Small group number 2216 of order 128
G is the group 128gp2216
G has 5 minimal generators, rank 5 and exponent 4.
The centre has rank 2.
There are 6 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 4, 4, 4, 5, 5.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 8 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- y5 in degree 1
- x in degree 2, a regular element
- w in degree 3
- v in degree 4, a regular element
There are 5 minimal relations:
- y4.y5 =
y1.y5
+ y1.y3
- y2.y5 =
0
- y1.y3.y5 =
y1.y3.y4
+ y1.y32
- y5.w =
0
- w2 =
y2.y42.w
+ y2.y3.y4.w
+ y1.y2.y4.w
+ y44.x
+ y32.y42.x
+ y22.v
+ y12.y42.x
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y2.y3 =
0
- y1.y3.y42 =
y1.y32.y4
+ y12.y3.y4
- y1.y3.w =
0
This cohomology ring was obtained from a calculation
out to degree 13. The cohomology ring approximation
is stable from degree 6 onwards, and
Benson's tests detect stability from degree 13
onwards.
This cohomology ring has dimension 5 and depth 4.
Here is a homogeneous system of parameters:
- h1 =
x
in degree 2
- h2 =
v
in degree 4
- h3 =
y54
+ y44
+ y32.y52
+ y32.y42
+ y34
+ y2.y3.y42
+ y2.y32.y4
+ y22.y42
+ y22.y3.y4
+ y22.y32
+ y24
+ y1.y2.y42
+ y1.y22.y4
+ y12.y52
+ y12.y42
+ y12.y2.y4
+ y12.y22
+ y14
in degree 4
- h4 =
y32.y54
+ y32.y44
+ y34.y52
+ y34.y42
+ y2.y3.y44
+ y2.y34.y4
+ y22.y44
+ y22.y32.y42
+ y22.y34
+ y24.y42
+ y24.y3.y4
+ y24.y32
+ y1.y2.y44
+ y1.y24.y4
+ y12.y54
+ y12.y44
+ y12.y22.y42
+ y12.y24
+ y14.y52
+ y14.y42
+ y14.y2.y4
+ y14.y22
in degree 6
- h5 =
y3
+ y1
in degree 1
The first
4 terms h1, h2, h3, h4 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, -1, 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
-
y5
in degree 1
-
y4
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y52
in degree 2
-
y42
in degree 2
-
y2.y4
in degree 2
-
y22
in degree 2
-
y1.y5
in degree 2
-
y1.y4
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w
in degree 3
-
y53
in degree 3
-
y43
in degree 3
-
y2.y42
in degree 3
-
y22.y4
in degree 3
-
y23
in degree 3
-
y1.y52
in degree 3
-
y1.y42
in degree 3
-
y1.y2.y4
in degree 3
-
y1.y22
in degree 3
-
y12.y4
in degree 3
-
y13
in degree 3
-
y4.w
in degree 4
-
y44
in degree 4
-
y2.w
in degree 4
-
y2.y43
in degree 4
-
y22.y42
in degree 4
-
y23.y4
in degree 4
-
y24
in degree 4
-
y1.w
in degree 4
-
y1.y53
in degree 4
-
y1.y43
in degree 4
-
y1.y2.y42
in degree 4
-
y1.y22.y4
in degree 4
-
y1.y23
in degree 4
-
y13.y4
in degree 4
-
y14
in degree 4
-
y42.w
in degree 5
-
y2.y4.w
in degree 5
-
y22.w
in degree 5
-
y22.y43
in degree 5
-
y23.y42
in degree 5
-
y24.y4
in degree 5
-
y25
in degree 5
-
y1.y4.w
in degree 5
-
y1.y44
in degree 5
-
y1.y2.w
in degree 5
-
y1.y2.y43
in degree 5
-
y1.y22.y42
in degree 5
-
y1.y23.y4
in degree 5
-
y1.y24
in degree 5
-
y14.y4
in degree 5
-
y15
in degree 5
-
y43.w
in degree 6
-
y2.y42.w
in degree 6
-
y22.y4.w
in degree 6
-
y23.w
in degree 6
-
y23.y43
in degree 6
-
y24.y42
in degree 6
-
y25.y4
in degree 6
-
y1.y42.w
in degree 6
-
y1.y2.y4.w
in degree 6
-
y1.y22.w
in degree 6
-
y1.y22.y43
in degree 6
-
y1.y23.y42
in degree 6
-
y1.y24.y4
in degree 6
-
y1.y25
in degree 6
-
y15.y4
in degree 6
-
y2.y43.w
in degree 7
-
y22.y42.w
in degree 7
-
y23.y4.w
in degree 7
-
y24.w
in degree 7
-
y24.y43
in degree 7
-
y25.y42
in degree 7
-
y1.y43.w
in degree 7
-
y1.y2.y42.w
in degree 7
-
y1.y22.y4.w
in degree 7
-
y1.y23.w
in degree 7
-
y1.y23.y43
in degree 7
-
y1.y24.y42
in degree 7
-
y1.y25.y4
in degree 7
-
y22.y43.w
in degree 8
-
y23.y42.w
in degree 8
-
y24.y4.w
in degree 8
-
y25.w
in degree 8
-
y25.y43
in degree 8
-
y1.y2.y43.w
in degree 8
-
y1.y22.y42.w
in degree 8
-
y1.y23.y4.w
in degree 8
-
y1.y24.w
in degree 8
-
y1.y24.y43
in degree 8
-
y1.y25.y42
in degree 8
-
y23.y43.w
in degree 9
-
y24.y42.w
in degree 9
-
y25.y4.w
in degree 9
-
y1.y22.y43.w
in degree 9
-
y1.y23.y42.w
in degree 9
-
y1.y24.y4.w
in degree 9
-
y1.y25.w
in degree 9
-
y1.y25.y43
in degree 9
-
y24.y43.w
in degree 10
-
y25.y42.w
in degree 10
-
y1.y23.y43.w
in degree 10
-
y1.y24.y42.w
in degree 10
-
y1.y25.y4.w
in degree 10
-
y25.y43.w
in degree 11
-
y1.y24.y43.w
in degree 11
-
y1.y25.y42.w
in degree 11
-
y1.y25.y43.w
in degree 12
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
-
y5.h5
in degree 6
-
y52.h5
in degree 7
-
y1.y5.h5
in degree 7
-
y12.h5
+ y1.h6
in degree 7
-
y53.h5
in degree 8
-
y1.y52.h5
in degree 8
-
y12.y4.h5
+ y1.y4.h6
in degree 8
-
y13.h5
+ y12.h6
in degree 8
-
y44.h5
+ y22.y42.h5
+ y24.h5
+ y2.y42.h6
+ y22.y4.h6
+ y42.h7
+ y2.y4.h7
+ y22.h7
+ h9
in degree 9
-
y1.y53.h5
in degree 9
-
y13.y4.h5
+ y12.y4.h6
in degree 9
-
y14.h5
+ y13.h6
in degree 9
-
y1.y44.h5
+ y1.y22.y42.h5
+ y1.y24.h5
+ y1.y2.y42.h6
+ y1.y22.y4.h6
+ y1.y42.h7
+ y1.y2.y4.h7
+ y1.y22.h7
+ y1.h9
in degree 10
-
y14.y4.h5
+ y13.y4.h6
in degree 10
-
y15.h5
+ y14.h6
in degree 10
-
y15.y4.h5
+ y14.y4.h6
in degree 11
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
- y5 restricts to
0
- x restricts to
y22
- w restricts to
0
- v restricts to
y14
Restriction to special subgroup number 2, which is 16gp14
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y4
- y4 restricts to
0
- y5 restricts to
y3
- x restricts to
y2.y3
+ y22
- w restricts to
0
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to special subgroup number 3, which is 16gp14
- y1 restricts to
y4
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y4
- y5 restricts to
y3
- x restricts to
y2.y3
+ y22
- w restricts to
0
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to special subgroup number 4, which is 16gp14
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y4
- y4 restricts to
0
- y5 restricts to
y4
- x restricts to
y2.y4
+ y22
- w restricts to
0
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to special subgroup number 5, which is 16gp14
- y1 restricts to
y4
- y2 restricts to
0
- y3 restricts to
y4
+ y3
- y4 restricts to
y3
- y5 restricts to
y4
- x restricts to
y2.y4
+ y22
- w restricts to
0
- v restricts to
y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to special subgroup number 6, which is 32gp51
- y1 restricts to
y3
- y2 restricts to
y4
- y3 restricts to
0
- y4 restricts to
y5
- y5 restricts to
0
- x restricts to
y2.y4
+ y22
- w restricts to
y2.y52
+ y2.y3.y5
+ y1.y3.y4
+ y12.y4
- v restricts to
y1.y3.y52
+ y1.y32.y5
+ y12.y52
+ y12.y3.y5
+ y12.y32
+ y14
Restriction to special subgroup number 7, which is 32gp51
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y4
- y4 restricts to
y5
- y5 restricts to
0
- x restricts to
y2.y3
+ y22
- w restricts to
y2.y52
+ y2.y4.y5
+ y1.y3.y4
+ y12.y3
- v restricts to
y1.y4.y52
+ y1.y42.y5
+ y12.y52
+ y12.y4.y5
+ y12.y42
+ y14
(1 + 4t + 8t2
+ 12t3 + 15t4 + 16t5
+ 15t6 + 12t7 + 8t8
+ 4t9 + t10) /
(1 - t) (1 - t2) (1 - t4)2 (1 - t6\
)
Back to the groups of order 128