Small group number 152 of order 64
G is the group 64gp152
The Hall-Senior number of this group is 257.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 10 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x in degree 2
- v in degree 4
- u1 in degree 5
- u2 in degree 5
- u3 in degree 5
- r1 in degree 8
- r2 in degree 8, a regular element
There are 29 minimal relations:
- y2.y3 =
0
- y1.y3 =
y12
- y2.x =
y13
- y2.v =
0
- y1.v =
0
- y13.x =
0
- x.v =
y3.u2
+ y1.u2
- y3.u1 =
y32.v
+ y32.x2
+ y34.x
- y2.u3 =
y1.u1
+ y12.x2
- y2.u2 =
y1.u1
+ y12.x2
- y1.u3 =
0
- x.u1 =
y3.x3
+ y32.u2
+ y33.x2
- v2 =
y32.x3
+ y33.u3
+ y33.u2
+ y12.x3
- v.u3 =
y3.r1
+ y32.x.u2
+ y34.u3
+ y35.v
+ y12.x.u2
- v.u2 =
y3.x4
+ y32.x.u3
+ y32.x.u2
+ y1.x4
+ y12.x.u2
- v.u1 =
y32.x.u2
+ y33.x3
+ y34.u3
+ y12.x.u2
- y2.r1 =
y1.y23.u1
- y1.r1 =
0
- u32 =
y32.r1
+ y32.x4
+ y33.x.u2
+ y35.u3
+ y35.u2
+ y36.x2
+ y12.x4
+ y32.r2
+ y12.r2
- u2.u3 =
x.r1
+ y3.x2.u2
+ y33.x.u3
+ y35.u2
+ y1.x2.u2
+ y12.x4
- u22 =
x5
+ y3.x2.u3
+ y3.x2.u2
+ y12.x4
+ y12.r2
- u1.u3 =
y3.x2.u3
+ y32.r1
+ y33.x.u3
+ y33.x.u2
+ y35.u3
+ y36.v
+ y1.y2.r2
- u1.u2 =
y3.x2.u2
+ y32.x4
+ y33.x.u3
+ y1.y2.r2
- u12 =
y32.x4
+ y34.x3
+ y35.u3
+ y35.u2
+ y36.x2
+ y1.y24.u1
+ y22.r2
- v.r1 =
y3.x3.u3
+ y32.x.r1
+ y32.x5
+ y33.x2.u3
+ y34.x4
+ y35.x.u3
+ y36.x3
+ y37.u3
+ y37.u2
+ y38.v
+ y38.x2
+ y12.x5
+ y34.r2
- u3.r1 =
y3.x2.r1
+ y32.x3.u3
+ y33.x.r1
+ y34.x2.u3
+ y34.x2.u2
+ y35.r1
+ y35.x4
+ y37.x3
+ y39.v
+ y3.v.r2
- u2.r1 =
x4.u3
+ y3.x2.r1
+ y3.x6
+ y32.x3.u3
+ y33.x5
+ y34.x2.u3
+ y35.x4
+ y36.x.u3
+ y36.x.u2
+ y37.x3
+ y38.u2
+ y1.x6
+ y12.x3.u2
+ y33.x.r2
- u1.r1 =
y3.x2.r1
+ y32.x3.u3
+ y33.x5
+ y34.x2.u3
+ y35.x4
+ y36.x.u3
+ y37.x3
+ y38.u3
+ y38.u2
+ y39.v
+ y39.x2
+ y35.r2
+ y1.y24.r2
- r12 =
y32.x3.r1
+ y33.x4.u3
+ y33.x4.u2
+ y34.x2.r1
+ y35.x3.u2
+ y36.x.r1
+ y36.x5
+ y37.x2.u3
+ y37.x2.u2
+ y38.r1
+ y39.x.u3
+ y310.x3
+ y311.u3
+ y312.v
+ y32.x3.r2
+ y33.u3.r2
+ y33.u2.r2
+ y34.v.r2
+ y36.x.r2
+ y12.x3.r2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y12.y2 =
0
- y14 =
0
- y12.u1 =
0
- y13.u2 =
0
This cohomology ring was obtained from a calculation
out to degree 16. The cohomology ring approximation
is stable from degree 16 onwards, and
Benson's tests detect stability from degree 16
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r2
in degree 8
- h2 =
x
+ y32
+ y22
in degree 2
- h3 =
y3
in degree 1
The first
term h1 forms
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, 3, 7, 8.
-
Filter degree type:
-1, -2, -3, -3.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3) is as follows.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y1.y22
in degree 3
-
v
in degree 4
-
u3
in degree 5
-
u2
in degree 5
-
u1
in degree 5
-
y2.u1
in degree 6
-
y1.u1
in degree 6
-
y1.y2.u1
in degree 7
-
r
in degree 8
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y2
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y1.y22
in degree 3
-
y1.h2
in degree 3
-
u1
+ v.h
in degree 5
-
y2.u1
in degree 6
-
y1.u1
+ y1.v.h
in degree 6
-
y1.y2.u1
in degree 7
-
u2.h2
+ v.h3
in degree 7
A basis for AnnR/(h1)(h2) is as follows.