Small group number 136 of order 64
G is the group 64gp136
The Hall-Senior number of this group is 262.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
There are 4 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3, 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
- x1 in degree 2
- x2 in degree 2
- u1 in degree 5
- u2 in degree 5
- u3 in degree 5
- t in degree 6
- r in degree 8, a regular element
There are 27 minimal relations:
- y1.y3 =
0
- y1.y2 =
0
- y3.x1 =
y13
- y1.x2 =
0
- x22 =
y2.y3.x2
+ y22.x2
- y13.x1 =
0
- x12.x2 =
y2.u2
- y3.u2 =
0
- y3.u1 =
y2.u3
- y1.u3 =
0
- y1.u1 =
0
- x2.u3 =
y3.t
+ y32.u3
+ y2.y3.u3
- x2.u2 =
y22.u2
- x2.u1 =
y2.t
+ y2.y3.u3
+ y22.u3
+ y23.x1.x2
- x1.u3 =
y12.u2
- y1.t =
y12.u2
- x2.t =
y32.t
+ y33.u3
+ y2.y32.u3
+ y22.t
+ y22.y3.u3
+ y23.u3
- u32 =
y2.y34.u3
+ y22.y33.u3
+ y24.y3.u3
+ y32.r
- u2.u3 =
0
- u22 =
y2.x12.u2
+ y1.x12.u2
+ y12.r
- u1.u3 =
y22.y33.u3
+ y23.y32.u3
+ y25.u3
+ y2.y3.r
- u1.u2 =
x12.t
+ y23.x1.u2
+ y1.x12.u2
- u12 =
y22.x1.t
+ y22.x14
+ y23.x1.u2
+ y23.y32.u3
+ y24.y3.u3
+ y25.u1
+ y26.x1.x2
+ y22.r
- u3.t =
y2.y34.t
+ y22.y33.t
+ y24.y3.t
+ y3.x2.r
+ y33.r
+ y2.y32.r
- u2.t =
y2.x12.t
+ y12.x12.u2
+ y13.r
- u1.t =
y22.x12.u2
+ y22.y33.t
+ y23.y32.t
+ y24.x1.u2
+ y25.t
+ y26.u2
+ y2.x2.r
+ y2.y32.r
+ y22.y3.r
- t2 =
y2.y36.u3
+ y22.y34.t
+ y23.x12.u2
+ y24.x1.t
+ y24.y32.t
+ y24.y33.u3
+ y25.x1.u2
+ y25.y3.t
+ y26.t
+ y26.y3.u3
+ y27.u3
+ y27.u2
+ y34.r
+ y2.y3.x2.r
+ y22.x2.r
+ y22.y32.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 12 onwards, and
Benson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x1
+ y32
+ y2.y3
+ y22
in degree 2
- h3 =
y3
+ y2
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
-
x2
in degree 2
-
y22
in degree 2
-
y12
in degree 2
-
y2.x2
in degree 3
-
y13
in degree 3
-
y22.x2
in degree 4
-
u3
in degree 5
-
u2
in degree 5
-
u1
in degree 5
-
t
in degree 6
-
y2.u1
in degree 6
-
y1.u2
in degree 6
-
y2.t
in degree 7
-
y12.u2
in degree 7
-
y22.t
in degree 8
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1
in degree 1
-
y12
in degree 2
-
y13
in degree 3
-
u2
+ y22.x2.h
+ y2.x2.h2
+ x2.h3
in degree 5
-
y1.u2
in degree 6
-
y12.u2
in degree 7
A basis for AnnR/(h1)(h2) is as follows.