Small group number 163 of order 64
G is the group 64gp163
The Hall-Senior number of this group is 223.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 2.
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 10 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- w1 in degree 3, a nilpotent element
- w2 in degree 3
- w3 in degree 3
- w4 in degree 3
- v1 in degree 4, a regular element
- v2 in degree 4, a regular element
- u in degree 5
There are 27 minimal relations:
- y2.y3 =
y12
- y1.y3 =
0
- y1.y22 =
0
- y12.y2 =
0
- y3.w4 =
y3.w3
+ y3.w2
+ y1.w1
- y2.w4 =
y24
+ y1.w1
- y2.w2 =
y2.w1
+ y1.w3
- y3.w1 =
0
- y1.w4 =
0
- y1.w2 =
0
- y1.y2.w3 =
0
- w42 =
y26
+ y32.v1
- w3.w4 =
y3.u
+ y23.w3
+ y12.v2
- w32 =
y26
+ y23.w1
+ y32.v2
+ y32.v1
+ y22.v1
+ y12.v2
- w2.w4 =
y3.u
+ y23.w1
+ y32.v1
+ y12.v2
+ y12.v1
- w2.w3 =
y3.u
+ w1.w3
+ y32.v2
+ y32.v1
+ y1.y2.v1
+ y12.v2
+ y12.v1
- w22 =
y32.v2
- y2.u =
y26
+ w1.w3
+ y23.w1
+ y22.v2
+ y22.v1
+ y1.y2.v2
+ y12.v1
- w1.w4 =
y23.w1
- w1.w2 =
0
- y1.u =
y1.y2.v2
+ y1.y2.v1
+ y12.v2
+ y12.v1
- w12 =
y12.v1
- w4.u =
y28
+ y22.w1.w3
+ y25.w1
+ y3.w3.v1
+ y24.v2
+ y24.v1
+ y1.w1.v2
- w3.u =
y25.w3
+ y22.w1.w3
+ y25.w1
+ y3.w3.v2
+ y3.w3.v1
+ y3.w2.v2
+ y3.w2.v1
+ y2.w3.v2
+ y2.w3.v1
+ y2.w1.v1
+ y1.w3.v2
+ y1.w1.v2
- w2.u =
y25.w1
+ y3.w3.v2
+ y3.w2.v2
+ y3.w2.v1
+ y2.w1.v2
+ y2.w1.v1
+ y1.w3.v2
+ y1.w3.v1
+ y1.w1.v2
- w1.u =
y25.w1
+ y2.w1.v2
+ y2.w1.v1
+ y1.w1.v2
+ y1.w1.v1
- u2 =
y210
+ y32.v1.v2
+ y32.v12
+ y22.v22
+ y22.v12
+ y12.v22
+ y12.v12
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y13 =
0
- y1.y2.w1 =
y12.w3
- y12.w1 =
0
- y1.w1.w3 =
0
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 10 onwards, and
Benson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v1
in degree 4
- h2 =
v2
in degree 4
- h3 =
y32
+ y22
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The first
2 terms h1, h2 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, 5, 7.
-
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
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w4
in degree 3
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
y3.w3
in degree 4
-
y3.w2
in degree 4
-
y2.w3
in degree 4
-
y2.w1
in degree 4
-
y1.w3
in degree 4
-
y1.w1
in degree 4
-
u
in degree 5
-
y12.w3
in degree 5
-
y3.u
in degree 6
-
w1.w3
in degree 6
-
y2.w1.w3
in degree 7
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y1
in degree 1
-
y1.y2
in degree 2
-
y12
in degree 2
-
y1.w3
in degree 4
-
y1.w1
in degree 4
-
y12.w3
in degree 5