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.
The 3 maximal subgroups are:
32gp35, 32gp8 (2x).
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 =
y23.w1
- y1.u2 =
y23.w1
- y1.u1 =
0
- w1.v =
y2.t
+ x1.u2
+ y23.v
- 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
+ y23.u2
- w1.u2 =
y2.s
+ y22.t
+ y23.u2
- w1.u1 =
y23.u2
- y1.s =
0
- x2.w22 =
y23.u2
- v.u1 =
w2.t
+ x1.s
+ y22.s
- w1.t =
y2.v2
+ x1.s
- 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
+ y23.s
- u1.u2 =
w2.s
+ w22.v
- u12 =
w22.v
- w1.s =
y2.v.u2
+ y22.v2
- u2.t =
v.s
+ w2.v2
+ y2.v.t
+ y1.x2.r
+ w22.u2
+ y23.v2
- u1.t =
w2.v2
+ w22.u2
+ y22.v.u2
+ y23.v2
- t2 =
v3
+ w22.t
+ y1.w2.r
+ y23.v.u2
- u2.s =
w22.t
+ y22.v.t
+ y2.w1.r
- u1.s =
w2.v.u2
+ w22.t
+ y23.v.u2
- t.s =
v2.u2
+ w2.v.t
+ y2.v3
+ y23.v.t
- s2 =
w2.v.s
+ w22.v2
+ y2.v2.u2
+ 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
Essential ideal:
There is one minimal generator:
Nilradical:
There are 9 minimal generators:
-
y2
-
y1
-
x2
-
x1
-
w2
-
w1
-
u2
-
u1
-
s
This cohomology ring was obtained from a calculation
out to degree 14. The cohomology ring approximation
is stable from degree 14 onwards, and
Carlson'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 remaining
term h2 is
annihilated by the class
y1.
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.
The ideal of essential classes is
free of rank 2 as a module over the polynomial algebra
on h1.
These free generators are:
- G1 =
y1.x2
in degree 3
- G2 =
y23.w1
in degree 6
The essential ideal squares to zero.
A basis for R/(h1, h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 12.
-
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.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y1
in degree 1
-
y1.x2
in degree 3
-
y1.w2
in degree 4
-
y23.w1
in degree 6
Restriction to maximal subgroup number 1, which is 32gp35
- y1 restricts to
0
- y2 restricts to
y3
+ y1
- x1 restricts to
y1.y3
- x2 restricts to
y2.y3
- w1 restricts to
y3.x
+ y1.x
+ y12.y3
+ y12.y2
- w2 restricts to
y3.x
+ y12.y2
- v restricts to
x2
+ y1.y3.x
- u1 restricts to
y3.x2
- u2 restricts to
y3.v
+ y1.v
+ y2.x2
+ y1.x2
+ y1.y2.y3.x
+ y12.y2.x
- t restricts to
x3
+ y1.y3.v
+ y2.y3.x2
+ y1.y2.x2
+ y12.x2
+ y12.y2.y3.x
- s restricts to
y3.x.v
+ y1.x.v
+ y2.x3
+ y12.y2.v
- r restricts to
v2
+ x2.v
+ y12.x.v
+ y2.y3.x3
+ y1.y3.x3
+ y12.x3
+ y12.y2.y3.v
Restriction to maximal subgroup number 2, which is 32gp8
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
x1
- x2 restricts to
y22
- w1 restricts to
y2.x1
- w2 restricts to
w
+ y2.x2
+ y2.x1
+ y23
- v restricts to
x22
+ y22.x2
+ y22.x1
- u1 restricts to
u1
+ x2.w
+ y2.x22
+ y22.w
- u2 restricts to
y2.x22
+ y22.w
- t restricts to
x23
+ t
- s restricts to
x22.w
+ y2.t
- r restricts to
x24
+ r
+ y2.x22.w
+ y22.x23
+ y22.t
Restriction to maximal subgroup number 3, which is 32gp8
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
x1
+ y22
- x2 restricts to
x1
- w1 restricts to
y2.x1
+ y23
- w2 restricts to
w
+ y2.x1
- v restricts to
x22
+ y22.x2
- u1 restricts to
u1
+ x2.w
+ y22.w
- u2 restricts to
u1
+ x2.w
+ y2.x22
- t restricts to
x23
+ t
+ y2.u2
+ y22.x22
+ y23.w
- s restricts to
y2.x23
+ y2.t
+ y22.u2
+ y22.x2.w
- r restricts to
x24
+ r
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y24
- u1 restricts to
0
- u2 restricts to
0
- t restricts to
y26
- s restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- t restricts to
0
- s restricts to
0
- r restricts to
y8
(1 + 2t + 3t2
+ 4t3 + 3t4 + 3t5
+ 3t6 + 2t7 + 2t8
+ t9) /
(1 - t4) (1 - t8)
Back to the groups of order 64