Small group number 37 of order 64
G is the group 64gp37
The Hall-Senior number of this group is 255.
G has 2 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 2.
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, a nilpotent element
- x2 in degree 2, a nilpotent element
- w1 in degree 3, a nilpotent element
- w2 in degree 3, a nilpotent element
- v in degree 4
- u1 in degree 5, a nilpotent element
- u2 in degree 5, a nilpotent element
- t in degree 6
- s in degree 7, a nilpotent element
- r in degree 8, a regular element
There are 47 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x2 =
0
- y2.x1 =
y1.x2
- y1.x1 =
0
- x22 =
y2.w2
- x1.x2 =
y2.w2
+ y1.w2
- x12 =
y1.w2
- y1.w1 =
0
- y24 =
0
- y1.v =
0
- x2.w1 =
0
- x1.w2 =
y22.w1
- x1.w1 =
0
- x1.v =
w22
+ y23.w1
- w1.w2 =
0
- w12 =
y22.v
+ y23.w1
- y2.u1 =
y22.v
+ y23.w1
- y1.u2 =
y23.w1
- y1.u1 =
0
- w1.v =
y2.t
+ x1.u2
- y1.t =
0
- x2.u2 =
y23.v
- x2.u1 =
x1.u2
+ y23.v
- x1.u1 =
y23.v
- x2.t =
w2.u2
+ w2.u1
- x1.t =
w2.u1
- w1.u2 =
y2.s
+ y23.u2
- w1.u1 =
y22.t
+ y23.u2
- y1.s =
0
- x2.w22 =
y23.u2
- v.u1 =
w2.t
+ y2.v2
+ x1.s
+ y22.s
- w1.t =
y2.v2
+ x1.s
+ y23.t
- x2.s =
x1.s
+ y23.t
- x2.w2.v =
x1.s
- x2.v2 =
w2.s
- u22 =
w2.s
+ y2.v.u2
+ y22.v2
+ y22.r
- u1.u2 =
w2.s
+ w22.v
+ y2.v.u2
- u12 =
w22.v
+ y22.v2
- w1.s =
y2.v.u2
- u2.t =
v.s
+ w2.v2
+ w22.u2
+ y22.v.u2
+ y23.v2
+ y1.x2.r
- u1.t =
w2.v2
+ y2.v.t
+ w22.u2
+ y22.v.u2
- t2 =
v3
+ w22.t
+ y23.v.u2
+ y1.w2.r
- u2.s =
w22.t
+ y2.v.s
+ y22.v.t
+ y23.v.u2
+ y2.w1.r
- u1.s =
w2.v.u2
+ w22.t
+ y2.v.s
- t.s =
v2.u2
+ w2.v.t
+ y22.v.s
+ y23.v.t
- s2 =
w2.v.s
+ w22.v2
+ y2.v2.u2
+ y22.v3
+ y23.v.s
+ y22.v.r
+ y23.w1.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y22.w2 =
0
- y1.x2.w2 =
y23.w1
- y2.w22 =
0
- y1.w22 =
0
- y2.w2.v =
x2.w22
- w23 =
y23.t
- y2.w2.u2 =
0
- y2.w2.t =
y23.s
- w22.u1 =
y23.v2
- y2.w2.s =
0
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 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
v
in degree 4
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, 6, 10.
-
Filter degree type:
-1, -2, -2.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2) is as follows.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
x1
in degree 2
-
y22
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y1.x2
in degree 3
-
y23
in degree 3
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y1.w2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
x2.w2
in degree 5
-
y22.w1
in degree 5
-
t
in degree 6
-
y2.u2
in degree 6
-
y23.w1
in degree 6
-
s
in degree 7
-
x1.u2
in degree 7
-
y22.u2
in degree 7
-
w2.u2
in degree 8
-
w2.u1
in degree 8
-
y2.s
in degree 8
-
y22.s
in degree 9
-
y23.s
in degree 10
A basis for AnnR/(h1)(h2) is as follows.
-
y1
in degree 1
-
y1.x2
in degree 3
-
y1.w2
in degree 4
-
y23.w1
in degree 6