Small group number 1578 of order 128
G = V8wrC2 is Wreath product V_8 wr C_2
G has 4 minimal generators, rank 6 and exponent 4.
The centre has rank 3.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 6.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 11 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
- x3 in degree 2
- x4 in degree 2, a regular element
- x5 in degree 2, a regular element
- x6 in degree 2, a regular element
- w in degree 3
There are 18 minimal relations:
- y3.y4 =
0
- y2.y4 =
0
- y1.y4 =
0
- y4.x3 =
0
- y4.x2 =
0
- y4.x1 =
0
- y3.x3 =
y2.x2
+ y1.x1
- x32 =
y1.y2.x3
+ y22.x4
+ y12.x5
- x2.x3 =
y1.w
+ y1.y2.x2
+ y12.x1
+ y2.y3.x4
- x22 =
y1.y3.x2
+ y32.x4
+ y12.x6
- x1.x3 =
y2.w
+ y1.y3.x5
- x1.x2 =
y3.w
+ y1.y3.x1
+ y1.y2.x6
- x12 =
y2.y3.x1
+ y32.x5
+ y22.x6
- y4.w =
0
- x3.w =
y1.y2.w
+ y2.x1.x4
+ y1.x2.x5
+ y12.y3.x5
- x2.w =
y3.x1.x4
+ y1.x3.x6
- x1.w =
y2.y3.w
+ y3.x2.x5
+ y2.x3.x6
+ y1.y32.x5
- w2 =
y1.y2.y3.w
+ y2.y3.x1.x4
+ y1.y3.x2.x5
+ y1.y2.x3.x6
+ y12.y32.x5
+ y32.x4.x5
+ y22.x4.x6
+ y12.x5.x6
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 6 onwards, and
Benson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 6 and depth 4.
Here is a homogeneous system of parameters:
- h1 =
x4
in degree 2
- h2 =
x5
in degree 2
- h3 =
x6
in degree 2
- h4 =
y44
+ y34
+ y22.y32
+ y24
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
in degree 4
- h5 =
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
- h6 =
y1
in degree 1
The first
4 terms h1, h2, h3, h4 form
a regular sequence of maximum length.
The first
3 terms h1, h2, h3 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, 4, 10, 11.
-
Filter degree type:
-1, -2, -3, -4, -5, -6, -6.
-
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, h6) is as follows.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y42
in degree 2
-
y32
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
w
in degree 3
-
y43
in degree 3
-
y3.x2
in degree 3
-
y3.x1
in degree 3
-
y33
in degree 3
-
y2.x3
in degree 3
-
y2.x2
in degree 3
-
y2.x1
in degree 3
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y3.w
in degree 4
-
y32.x2
in degree 4
-
y32.x1
in degree 4
-
y34
in degree 4
-
y2.w
in degree 4
-
y2.y3.x2
in degree 4
-
y2.y3.x1
in degree 4
-
y2.y33
in degree 4
-
y22.x3
in degree 4
-
y22.x2
in degree 4
-
y22.x1
in degree 4
-
y22.y32
in degree 4
-
y23.y3
in degree 4
-
y24
in degree 4
-
y32.w
in degree 5
-
y33.x2
in degree 5
-
y33.x1
in degree 5
-
y2.y3.w
in degree 5
-
y2.y32.x2
in degree 5
-
y2.y32.x1
in degree 5
-
y22.w
in degree 5
-
y22.y3.x2
in degree 5
-
y22.y3.x1
in degree 5
-
y22.y33
in degree 5
-
y23.x3
in degree 5
-
y23.x2
in degree 5
-
y23.x1
in degree 5
-
y23.y32
in degree 5
-
y24.y3
in degree 5
-
y25
in degree 5
-
y33.w
in degree 6
-
y2.y32.w
in degree 6
-
y2.y33.x1
in degree 6
-
y22.y3.w
in degree 6
-
y22.y32.x2
in degree 6
-
y22.y32.x1
in degree 6
-
y23.w
in degree 6
-
y23.y3.x2
in degree 6
-
y23.y3.x1
in degree 6
-
y23.y33
in degree 6
-
y24.x3
in degree 6
-
y24.x2
in degree 6
-
y24.x1
in degree 6
-
y24.y32
in degree 6
-
y25.y3
in degree 6
-
y2.y33.w
in degree 7
-
y22.y32.w
in degree 7
-
y22.y33.x1
in degree 7
-
y23.y3.w
in degree 7
-
y23.y32.x2
in degree 7
-
y23.y32.x1
in degree 7
-
y24.w
in degree 7
-
y24.y3.x2
in degree 7
-
y24.y3.x1
in degree 7
-
y24.y33
in degree 7
-
y25.x3
in degree 7
-
y25.x2
in degree 7
-
y25.x1
in degree 7
-
y25.y32
in degree 7
-
y22.y33.w
in degree 8
-
y23.y32.w
in degree 8
-
y23.y33.x1
in degree 8
-
y24.y3.w
in degree 8
-
y24.y32.x2
in degree 8
-
y24.y32.x1
in degree 8
-
y25.w
in degree 8
-
y25.y3.x2
in degree 8
-
y25.y3.x1
in degree 8
-
y25.y33
in degree 8
-
y23.y33.w
in degree 9
-
y24.y32.w
in degree 9
-
y24.y33.x1
in degree 9
-
y25.y3.w
in degree 9
-
y25.y32.x2
in degree 9
-
y25.y32.x1
in degree 9
-
y24.y33.w
in degree 10
-
y25.y32.w
in degree 10
-
y25.y33.x1
in degree 10
-
y25.y33.w
in degree 11
A basis for AnnR/(h1, h2, h3, h4, h5)(h6) is as follows.
-
y4
in degree 1
-
y42
in degree 2
-
y43
in degree 3
-
y34
+ y22.y32
+ y24
+ y2.y32.h
+ y22.y3.h
+ y32.h2
+ y2.y3.h2
+ y22.h2
+ h4
in degree 4
-
y25.y33.x1
+ y24.y33.x1.h
+ y23.y33.x1.h2
+ y25.y3.x1.h2
+ y22.y33.x1.h3
+ y24.y3.x1.h3
+ y25.x3.h3
+ y2.y33.x1.h4
+ y23.y3.x1.h4
+ y24.x3.h4
+ y33.x2.h5
+ y33.x1.h5
+ y22.y3.x2.h5
+ y22.y3.x1.h5
+ y2.y3.x2.h6
+ y22.x3.h6
+ y22.x2.h6
+ y3.x2.h7
+ y2.x2.h7
+ y2.x1.h7
+ x3.h8
+ x1.h8
in degree 10
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
-
y4
in degree 1
-
y42
in degree 2
-
y43
in degree 3
-
y34
+ y22.y32
+ y24
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
in degree 4
Restriction to special subgroup number 1, which is 8gp5
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
y12
- x5 restricts to
y22
- x6 restricts to
y32
- w restricts to
0
Restriction to special subgroup number 2, which is 16gp14
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y4
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
y1.y4
+ y12
- x5 restricts to
y2.y4
+ y22
- x6 restricts to
y3.y4
+ y32
- w restricts to
0
Restriction to special subgroup number 3, which is 64gp267
- y1 restricts to
y4
- y2 restricts to
y5
- y3 restricts to
y6
- y4 restricts to
0
- x1 restricts to
y3.y5
+ y2.y6
- x2 restricts to
y3.y4
+ y1.y6
- x3 restricts to
y2.y4
+ y1.y5
- x4 restricts to
y1.y4
+ y12
- x5 restricts to
y2.y5
+ y22
- x6 restricts to
y3.y6
+ y32
- w restricts to
y2.y4.y6
+ y2.y3.y4
+ y1.y3.y5
+ y1.y2.y6
(1 + 3t + 6t2
+ 10t3 + 13t4 + 15t5
+ 15t6 + 13t7 + 10t8
+ 6t9 + 3t10 + t11) /
(1 - t) (1 - t2)3 (1 - t4) (1 - t6\
)
Back to the groups of order 128