Small group number 2163 of order 128
G is the group 128gp2163
G has 5 minimal generators, rank 6 and exponent 4.
The centre has rank 4.
The 31 maximal subgroups are:
64gp193 (3x), 64gp202 (24x), 64gp261 (3x), V64.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
5, 6.
This cohomology ring calculation is only complete
out to degree 4.
Ring structure
| Restriction information
The cohomology ring has 8 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1, a regular element
- y5 in degree 1, a regular element
- x1 in degree 2
- x2 in degree 2, a regular element
- x3 in degree 2, a regular element
There are 4 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y3.x1 =
0
- x12 =
y1.y2.x1
+ y22.x3
+ y12.x2
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
Restriction to maximal subgroup number 1, which is V64
- y1 restricts to
y3
- y2 restricts to
y4
- y3 restricts to
0
- y4 restricts to
y2
- y5 restricts to
y1
- x1 restricts to
y4.y5
+ y3.y6
- x2 restricts to
y62
+ y4.y6
- x3 restricts to
y52
+ y3.y5
Restriction to maximal subgroup number 2, which is 64gp261
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
y2
+ y3
- y5 restricts to
y1
+ y5
+ y4
- x1 restricts to
y1.y5
+ y1.y3
- x2 restricts to
y2.y5
+ y2.y3
+ y52
+ y32
- x3 restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 3, which is 64gp193
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
- y4 restricts to
y2
+ y4
+ y1
- y5 restricts to
y2
+ y4
+ y3
+ y1
- x1 restricts to
x1
+ y22
+ y2.y3
+ y1.y4
+ y1.y3
- x2 restricts to
y22
+ x2
+ y32
- x3 restricts to
x3
+ y2.y4
+ y2.y3
+ y42
+ y32
+ y1.y4
+ y1.y3
Restriction to maximal subgroup number 4, which is 64gp261
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y5
+ y3
- y5 restricts to
y4
+ y3
- x1 restricts to
y22
+ y2.y5
+ y2.y4
+ y2.y3
- x2 restricts to
x
+ y2.y5
+ y2.y3
+ y1.y5
+ y1.y3
+ y52
+ y32
- x3 restricts to
y22
+ y1.y5
+ y1.y4
+ y1.y3
+ y52
+ y42
+ y32
Restriction to maximal subgroup number 5, which is 64gp193
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y4
+ y1
- y5 restricts to
y3
- x1 restricts to
x1
+ y1.y3
- x2 restricts to
x1
+ x3
+ x2
+ y2.y3
+ y32
- x3 restricts to
x2
Restriction to maximal subgroup number 6, which is 64gp261
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y5
+ y4
- y5 restricts to
y2
+ y5
- x1 restricts to
y2.y5
+ y2.y4
+ y2.y3
- x2 restricts to
x
+ y2.y4
+ y2.y3
+ y1.y4
+ y1.y3
+ y42
+ y32
- x3 restricts to
x
+ y2.y5
+ y1.y5
+ y52
Restriction to maximal subgroup number 7, which is 64gp193
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y4
+ y1
- y5 restricts to
y4
+ y3
+ y1
- x1 restricts to
x1
+ y22
+ y2.y4
+ y2.y3
+ y1.y3
- x2 restricts to
x3
+ y2.y3
+ y32
+ y1.y3
- x3 restricts to
x1
+ x3
+ x2
+ y2.y4
+ y42
Restriction to maximal subgroup number 8, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- y5 restricts to
y4
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 9, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
- y5 restricts to
y3
+ y4
- x1 restricts to
x1
+ y22
+ y2.y4
+ y1.y4
- x2 restricts to
x2
+ y3.y4
+ y2.y4
+ y42
- x3 restricts to
x1
+ y1.y2
+ x3
+ x2
Restriction to maximal subgroup number 10, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y1
- y5 restricts to
y4
- x1 restricts to
x1
- x2 restricts to
x3
- x3 restricts to
x2
Restriction to maximal subgroup number 11, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
+ y1
- y5 restricts to
y2
+ y4
- x1 restricts to
x1
+ y1.y2
+ y12
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
y1.y2
+ y12
+ x2
Restriction to maximal subgroup number 12, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y1
- y5 restricts to
y4
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 13, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
- y5 restricts to
y3
+ y4
- x1 restricts to
x1
+ y22
+ y12
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
y1.y2
+ y12
+ x2
Restriction to maximal subgroup number 14, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y1
- y5 restricts to
y4
- x1 restricts to
x1
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
x2
Restriction to maximal subgroup number 15, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
+ y1
- y5 restricts to
y1
+ y4
- x1 restricts to
x1
+ y12
- x2 restricts to
x3
- x3 restricts to
y1.y2
+ y12
+ x2
Restriction to maximal subgroup number 16, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
0
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 17, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y3
- x1 restricts to
x1
+ y22
+ y2.y4
+ y1.y4
- x2 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
- x3 restricts to
x1
+ y1.y2
+ x3
+ x2
Restriction to maximal subgroup number 18, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y1
- x1 restricts to
x1
+ y22
+ y1.y2
+ y12
+ y2.y4
+ y1.y4
- x2 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
- x3 restricts to
y1.y2
+ y12
+ x2
+ y3.y4
+ y2.y4
+ y42
Restriction to maximal subgroup number 19, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
+ y1
+ y4
- y5 restricts to
y3
+ y2
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 20, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y1
+ y4
- y5 restricts to
y1
- x1 restricts to
x1
+ y1.y2
+ y1.y4
- x2 restricts to
x1
+ y1.y2
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
- x3 restricts to
x3
Restriction to maximal subgroup number 21, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y4
- y5 restricts to
y3
+ y2
+ y1
- x1 restricts to
x1
+ y22
- x2 restricts to
y22
+ y1.y2
+ x3
- x3 restricts to
x1
+ y1.y2
+ x3
+ x2
Restriction to maximal subgroup number 22, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y2
+ y1
+ y4
- y5 restricts to
y1
- x1 restricts to
x1
+ y22
+ y1.y2
+ y12
+ y2.y4
- x2 restricts to
x1
+ y1.y2
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
- x3 restricts to
y1.y2
+ y12
+ x2
Restriction to maximal subgroup number 23, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y3
+ y2
+ y1
- x1 restricts to
x1
+ y22
+ y1.y2
+ y2.y4
- x2 restricts to
x2
- x3 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
Restriction to maximal subgroup number 24, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y4
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 25, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y2
+ y1
+ y4
- y5 restricts to
y3
+ y2
+ y1
+ y4
- x1 restricts to
x1
+ y2.y4
- x2 restricts to
x2
- x3 restricts to
x1
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
Restriction to maximal subgroup number 26, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y2
+ y1
+ y4
- x1 restricts to
x1
+ y22
+ y12
+ y1.y4
- x2 restricts to
x1
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
- x3 restricts to
y22
+ y1.y2
+ x3
Restriction to maximal subgroup number 27, which is 64gp202
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
+ y4
- y5 restricts to
y2
+ y4
- x1 restricts to
x1
+ y22
+ y1.y4
- x2 restricts to
x2
+ y3.y4
+ y2.y4
+ y42
- x3 restricts to
y22
+ y1.y2
+ x3
Restriction to maximal subgroup number 28, which is 64gp202
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y4
- y5 restricts to
y2
+ y4
- x1 restricts to
x1
+ y22
+ y2.y4
- x2 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
- x3 restricts to
x2
Restriction to maximal subgroup number 29, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y2
+ y1
+ y4
- y5 restricts to
y3
+ y4
- x1 restricts to
x1
+ y22
+ y2.y4
- x2 restricts to
x2
- x3 restricts to
x1
+ y1.y2
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
Restriction to maximal subgroup number 30, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y1
+ y4
- y5 restricts to
y4
- x1 restricts to
x1
+ y22
+ y1.y4
- x2 restricts to
x2
+ y3.y4
+ y2.y4
+ y42
- x3 restricts to
x1
+ y1.y2
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
Restriction to maximal subgroup number 31, which is 64gp202
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
+ y4
- y5 restricts to
y4
- x1 restricts to
x1
+ y2.y4
- x2 restricts to
x3
+ y3.y4
+ y1.y4
+ y42
- x3 restricts to
x1
+ x3
+ x2
+ y3.y4
+ y2.y4
+ y1.y4
+ y42
Restriction to maximal elementary abelian number 1, which is V32
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y5
- y4 restricts to
y1
- y5 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y3.y5
+ y32
- x3 restricts to
y4.y5
+ y42
Restriction to maximal elementary abelian number 2, which is V64
- y1 restricts to
y6
- y2 restricts to
y5
- y3 restricts to
0
- y4 restricts to
y1
- y5 restricts to
y2
- x1 restricts to
y4.y5
+ y3.y6
- x2 restricts to
y3.y5
+ y32
- x3 restricts to
y4.y6
+ y42
Restriction to the greatest central elementary abelian, which is V16
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y1
- y5 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y32
- x3 restricts to
y42
Back to the groups of order 128