Small group number 217 of order 64
G is the group 64gp217
The Hall-Senior number of this group is 172.
G has 4 minimal generators, rank 3 and exponent 4.
The centre has rank 2.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 8 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- x in degree 2, a regular element
- u1 in degree 5
- u2 in degree 5
- r in degree 8, a regular element
There are 10 minimal relations:
- y3.y4 =
y2.y4
+ y22
+ y1.y2
+ y12
- y1.y4 =
0
- y22.y4 =
y22.y3
+ y23
+ y1.y2.y3
+ y12.y3
- y13.y32 =
0
- y4.u1 =
y13.y2.x
+ y2.y4.x2
+ y22.x2
+ y1.y2.x2
+ y12.x2
- y1.u2 =
y1.y33.x
+ y1.y2.y32.x
+ y12.y2.y3.x
+ y13.y3.x
+ y12.x2
- y32.u2 =
y12.u1
+ y35.x
+ y2.y34.x
+ y1.y2.y33.x
+ y12.y33.x
+ y12.y2.y32.x
+ y1.y32.x2
+ y12.y3.x2
+ y13.x2
- u22 =
y2.y44.u2
+ y2.y49
+ y25.u2
+ y29.y3
+ y42.r
+ y43.x.u2
+ y2.y42.x.u2
+ y2.y47.x
+ y27.y3.x
+ y28.x
+ y36.x2
+ y2.y45.x2
+ y25.y3.x2
+ y26.x2
+ y1.y2.y34.x2
+ y12.y34.x2
+ y2.y43.x3
+ y12.x4
- u1.u2 =
y33.x.u1
+ y2.y32.x.u1
+ y1.y2.y3.x.u1
+ y12.y3.x.u1
+ y12.y36.x
+ y12.y2.x.u1
+ y3.x2.u2
+ y1.x2.u1
+ y34.x3
+ y2.y33.x3
+ y1.y2.y32.x3
+ y12.y32.x3
+ y12.y2.y3.x3
+ y13.y2.x3
+ y1.y3.x4
- u12 =
y1.y34.u1
+ y12.y2.y32.u1
+ y38.x
+ y1.y2.y36.x
+ y12.r
+ y12.y2.x.u1
+ y1.y35.x2
+ y1.y2.y34.x2
+ y12.y34.x2
+ y12.y32.x3
+ y12.y2.y3.x3
+ y32.x4
+ y12.x4
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y22 =
y12.y2
+ y13
- y22.y32 =
y1.y2.y32
+ y12.y32
+ y13.y3
+ y13.y2
- y14 =
0
- y22.u1 =
y1.y2.u1
+ y12.u1
+ y13.y2.y3.x
+ y22.y3.x2
+ y1.y2.y3.x2
+ y12.y3.x2
- y13.u1 =
y13.y3.x2
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 =
x
in degree 2
- h2 =
r
in degree 8
- h3 =
y42
+ y32
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, 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
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
y2.y4
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y1.y2.y3
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y23.y3
in degree 4
-
y12.y2.y3
in degree 4
-
y13.y3
in degree 4
-
y13.y2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y13.y2.y3
in degree 5
-
y4.u2
in degree 6
-
y3.u2
in degree 6
-
y3.u1
in degree 6
-
y2.u2
in degree 6
-
y2.u1
in degree 6
-
y1.u1
in degree 6
-
y2.y4.u2
in degree 7
-
y2.y3.u2
in degree 7
-
y2.y3.u1
in degree 7
-
y22.u2
in degree 7
-
y1.y3.u1
in degree 7
-
y1.y2.u1
in degree 7
-
y12.u1
in degree 7
-
y23.u2
in degree 8
-
y1.y2.y3.u1
in degree 8
-
y12.y3.u1
in degree 8
-
y12.y2.u1
in degree 8
-
y12.y2.y3.u1
in degree 9
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y13
in degree 3
-
y13.y3
in degree 4
-
y13.y2
in degree 4
-
y13.y2.y3
in degree 5