Small group number 30 of order 64
G is the group 64gp30
The Hall-Senior number of this group is 133.
G has 2 minimal generators, rank 3 and exponent 16.
The centre has rank 1.
The 3 maximal subgroups are:
32gp17 (2x), 32gp37.
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 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
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- w in degree 3, a nilpotent element
- v in degree 4, a nilpotent element
- u1 in degree 5, a nilpotent element
- u2 in degree 5
- t in degree 6, a nilpotent element
- s in degree 7, a nilpotent element
- r1 in degree 8, a nilpotent element
- r2 in degree 8, a regular element
There are 52 minimal relations:
- y1.y2 =
0
- y12 =
0
- y2.x1 =
y1.x2
- y1.x1 =
0
- x1.x2 =
y2.w
- x12 =
0
- y1.w =
0
- y1.x22 =
0
- x1.w =
0
- y1.v =
0
- y2.u1 =
0
- y1.u2 =
0
- w2 =
0
- x1.v =
0
- y1.u1 =
0
- x2.u1 =
y2.t
- x1.u2 =
y2.t
- w.v =
0
- x1.u1 =
0
- y1.t =
0
- w.u2 =
x22.v
+ y2.s
+ y2.x22.w
- x2.t =
x22.v
+ y2.s
- v2 =
0
- w.u1 =
0
- x1.t =
0
- y1.s =
0
- v.u2 =
x23.w
+ y2.r1
- v.u1 =
0
- w.t =
0
- x1.s =
0
- y1.r1 =
0
- u22 =
x25
+ y2.x22.u2
+ y22.x24
+ y24.x23
+ y25.u2
+ y26.x22
+ y22.r2
+ y22.r1
+ y24.x2.v
- u1.u2 =
y2.x23.w
- u12 =
0
- v.t =
0
- w.s =
0
- x1.r1 =
0
- u2.t =
x24.w
+ y1.x2.r2
- u1.t =
0
- v.s =
0
- w.r1 =
0
- u2.s =
x22.r1
+ y2.x24.w
+ y2.w.r2
- t2 =
0
- u1.s =
0
- v.r1 =
0
- u2.r1 =
x23.s
+ y2.x22.r1
+ y2.x24.v
+ y24.x2.s
+ y25.r1
+ y26.s
+ y2.v.r2
- t.s =
0
- u1.r1 =
0
- s2 =
0
- t.r1 =
0
- s.r1 =
0
- r12 =
0
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y22.w =
y1.x22
- y22.t =
0
- y22.x22.v =
y23.s
Essential ideal:
There is one minimal generator:
Nilradical:
There are 8 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 16. The cohomology ring approximation
is stable from degree 16 onwards, and
Carlson's tests detect stability from degree 16
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r2
in degree 8
- h2 =
y22
in degree 2
- h3 =
x2
in degree 2
The first
term h1 forms
a regular sequence of maximum length.
The remaining
2 terms h2, h3 are all
annihilated by the class
y1.x2.
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 1 as a module over the polynomial algebra
on h1.
These free generators are:
The essential ideal squares to zero.
A basis for R/(h1, h2, h3) 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
-
x1
in degree 2
-
w
in degree 3
-
v
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y2.v
in degree 5
-
y2.u2
in degree 6
-
t
in degree 6
-
s
in degree 7
-
r1
in degree 8
-
y2.r1
in degree 9
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
y1.x2
in degree 3
-
y2.t
+ y2.x2.v
in degree 7
A basis for AnnR/(h1)(h2, h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
A basis for AnnR/(h1)(h2)
/ h3 AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
y1
in degree 1
-
x1
in degree 2
-
w
in degree 3
-
u1
in degree 5
-
t
in degree 6
Restriction to maximal subgroup number 1, which is 32gp37
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- x1 restricts to
y1.y2
- x2 restricts to
y2.y3
+ y32
+ y1.y3
- w restricts to
y1.y32
- v restricts to
y2.w
+ y1.y33
- u1 restricts to
y1.y2.y33
- u2 restricts to
y2.v
+ y22.y33
+ y35
+ y1.v
- t restricts to
y1.y2.v
+ y1.y35
- s restricts to
y22.y32.w
+ y1.y32.v
+ y34.w
+ y1.y36
- r1 restricts to
y2.w.v
+ y22.y33.w
+ y1.y33.v
+ y35.w
+ y1.y37
- r2 restricts to
y24.v
+ v2
+ y26.y32
+ y22.y32.v
+ y24.y34
+ y34.v
+ y22.y36
+ y38
+ y2.w.v
+ y24.y3.w
+ y23.y32.w
+ y22.y33.w
+ y1.y2.y32.v
+ y1.y33.v
+ y35.w
+ y1.y2.y36
Restriction to maximal subgroup number 2, which is 32gp17
- y1 restricts to
y1
- y2 restricts to
0
- x1 restricts to
y1.y2
- x2 restricts to
y22
+ y1.y2
- w restricts to
w
- v restricts to
y2.w
- u1 restricts to
y1.v
- u2 restricts to
y25
- t restricts to
y23.w
+ y1.y2.v
- s restricts to
y24.w
+ w.v
- r1 restricts to
y25.w
+ y2.w.v
- r2 restricts to
y28
+ v2
+ y2.w.v
Restriction to maximal subgroup number 3, which is 32gp17
- y1 restricts to
y1
- y2 restricts to
y1
- x1 restricts to
y1.y2
- x2 restricts to
y22
- w restricts to
w
- v restricts to
y2.w
- u1 restricts to
y1.v
- u2 restricts to
y25
+ y22.w
+ y1.v
- t restricts to
y23.w
+ y1.y2.v
- s restricts to
y24.w
+ w.v
- r1 restricts to
y25.w
+ y2.w.v
- r2 restricts to
y28
+ v2
+ y2.w.v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- w restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
y35
+ y22.y33
+ y12.y23
+ y14.y2
- t restricts to
0
- s restricts to
0
- r1 restricts to
0
- r2 restricts to
y38
+ y22.y36
+ y24.y34
+ y26.y32
+ y12.y22.y34
+ y12.y24.y32
+ y12.y26
+ y14.y34
+ y14.y22.y32
+ 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
- w restricts to
0
- v restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- t restricts to
0
- s restricts to
0
- r1 restricts to
0
- r2 restricts to
y8
(1 + 2t + t2
+ t5 + 2t6 + t7) /
(1 - t2)2 (1 - t8)
Back to the groups of order 64