Small group number 35 of order 64
G is the group 64gp35
The Hall-Senior number of this group is 253.
G has 2 minimal generators, rank 3 and exponent 4.
The centre has rank 1.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 12 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- w in degree 3
- u1 in degree 5, a nilpotent element
- u2 in degree 5
- t1 in degree 6
- t2 in degree 6
- s in degree 7
- r in degree 8, a regular element
There are 44 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x3 =
y1.x1
- y2.x1 =
y1.x1
- y1.x2 =
y23
- x1.x3 =
x12
- y2.w =
0
- y1.w =
0
- y23.x2 =
0
- w2 =
x22.x3
- y1.u2 =
0
- y2.u1 =
y22.x22
- y1.u1 =
0
- x3.u2 =
x2.x3.w
+ x1.x2.w
+ x12.w
+ y1.t1
- x1.u2 =
x12.w
+ y1.t1
- x3.u1 =
y1.t2
+ y1.x13
- x2.u1 =
y2.x23
+ y22.u2
- x1.u1 =
y1.t1
+ y1.x13
- y2.t2 =
y1.t1
+ y22.u2
- y2.t1 =
y2.x23
+ y1.t1
- w.u2 =
x23.x3
+ x1.x23
+ x12.x22
- x3.t1 =
x23.x3
+ x1.t1
+ x1.x23
- x1.t2 =
x1.t1
+ x1.x23
- w.u1 =
0
- y2.s =
y2.x2.u2
+ y22.x23
- y1.s =
0
- w.t2 =
x3.s
+ x33.w
+ x2.x32.w
+ x1.x22.w
+ x12.x2.w
+ y1.x3.t2
+ y1.x1.t1
- w.t1 =
x23.w
+ x1.s
+ x1.x22.w
+ x13.w
- u22 =
x24.x3
+ x1.x24
+ x13.x22
+ y2.x22.u2
+ y22.r
- w.s =
x22.t2
+ x22.x33
+ x23.x32
+ x1.x24
+ x12.x23
+ y2.x22.u2
+ y22.x24
- u1.u2 =
y2.x22.u2
- u12 =
y22.x24
- u2.t2 =
x2.x3.s
+ x2.x33.w
+ x22.x32.w
+ x1.x2.s
+ x12.s
+ x13.x2.w
+ x14.w
+ y1.x15
+ y1.x1.r
+ y23.r
- u2.t1 =
x23.u2
+ x12.s
+ x12.x22.w
+ x14.w
+ y1.x15
+ y22.x22.u2
+ y1.x1.r
- u1.t2 =
y1.x32.t2
+ y1.x15
+ y22.x22.u2
+ y1.x3.r
- u1.t1 =
y2.x25
+ y1.x12.t1
+ y1.x15
+ y22.x22.u2
+ y1.x1.r
- t22 =
x33.t2
+ x2.x32.t2
+ x2.x35
+ x13.t1
+ x14.x22
+ x15.x2
+ x16
+ x32.r
- t1.t2 =
x23.t2
+ x12.x2.t1
+ x12.x24
+ x13.x23
+ x14.x22
+ x16
+ x12.r
- t12 =
x26
+ x12.x2.t1
+ x12.x24
+ x13.x23
+ x14.x22
+ x16
+ x12.r
- u2.s =
x23.t2
+ x23.t1
+ x23.x33
+ x24.x32
+ x26
+ x1.x22.t1
+ x1.x25
+ x13.x23
+ x14.x22
+ y2.x23.u2
+ y22.x2.r
- u1.s =
y2.x23.u2
+ y22.x25
- t2.s =
x2.x34.w
+ x1.x22.s
+ x1.x24.w
+ x12.x2.s
+ x13.s
+ x13.x22.w
+ y1.x33.t2
+ y1.x13.t1
+ y22.x23.u2
+ x3.w.r
+ y1.x32.r
+ y1.x12.r
- t1.s =
x23.s
+ x1.x22.s
+ x1.x24.w
+ x12.x2.s
+ x13.s
+ x13.x22.w
+ x14.x2.w
+ y22.x23.u2
+ x1.w.r
- s2 =
x22.x32.t2
+ x22.x35
+ x23.x3.t2
+ x23.x34
+ x24.x33
+ x1.x26
+ x12.x22.t1
+ x14.x23
+ x15.x22
+ y2.x24.u2
+ y22.x26
+ x22.x3.r
+ y22.x22.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 14. The cohomology ring approximation
is stable from degree 14 onwards, and
Benson's tests detect stability from degree 14
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x3
+ x2
in degree 2
- h3 =
x3
in degree 2
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, 9.
-
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
-
x1
in degree 2
-
y22
in degree 2
-
w
in degree 3
-
u2
in degree 5
-
x1.w
in degree 5
-
u1
in degree 5
-
t2
in degree 6
-
t1
in degree 6
-
y2.u2
in degree 6
-
s
in degree 7
-
x1.s
in degree 9
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y22
in degree 2
-
y2.h
in degree 3
-
y1.h
in degree 3
-
y2.u2
in degree 6
-
u2.h
+ w.h2
in degree 7
-
u1.h
in degree 7
A basis for AnnR/(h1)(h2) is as follows.