y5
+ y22
- x8 restricts to
y3.y5
+ y32
- w restricts to
y1.y4.y5
+ y12.y5
Restriction to the greatest central elementary abelian, which is V16
- 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
y42
- x6 restricts to
y12
- x7 restricts to
y22
- x8 restricts to
y32
- w restricts to
0
Back to the groups of order 128