Small group number 136 of order 64
G is the group 64gp136
The Hall-Senior number of this group is 262.
G has 3 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
32gp11 (2x), 32gp31, 32gp44 (2x), E32+, 32gp7.
There are 4 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 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
- x1 in degree 2
- x2 in degree 2
- u1 in degree 5
- u2 in degree 5
- u3 in degree 5
- t in degree 6
- r in degree 8, a regular element
There are 27 minimal relations:
- y1.y3 =
0
- y1.y2 =
0
- y3.x1 =
y13
- y1.x2 =
0
- x22 =
y2.y3.x2
+ y22.x2
- y13.x1 =
0
- x12.x2 =
y2.u2
- y3.u2 =
0
- y3.u1 =
y2.u3
- y1.u3 =
0
- y1.u1 =
0
- x2.u3 =
y3.t
+ y32.u3
+ y2.y3.u3
- x2.u2 =
y22.u2
- x2.u1 =
y2.t
+ y2.y3.u3
+ y22.u3
+ y23.x1.x2
- x1.u3 =
y12.u2
- y1.t =
y12.u2
- x2.t =
y32.t
+ y33.u3
+ y2.y32.u3
+ y22.t
+ y22.y3.u3
+ y23.u3
- u32 =
y2.y34.u3
+ y22.y33.u3
+ y24.y3.u3
+ y32.r
- u2.u3 =
0
- u22 =
y2.x12.u2
+ y1.x12.u2
+ y12.r
- u1.u3 =
y22.y33.u3
+ y23.y32.u3
+ y25.u3
+ y2.y3.r
- u1.u2 =
x12.t
+ y23.x1.u2
+ y1.x12.u2
- u12 =
y22.x1.t
+ y22.x14
+ y23.x1.u2
+ y23.y32.u3
+ y24.y3.u3
+ y25.u1
+ y26.x1.x2
+ y22.r
- u3.t =
y2.y34.t
+ y22.y33.t
+ y24.y3.t
+ y3.x2.r
+ y33.r
+ y2.y32.r
- u2.t =
y2.x12.t
+ y12.x12.u2
+ y13.r
- u1.t =
y22.x12.u2
+ y22.y33.t
+ y23.y32.t
+ y24.x1.u2
+ y25.t
+ y26.u2
+ y2.x2.r
+ y2.y32.r
+ y22.y3.r
- t2 =
y2.y36.u3
+ y22.y34.t
+ y23.x12.u2
+ y24.x1.t
+ y24.y32.t
+ y24.y33.u3
+ y25.x1.u2
+ y25.y3.t
+ y26.t
+ y26.y3.u3
+ y27.u3
+ y27.u2
+ y34.r
+ y2.y3.x2.r
+ y22.x2.r
+ y22.y32.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
Essential ideal:
Zero ideal
Nilradical:
There is one minimal generator:
This cohomology ring was obtained from a calculation
out to degree 16. The cohomology ring approximation
is stable from degree 12 onwards, and
Carlson's tests detect stability from degree 12
onwards.
This cohomology ring has dimension 3 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
y22
in degree 2
- h3 =
x1
+ y32
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
y13.
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
the zero ideal.
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
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x2
in degree 2
-
y32
in degree 2
-
y2.y3
in degree 2
-
y12
in degree 2
-
y3.x2
in degree 3
-
y2.x2
in degree 3
-
y2.y32
in degree 3
-
y13
in degree 3
-
y32.x2
in degree 4
-
y2.y3.x2
in degree 4
-
u3
in degree 5
-
u2
in degree 5
-
u1
in degree 5
-
y2.y32.x2
in degree 5
-
t
in degree 6
-
y3.u3
in degree 6
-
y2.u3
in degree 6
-
y2.u1
in degree 6
-
y1.u2
in degree 6
-
y3.t
in degree 7
-
y2.t
in degree 7
-
y2.y3.u3
in degree 7
-
y12.u2
in degree 7
-
y32.t
in degree 8
-
y2.y3.t
in degree 8
-
y2.y32.t
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.
-
y13
in degree 3
-
y2.x1.x2
in degree 5
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
-
y12
in degree 2
-
y13
in degree 3
-
y1.u2
in degree 6
-
y12.u2
in degree 7
Restriction to maximal subgroup number 1, which is 32gp49
- y1 restricts to
0
- y2 restricts to
y4
+ y3
+ y1
- y3 restricts to
y1
- x1 restricts to
y2.y4
+ y2.y3
+ y22
- x2 restricts to
y32
+ y1.y3
- u1 restricts to
y35
+ y2.y34
+ y22.y33
+ y12.y22.y3
+ y14.y3
+ y4.v
+ y3.v
+ y1.v
- u2 restricts to
y22.y33
+ y24.y3
+ y12.y22.y3
- u3 restricts to
y1.y34
+ y12.y33
+ y13.y32
+ y13.y2.y3
+ y14.y3
+ y1.v
- t restricts to
y36
+ y1.y35
+ y12.y34
+ y13.y33
+ y32.v
+ y1.y4.v
- r restricts to
y2.y37
+ y23.y35
+ y24.y44
+ y24.y34
+ y25.y33
+ y26.y32
+ y28
+ y1.y37
+ y1.y26.y3
+ y12.y36
+ y12.y25.y3
+ y13.y35
+ y14.y23.y3
+ y15.y33
+ y16.y32
+ y17.y3
+ y44.v
+ y34.v
+ y2.y33.v
+ y22.y32.v
+ y1.y22.y3.v
+ y12.y42.v
+ y12.y32.v
+ y12.y2.y3.v
+ y13.y4.v
+ y13.y3.v
+ y14.v
+ v2
Restriction to maximal subgroup number 2, which is 32gp31
- y1 restricts to
y1
- y2 restricts to
y3
+ y1
- y3 restricts to
0
- x1 restricts to
x
- x2 restricts to
y32
+ y22
+ y1.y2
- u1 restricts to
y35
+ y22.w
+ x.w
+ y3.v
+ y23.x
+ y3.x2
+ y2.x2
+ y1.v
+ y1.x2
- u2 restricts to
y3.x2
+ y2.x2
+ y1.v
- u3 restricts to
y12.y2.x
- t restricts to
y36
+ y26
+ y3.x.w
+ y32.v
+ y34.x
+ y2.x.w
+ y22.v
+ y24.x
+ y32.x2
+ y22.x2
+ y1.y2.v
+ y12.v
- r restricts to
y33.x.w
+ y34.v
+ y24.v
+ y26.x
+ v2
+ y3.x2.w
+ y32.x.v
+ y22.x.v
+ y24.x2
+ x2.v
+ y32.x3
+ y22.x3
+ x4
+ y1.y2.x.v
+ y1.y2.x3
+ y12.x.v
Restriction to maximal subgroup number 3, which is 32gp7
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x1
- x2 restricts to
x2
+ y22
- u1 restricts to
y2.v1
+ y2.x12
+ y23.x1
+ y25
+ y2.v2
+ y1.v2
- u2 restricts to
x1.w2
+ y2.x12
+ y1.v2
- u3 restricts to
y1.v2
- t restricts to
y24.x2
+ y26
+ x2.v2
+ y22.v2
- r restricts to
x14
+ y22.x1.v1
+ y24.x12
+ y26.x1
+ y2.w2.v2
+ y22.x2.v2
+ y22.x1.v2
+ y24.v2
+ v22
Restriction to maximal subgroup number 4, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
y2.y3
- u1 restricts to
y12.y23
- u2 restricts to
u1
+ y1.y24
- u3 restricts to
u2
- t restricts to
y3.u2
+ y2.u2
+ y1.u1
+ y12.y24
- r restricts to
y28
+ r
+ y23.u1
+ y12.y26
+ y12.y2.u1
Restriction to maximal subgroup number 5, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x2
- x2 restricts to
x1
- u1 restricts to
y1.v
- u2 restricts to
x2.w
+ y1.x22
+ y1.v
- u3 restricts to
y2.v
+ y1.v
- t restricts to
y22.v
+ x1.v
+ y1.x2.w
- r restricts to
x24
+ v2
+ y1.x22.w
+ y1.w.v
Restriction to maximal subgroup number 6, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y3
- x1 restricts to
y22
+ y1.y2
+ y12
- x2 restricts to
y2.y3
- u1 restricts to
u2
+ y2.y34
+ y12.y23
- u2 restricts to
u1
- u3 restricts to
u2
+ y2.y34
- t restricts to
y2.u2
+ y1.u1
- r restricts to
y33.u2
+ y2.y37
+ y28
+ r
+ y23.u1
+ y12.y26
+ y12.y2.u1
Restriction to maximal subgroup number 7, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- x1 restricts to
x2
- x2 restricts to
x1
- u1 restricts to
y2.v
+ y23.x1
- u2 restricts to
x2.w
+ y1.x22
+ y1.v
- u3 restricts to
y2.v
+ y23.x1
+ y1.v
- t restricts to
x1.v
+ y1.x2.w
- r restricts to
x24
+ y24.v
+ v2
+ y26.x1
+ y1.x22.w
+ y1.w.v
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y22
- u1 restricts to
y23.y32
+ y24.y3
+ y25
+ y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u2 restricts to
y2.y34
+ y23.y32
- u3 restricts to
0
- t restricts to
y26
+ y1.y23.y32
+ y1.y24.y3
+ y12.y22.y32
+ y12.y23.y3
+ y12.y24
+ y14.y22
- r restricts to
y38
+ y22.y36
+ y23.y35
+ y24.y34
+ y25.y33
+ y27.y3
+ y1.y23.y34
+ y1.y26.y3
+ y12.y26
+ y14.y34
+ y14.y23.y3
+ y18
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
0
- u1 restricts to
y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u2 restricts to
0
- u3 restricts to
0
- t restricts to
0
- r restricts to
y38
+ y24.y34
+ y1.y25.y32
+ y1.y26.y3
+ y12.y22.y34
+ y12.y25.y3
+ y12.y26
+ y14.y34
+ y14.y22.y32
+ y18
Restriction to maximal elementary abelian number 3, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
0
- u1 restricts to
y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u2 restricts to
0
- u3 restricts to
y1.y2.y33
+ y1.y22.y32
+ y12.y33
+ y12.y2.y32
+ y12.y22.y3
+ y14.y3
- t restricts to
y1.y2.y34
+ y1.y23.y32
+ y12.y34
+ y12.y23.y3
+ y14.y32
+ y14.y2.y3
- r restricts to
y1.y22.y35
+ y1.y24.y33
+ y1.y25.y32
+ y1.y26.y3
+ y12.y2.y35
+ y12.y22.y34
+ y12.y24.y32
+ y12.y25.y3
+ y12.y26
+ y14.y34
+ y14.y2.y33
+ y18
Restriction to maximal elementary abelian number 4, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
y2.y3
+ y22
- u1 restricts to
y24.y3
+ y25
+ y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u2 restricts to
0
- u3 restricts to
y23.y32
+ y24.y3
+ y1.y2.y33
+ y1.y22.y32
+ y12.y33
+ y12.y2.y32
+ y12.y22.y3
+ y14.y3
- t restricts to
y23.y33
+ y24.y32
+ y25.y3
+ y26
+ y1.y2.y34
+ y1.y22.y33
+ y1.y23.y32
+ y1.y24.y3
+ y12.y34
+ y12.y2.y33
+ y12.y23.y3
+ y12.y24
+ y14.y32
+ y14.y22
- r restricts to
y24.y34
+ y27.y3
+ y1.y22.y35
+ y1.y24.y33
+ y1.y25.y32
+ y1.y26.y3
+ y12.y2.y35
+ y12.y22.y34
+ y12.y24.y32
+ y12.y25.y3
+ y12.y26
+ y14.y34
+ y14.y2.y33
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- u3 restricts to
0
- t restricts to
0
- r restricts to
y8
(1 + 3t + 4t2
+ 3t3 + t4 + 2t5
+ 5t6 + 4t7 + t8) /
(1 - t2)2 (1 - t8)
Back to the groups of order 64