Small group number 852 of order 128
G is the group 128gp852
G has 3 minimal generators, rank 4 and exponent 8.
The centre has rank 1.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 14 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 in degree 2
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- v in degree 4
- u1 in degree 5
- u2 in degree 5
- t in degree 6
- s in degree 7
- r in degree 8, a regular element
There are 51 minimal relations:
- y1.y3 =
0
- y12 =
0
- y3.x2 =
y1.y22
- y22.y3 =
y1.y22
- y1.x1 =
y1.y22
- x1.x2 =
y24
- y3.w1 =
y1.w2
- y22.x2 =
y24
+ y1.w1
- y22.x1 =
y24
+ y1.w2
- y1.w3 =
0
- x2.w3 =
0
- x2.w2 =
y22.w1
+ y1.y24
- x1.w1 =
y22.w1
- y22.w3 =
0
- y22.w2 =
y22.w1
+ y1.y24
- y1.v =
0
- w32 =
x13
+ y3.x1.w3
+ y32.v
+ y33.w2
+ y26
+ y1.y22.w1
- w22 =
x1.v
+ y3.u1
- w1.w3 =
0
- w1.w2 =
y22.v
+ y1.y22.w1
- w12 =
x2.v
+ y1.x2.w1
+ y1.y22.w1
- y3.u2 =
y1.y22.w1
- y1.u2 =
y1.x2.w1
- y1.u1 =
0
- w3.v =
x1.u1
+ y3.t
+ y1.y26
- x2.u2 =
x2.u1
+ x22.w1
+ y22.u2
+ y24.w1
- x1.u2 =
y22.u2
- y22.u1 =
0
- y1.t =
y1.y26
- w3.u2 =
0
- w3.u1 =
x12.v
+ y3.s
+ y3.x1.u1
+ y3.x12.w2
+ y32.x1.v
+ y24.v
- w2.u2 =
y22.t
+ y28
- w1.u2 =
x2.t
+ y28
+ y1.x22.w1
+ y1.y24.w1
- w1.u1 =
x2.t
+ x22.v
+ y22.t
+ y24.v
+ y1.y24.w1
- y1.s =
y1.y24.w1
- v.u2 =
w1.t
+ y26.w1
- w3.t =
x1.s
+ x13.w2
+ y3.v2
+ y3.x1.t
+ y3.x12.v
+ y32.w2.v
- x2.s =
x2.w1.v
+ x22.u1
+ y22.w1.v
+ y26.w1
- y22.s =
y26.w1
- u22 =
x2.v2
+ x23.v
+ y1.x23.w1
+ y1.y26.w1
- u1.u2 =
x2.v2
+ x22.t
+ x23.v
+ y22.v2
+ y24.t
+ y26.v
- u12 =
x2.v2
+ x1.v2
+ y3.v.u1
+ y32.v2
+ y32.x1.t
+ y32.x1.w2.w3
+ y32.x12.v
+ y32.x14
+ y33.s
+ y33.x1.u1
+ y33.x12.w3
+ y34.w2.w3
+ y34.x1.v
+ y35.u1
+ y35.x1.w3
+ y35.x1.w2
+ y36.v
+ y32.r
- w3.s =
x12.t
+ x12.w2.w3
+ y3.v.u1
+ y3.x12.u1
+ y32.w2.u1
+ y32.x1.t
+ y24.t
- w1.s =
x2.v2
+ x22.t
+ x23.v
+ y22.v2
+ y24.t
- u2.t =
w1.v2
+ x22.w1.v
+ y26.u2
- u1.t =
v.s
+ x12.w2.v
+ y3.x12.t
+ y3.x12.w2.w3
+ y3.x13.v
+ y3.x15
+ y32.x1.s
+ y32.x12.u1
+ y32.x13.w3
+ y33.x1.w2.w3
+ y33.x12.v
+ y34.x1.u1
+ y34.x12.w3
+ y34.x12.w2
+ y35.x1.v
+ y3.x1.r
+ y1.y22.r
- t2 =
v3
+ x22.v2
+ x1.v.t
+ x12.v2
+ x13.t
+ x13.w2.w3
+ x14.v
+ x16
+ y3.w2.v2
+ y3.x12.s
+ y3.x13.u1
+ y3.x14.w3
+ y32.x12.w2.w3
+ y32.x13.v
+ y33.x12.u1
+ y33.x13.w3
+ y33.x13.w2
+ y34.x12.v
+ y22.v.t
+ y24.v2
+ y26.t
+ y28.v
+ x12.r
+ y24.r
- u2.s =
x2.v.t
+ x22.v2
+ x23.t
+ x24.v
+ y22.v.t
+ y24.v2
+ y28.v
+ y212
- u1.s =
x2.v.t
+ x1.v.t
+ x12.w2.u1
+ y3.v.s
+ y3.x12.s
+ y3.x12.w2.v
+ y3.x13.u1
+ y3.x14.w3
+ y32.v.t
+ y32.x1.v2
+ y32.x12.w2.w3
+ y32.x13.v
+ y33.v.u1
+ y34.w2.u1
+ y34.x1.w2.w3
+ y34.x12.v
+ y34.x14
+ y35.s
+ y35.w2.v
+ y35.x12.w2
+ y36.t
+ y37.u1
+ y37.x1.w2
+ y1.y28.w1
+ y3.w3.r
+ y32.x1.r
- t.s =
v2.u1
+ x22.w1.t
+ x23.w1.v
+ x1.v.s
+ x12.v.u1
+ x12.w2.t
+ x13.s
+ x13.w2.v
+ x14.u1
+ x15.w3
+ y3.w2.v.u1
+ y3.x12.v2
+ y3.x13.w2.w3
+ y3.x14.v
+ y32.x1.v.u1
+ y33.x1.w2.u1
+ y33.x12.w2.w3
+ y33.x13.v
+ y33.x15
+ y34.x1.s
+ y34.x1.w2.v
+ y34.x13.w2
+ y35.x1.t
+ y36.x1.u1
+ y36.x12.w2
+ y24.w1.t
+ y26.w1.v
+ y210.w1
+ y1.y212
+ x1.w3.r
+ y3.x12.r
+ y1.y24.r
- s2 =
x2.v3
+ x23.v2
+ x1.v3
+ x12.v.t
+ x13.v2
+ x14.t
+ x14.w2.w3
+ x17
+ y3.v2.u1
+ y3.x1.v.s
+ y3.x1.w2.v2
+ y3.x12.v.u1
+ y3.x13.w2.v
+ y3.x14.u1
+ y32.v3
+ y32.w2.v.u1
+ y32.x12.v2
+ y32.x12.w2.u1
+ y32.x14.v
+ y33.v.s
+ y33.w2.v2
+ y33.x12.w2.v
+ y33.x13.u1
+ y33.x14.w3
+ y34.w2.s
+ y34.x1.v2
+ y34.x1.w2.u1
+ y34.x13.v
+ y34.x15
+ y35.v.u1
+ y35.w2.t
+ y35.x1.w2.v
+ y35.x12.u1
+ y36.v2
+ y36.w2.u1
+ y36.x1.t
+ y36.x1.w2.w3
+ y36.x12.v
+ y37.s
+ y37.w2.v
+ y37.x1.u1
+ y38.x1.v
+ y24.v.t
+ y28.t
+ y210.v
+ y214
+ y1.y210.w1
+ x13.r
+ y3.x1.w3.r
+ y32.v.r
+ y32.x12.r
+ y33.w2.r
+ y26.r
+ y1.y22.w1.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
- x2.v.u1 =
x2.w1.t
+ x22.w1.v
+ y22.w1.t
+ y24.w1.v
This cohomology ring was obtained from a calculation
out to degree 17. The cohomology ring approximation
is stable from degree 14 onwards, and
Benson's tests detect stability from degree 17
onwards.
This cohomology ring has dimension 4 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
v
+ x22
+ x12
+ y3.w2
+ y34
+ y24
in degree 4
- h3 =
x2.v
+ x1.v
+ y3.x1.w2
+ y32.v
+ y32.x12
+ y33.w2
+ y2.u2
+ y2.x2.w1
in degree 6
- h4 =
y3
in degree 1
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The first
term h1 forms
a complete Duflot regular sequence.
That is, its restriction to the greatest central elementary abelian
subgroup forms a regular sequence of maximal length.
Data for Benson's test:
-
Raw filter degree type:
-1, -1, 7, 14, 15.
-
Filter degree type:
-1, -2, -3, -4, -4.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3, h4) is as follows.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
y2.x2
in degree 3
-
y2.x1
in degree 3
-
y23
in degree 3
-
y1.x2
in degree 3
-
x22
in degree 4
-
x12
in degree 4
-
y2.w3
in degree 4
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y24
in degree 4
-
y1.w1
in degree 4
-
y1.y2.x2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
x2.w1
in degree 5
-
x1.w3
in degree 5
-
x1.w2
in degree 5
-
y2.x22
in degree 5
-
y2.x12
in degree 5
-
y22.w1
in degree 5
-
y25
in degree 5
-
y1.y2.w1
in degree 5
-
t
in degree 6
-
w2.w3
in degree 6
-
x13
in degree 6
-
y2.u2
in degree 6
-
y2.u1
in degree 6
-
y2.x2.w1
in degree 6
-
y2.x1.w3
in degree 6
-
y2.x1.w2
in degree 6
-
y23.w1
in degree 6
-
y26
in degree 6
-
y1.x2.w1
in degree 6
-
s
in degree 7
-
x2.u1
in degree 7
-
x22.w1
in degree 7
-
x12.w3
in degree 7
-
x12.w2
in degree 7
-
y2.t
in degree 7
-
y2.w2.w3
in degree 7
-
y2.x13
in degree 7
-
y22.u2
in degree 7
-
y24.w1
in degree 7
-
y27
in degree 7
-
y1.y2.x2.w1
in degree 7
-
w2.u1
in degree 8
-
x2.t
in degree 8
-
x1.t
in degree 8
-
x1.w2.w3
in degree 8
-
y2.s
in degree 8
-
y2.x2.u1
in degree 8
-
y2.x22.w1
in degree 8
-
y2.x12.w3
in degree 8
-
y2.x12.w2
in degree 8
-
y22.t
in degree 8
-
y25.w1
in degree 8
-
y28
in degree 8
-
w2.t
in degree 9
-
x22.u1
in degree 9
-
x1.s
in degree 9
-
y2.w2.u1
in degree 9
-
y2.x2.t
in degree 9
-
y2.x1.t
in degree 9
-
y2.x1.w2.w3
in degree 9
-
y23.t
in degree 9
-
y26.w1
in degree 9
-
y29
in degree 9
-
w2.s
in degree 10
-
x22.t
in degree 10
-
x12.t
in degree 10
-
x12.w2.w3
in degree 10
-
y2.w2.t
in degree 10
-
y2.x22.u1
in degree 10
-
y2.x1.s
in degree 10
-
y24.t
in degree 10
-
y27.w1
in degree 10
-
y210
in degree 10
-
x1.w2.t
in degree 11
-
x12.s
in degree 11
-
y2.w2.s
in degree 11
-
y2.x22.t
in degree 11
-
y2.x12.t
in degree 11
-
y2.x12.w2.w3
in degree 11
-
y28.w1
in degree 11
-
x1.w2.s
in degree 12
-
x13.t
in degree 12
-
y2.x1.w2.t
in degree 12
-
y2.x12.s
in degree 12
-
y29.w1
in degree 12
-
x12.w2.t
in degree 13
-
y2.x1.w2.s
in degree 13
-
y2.x13.t
in degree 13
-
y210.w1
in degree 13
-
x12.w2.s
in degree 14
-
y2.x12.w2.t
in degree 14
-
y2.x12.w2.s
in degree 15
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y1
in degree 1
-
x2
+ y22
in degree 2
-
y1.y2
in degree 2
-
y2.x2
+ y23
in degree 3
-
y1.x2
in degree 3
-
y22.h
in degree 3
-
x22
+ y22.x2
in degree 4
-
y1.w1
in degree 4
-
y1.y2.x2
in degree 4
-
w1.h
in degree 4
-
y23.h
in degree 4
-
u2
+ y22.w1
in degree 5
-
x2.w1
+ y22.w1
in degree 5
-
y2.x22
+ y23.x2
in degree 5
-
y1.y2.w1
in degree 5
-
y2.w1.h
in degree 5
-
y24.h
in degree 5
-
x13
+ x1.h4
+ h6
in degree 6
-
y2.u2
+ y23.w1
in degree 6
-
y2.x2.w1
+ y23.w1
in degree 6
-
y26
in degree 6
-
y1.x2.w1
in degree 6
-
y22.w1.h
in degree 6
-
y25.h
in degree 6
-
x2.u1
+ y22.u1
in degree 7
-
x22.w1
+ y22.x2.w1
in degree 7
-
y2.x13
+ y2.x1.h4
+ y2.h6
in degree 7
-
y22.u2
+ y24.w1
in degree 7
-
y24.w1
in degree 7
-
y27
in degree 7
-
y1.y2.x2.w1
in degree 7
-
y23.w1.h
in degree 7
-
x2.t
+ y22.t
in degree 8
-
y2.x2.u1
+ y23.u1
in degree 8
-
y2.x22.w1
+ y23.x2.w1
in degree 8
-
y22.t
in degree 8
-
y25.w1
in degree 8
-
y28
in degree 8
-
x22.u1
+ y22.x2.u1
in degree 9
-
y2.x2.t
+ y23.t
in degree 9
-
y23.t
in degree 9
-
y26.w1
in degree 9
-
y29
in degree 9
-
x22.t
+ y22.x2.t
in degree 10
-
y2.x22.u1
+ y23.x2.u1
in degree 10
-
y24.t
in degree 10
-
y27.w1
in degree 10
-
y210
in degree 10
-
y2.x22.t
+ y23.x2.t
in degree 11
-
y28.w1
in degree 11
-
x12.t.h
+ x12.w2.w3.h
+ x12.w3.h4
+ t.h5
+ w2.w3.h5
+ u1.h6
+ x1.w3.h6
+ w3.h8
in degree 11
-
x13.t
+ x1.t.h4
+ t.h6
in degree 12
-
y29.w1
in degree 12
-
y2.x12.t.h
+ y2.x12.w2.w3.h
+ y2.x12.w3.h4
+ y2.t.h5
+ y2.w2.w3.h5
+ y2.u1.h6
+ y2.x1.w3.h6
+ y2.w3.h8
in degree 12
-
x12.w2.t
+ x12.s.h2
+ x12.w2.w3.h3
+ w2.t.h4
+ w2.u1.h5
+ x1.w2.w3.h5
+ s.h6
+ x12.w3.h6
+ u1.h8
+ x12.h9
+ w3.h10
+ w2.h10
in degree 13
-
y2.x13.t
+ y2.x1.t.h4
+ y2.t.h6
in degree 13
-
y210.w1
in degree 13
-
y2.x12.w2.t
+ y2.x12.s.h2
+ y2.x12.w2.w3.h3
+ y2.w2.t.h4
+ y2.w2.u1.h5
+ y2.x1.w2.w3.h5
+ y2.s.h6
+ y2.x12.w3.h6
+ y2.u1.h8
+ y2.x12.h9
+ y2.w3.h10
+ y2.w2.h10
in degree 14
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1
in degree 1
-
y1.y2
in degree 2
-
y1.x2
in degree 3
-
y1.y22
in degree 3
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
y1.y2.x2
in degree 4
-
y1.y23
in degree 4
-
y1.y2.w2
in degree 5
-
y1.y2.w1
in degree 5
-
y1.y24
in degree 5
-
y1.x2.w1
in degree 6
-
y1.y22.w1
in degree 6
-
y1.y25
in degree 6
-
y1.y2.x2.w1
in degree 7
-
y1.y23.w1
in degree 7
Restriction to special subgroup number 1, which is 2gp1
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- w3 restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- t restricts to
0
- s restricts to
0
- r restricts to
y8
Restriction to special subgroup number 2, which is 8gp5
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y22
- w1 restricts to
y2.y32
+ y22.y3
- w2 restricts to
0
- w3 restricts to
0
- v restricts to
y34
+ y22.y32
- u1 restricts to
y2.y34
+ y23.y32
- u2 restricts to
y2.y34
+ y24.y3
- t restricts to
y36
+ y2.y35
+ y23.y33
+ y24.y32
- s restricts to
y2.y36
+ y22.y35
+ y24.y33
+ y25.y32
- r restricts to
y38
+ y24.y34
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y22.y32
+ y14.y24
+ y18
Restriction to special subgroup number 3, which is 8gp5
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y32
- x2 restricts to
y32
- w1 restricts to
y2.y32
+ y22.y3
- w2 restricts to
y2.y32
+ y22.y3
- w3 restricts to
0
- v restricts to
y22.y32
+ y24
- u1 restricts to
0
- u2 restricts to
y2.y34
+ y24.y3
- t restricts to
y36
+ y22.y34
+ y23.y33
+ y25.y3
+ y26
- s restricts to
y2.y36
+ y22.y35
- r restricts to
y22.y36
+ y28
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y22.y32
+ y14.y24
+ y18
Restriction to special subgroup number 4, which is 16gp14
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- w1 restricts to
0
- w2 restricts to
y3.y42
+ y32.y4
+ y2.y3.y4
+ y12.y2
- w3 restricts to
y33
+ y2.y42
+ y22.y4
+ y22.y3
- v restricts to
y44
+ y32.y42
+ y22.y42
+ y12.y22
- u1 restricts to
y3.y44
+ y33.y42
+ y22.y3.y42
+ y12.y2.y32
+ y12.y22.y3
+ y14.y2
- u2 restricts to
0
- t restricts to
y46
+ y34.y42
+ y2.y45
+ y2.y3.y44
+ y2.y32.y43
+ y2.y33.y42
+ y22.y44
+ y22.y32.y42
+ y23.y43
+ y23.y3.y42
+ y12.y34
+ y12.y2.y33
+ y12.y22.y42
+ y12.y22.y32
+ y12.y23.y4
+ y12.y23.y3
+ y14.y32
+ y14.y2.y3
- s restricts to
y3.y46
+ y36.y4
+ y2.y3.y45
+ y2.y33.y43
+ y2.y35.y4
+ y22.y33.y42
+ y22.y34.y4
+ y23.y3.y43
+ y23.y33.y4
+ y24.y3.y42
+ y12.y35
+ y12.y2.y32.y42
+ y12.y22.y3.y42
+ y12.y22.y32.y4
+ y12.y22.y33
+ y12.y23.y3.y4
+ y12.y23.y32
+ y12.y24.y3
+ y14.y33
+ y14.y2.y42
+ y14.y2.y32
+ y14.y22.y4
- r restricts to
y48
+ y32.y46
+ y36.y42
+ y37.y4
+ y38
+ y2.y32.y45
+ y2.y33.y44
+ y2.y36.y4
+ y2.y37
+ y22.y33.y43
+ y23.y35
+ y24.y32.y42
+ y24.y33.y4
+ y25.y3.y42
+ y26.y42
+ y26.y3.y4
+ y26.y32
+ y12.y32.y44
+ y12.y34.y42
+ y12.y36
+ y12.y22.y34
+ y12.y23.y3.y42
+ y12.y23.y32.y4
+ y12.y23.y33
+ y12.y24.y42
+ y12.y24.y3.y4
+ y12.y25.y4
+ y12.y25.y3
+ y12.y26
+ y14.y44
+ y14.y32.y42
+ y14.y2.y33
+ y14.y23.y4
+ y14.y23.y3
+ y16.y22
+ y18
(1 + 2t + 3t2
+ 5t3 + 5t4 + 5t5
+ 5t6 + 4t7 + 4t8
+ 3t9 + 3t10 + 3t11
+ 2t12 + 2t13 + t14) /
(1 - t) (1 - t4) (1 - t6) (1 - t8)
Back to the groups of order 128