Small group number 139 of order 64
G is the group 64gp139
The Hall-Senior number of this group is 260.
G has 3 minimal generators, rank 3 and exponent 4.
The centre has rank 1.
The 7 maximal subgroups are:
32gp30 (3x), E32+, 32gp6 (3x).
There are 5 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 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
- w in degree 3
- u1 in degree 5
- u2 in degree 5
- u3 in degree 5
- r in degree 8, a regular element
There are 27 minimal relations:
- y1.y3 =
0
- y1.y2 =
0
- y3.x1 =
y2.x2
- y13 =
0
- y3.w =
y12.x2
- y2.w =
y12.x1
- y2.y3.x2 =
y22.x2
- y2.x22 =
y2.x1.x2
- y12.w =
0
- w2 =
x1.x22
+ x12.x2
+ y1.u2
- y3.u2 =
y32.x22
+ y2.u3
+ y2.u1
+ y12.x22
+ y12.x1.x2
+ y12.x12
- y3.u1 =
y2.u3
+ y2.u1
+ y22.x1.x2
+ y24.x2
+ y12.x22
+ y12.x12
- y2.u2 =
y2.u1
+ y22.x1.x2
+ y22.x12
+ y24.x2
+ y12.x1.x2
- y1.u3 =
0
- y1.u1 =
y1.x2.w
+ y1.x1.w
+ y12.x1.x2
- x2.u1 =
x22.w
+ x1.u3
+ x1.u1
+ x12.w
+ y2.x12.x2
+ y23.x1.x2
- y12.u2 =
0
- w.u3 =
y1.x2.u2
+ y1.x22.w
+ y1.x1.u2
+ y1.x12.w
+ y12.x12.x2
- w.u1 =
x1.x23
+ x13.x2
+ y1.x2.u2
+ y1.x1.x2.w
+ y1.x12.w
- y2.x2.u3 =
y22.x12.x2
+ y24.x1.x2
- x1.x2.u3 =
y2.x13.x2
+ y23.x12.x2
+ y1.w.u2
+ y1.x1.x23
+ y1.x13.x2
- u32 =
y32.x24
+ y33.x2.u3
+ y35.u3
+ y36.x22
+ y24.x12.x2
+ y25.u3
+ y26.x12
+ y32.r
+ y22.r
- u2.u3 =
y3.x22.u3
+ y2.x12.u1
+ y2.y34.u3
+ y22.x13.x2
+ y22.y33.u3
+ y23.x1.u3
+ y23.x1.u1
+ y23.y32.u3
+ y24.x12.x2
+ y24.y3.u3
+ y25.u3
+ y26.x12
+ y28.x2
+ y2.y3.r
+ y22.r
+ y1.x22.u2
+ y1.x23.w
+ y1.x12.u2
+ y1.x13.w
+ y12.x14
- u22 =
x1.x24
+ x14.x2
+ y32.x24
+ y22.x14
+ y22.y33.u3
+ y24.y3.u3
+ y25.u1
+ y26.x1.x2
+ y26.x12
+ y22.r
+ y1.x22.u2
+ y1.x1.x2.u2
+ y1.x1.x22.w
+ y1.x12.u2
+ y1.x12.x2.w
+ y12.x24
+ y12.x13.x2
+ y12.r
- u1.u3 =
y2.x12.u3
+ y2.x12.u1
+ y2.y34.u3
+ y22.x13.x2
+ y22.y33.u3
+ y23.x1.u3
+ y23.x1.u1
+ y23.y32.u3
+ y24.y3.u3
+ y25.u3
+ y26.x1.x2
+ y26.x12
+ y28.x2
+ y2.y3.r
+ y22.r
+ y1.x22.u2
+ y1.x23.w
+ y1.x1.x22.w
+ y1.x12.u2
+ y1.x12.x2.w
+ y1.x13.w
+ y12.x13.x2
+ y12.x14
- u1.u2 =
x2.w.u2
+ x1.w.u2
+ y2.x12.u3
+ y22.x13.x2
+ y22.y33.u3
+ y23.x1.u3
+ y23.x1.u1
+ y24.y3.u3
+ y25.u1
+ y26.x1.x2
+ y26.x12
+ y22.r
+ y1.x1.x2.u2
+ y1.x1.x22.w
+ y1.x12.u2
+ y1.x12.x2.w
+ y1.x13.w
- u12 =
x1.x24
+ x12.x23
+ x13.x22
+ x14.x2
+ y22.y33.u3
+ y24.y3.u3
+ y25.u1
+ y26.x12
+ y22.r
+ y1.x22.u2
+ y1.x12.u2
+ y12.x13.x2
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
Essential ideal:
Zero ideal
Nilradical:
There is one minimal generator:
This cohomology ring was obtained from a calculation
out to degree 14. The cohomology ring approximation
is stable from degree 10 onwards, and
Carlson's tests detect stability from degree 14
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x22
+ x1.x2
+ x12
+ y32.x2
+ y34
+ y22.x2
+ y22.x1
+ y22.y32
+ y24
in degree 4
- h3 =
x2
+ y22
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y12.x1.x2
+ y12.x12.
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 14.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x1
in degree 2
-
y32
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y12
in degree 2
-
w
in degree 3
-
y33
in degree 3
-
y2.x1
in degree 3
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y1.x1
in degree 3
-
y34
in degree 4
-
y2.y33
in degree 4
-
y22.x1
in degree 4
-
y22.y32
in degree 4
-
y24
in degree 4
-
y1.w
in degree 4
-
y12.x1
in degree 4
-
u3
in degree 5
-
u2
in degree 5
-
u1
in degree 5
-
x1.w
in degree 5
-
y2.y34
in degree 5
-
y22.y33
in degree 5
-
y25
in degree 5
-
y3.u3
in degree 6
-
y2.u3
in degree 6
-
y2.u1
in degree 6
-
y22.y34
in degree 6
-
y26
in degree 6
-
y1.u2
in degree 6
-
y1.x1.w
in degree 6
-
x1.u2
in degree 7
-
x1.u1
in degree 7
-
y32.u3
in degree 7
-
y2.y3.u3
in degree 7
-
y22.u3
in degree 7
-
y22.u1
in degree 7
-
w.u2
in degree 8
-
y33.u3
in degree 8
-
y2.x1.u1
in degree 8
-
y2.y32.u3
in degree 8
-
y22.y3.u3
in degree 8
-
y23.u1
in degree 8
-
y1.x1.u2
in degree 8
-
y2.y33.u3
in degree 9
-
y22.y32.u3
in degree 9
-
y24.u1
in degree 9
-
y1.w.u2
in degree 9
-
x1.w.u2
in degree 10
-
y22.y33.u3
in degree 10
-
y25.u1
in degree 10
-
y1.x1.w.u2
in degree 11
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 12.
-
y12.x1.x2
+ y12.x12
in degree 6
-
y12.x12.x2
+ y12.x13
in degree 8
Restriction to maximal subgroup number 1, which is 32gp49
- y1 restricts to
0
- y2 restricts to
y4
+ y2
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
+ y1.y4
+ y1.y2
- x2 restricts to
y32
+ y2.y3
+ y1.y2
+ y12
- w restricts to
y1.y32
+ y1.y2.y3
+ y12.y3
- u1 restricts to
y2.y34
+ y24.y3
+ y1.y23.y3
+ y12.y22.y3
+ y13.y32
+ y14.y3
+ y4.v
+ y2.v
- u2 restricts to
y2.y34
+ y23.y32
+ y1.y34
+ y1.y23.y3
+ y12.y43
+ y14.y4
+ y14.y3
+ y4.v
+ y2.v
- u3 restricts to
y2.y34
+ y24.y3
+ y1.y23.y3
+ y12.y22.y3
+ y13.y2.y3
+ y4.v
- r restricts to
y38
+ y27.y3
+ y1.y26.y3
+ y12.y46
+ y12.y36
+ y14.y44
+ y14.y34
+ y14.y23.y3
+ y14.y24
+ y15.y22.y3
+ y18
+ y44.v
+ y2.y43.v
+ y24.v
+ y1.y22.y3.v
+ y1.y23.v
+ y12.y22.v
+ v2
Restriction to maximal subgroup number 2, which is 32gp30
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y22
+ x
+ y1.y2
+ y12
- x2 restricts to
y22
+ y1.y2
- w restricts to
w1
+ y2.x
- u1 restricts to
x.w2
+ x.w1
+ y3.v
+ y1.y2.w1
+ y1.y24
+ y1.y22.x
- u2 restricts to
y22.w1
+ x.w2
+ y3.v
+ y3.x2
+ y1.y2.w1
+ y1.y24
+ y1.v
- u3 restricts to
x.w2
+ y3.v
+ y2.x2
+ y1.y22.x
- r restricts to
y28
+ y33.x.w2
+ y34.v
+ y26.x
+ v2
+ y34.x2
+ y22.x.v
+ x2.v
+ y22.x3
+ y1.w1.v
+ y1.y23.x2
+ y1.y2.x3
+ y12.x.v
Restriction to maximal subgroup number 3, which is 32gp6
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x3
+ x2
- x2 restricts to
x3
- w restricts to
w2
+ y1.x3
- u1 restricts to
x1.w1
+ y2.v
+ y1.v
- u2 restricts to
x3.w2
+ x1.w2
+ x1.w1
+ y2.x22
+ y2.v
+ y1.x32
+ y1.x12
- u3 restricts to
y2.v
+ y1.x32
+ y1.x12
- r restricts to
x34
+ y24.x22
+ y24.v
+ v2
Restriction to maximal subgroup number 4, which is 32gp30
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
y22
+ y1.y2
- x2 restricts to
x
- w restricts to
w1
+ y2.x
- u1 restricts to
y22.w1
+ x.w1
+ y23.x
+ y2.x2
+ y1.y22.x
- u2 restricts to
y22.w1
+ y3.x2
+ y2.x2
+ y1.y2.w1
+ y1.v
+ y1.y22.x
- u3 restricts to
x.w2
+ y3.v
+ y2.x2
+ y1.y2.w1
- r restricts to
y33.x.w2
+ y34.v
+ y26.x
+ v2
+ y3.x2.w2
+ y32.x.v
+ y34.x2
+ y22.x.v
+ y24.x2
+ x2.v
+ x4
+ y1.w1.v
+ y1.y22.x.w1
+ y1.y25.x
+ y1.x2.w1
Restriction to maximal subgroup number 5, which is 32gp6
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x3
- x2 restricts to
x3
+ x2
- w restricts to
w2
+ y1.x3
- u1 restricts to
x1.w1
+ y1.v
- u2 restricts to
x3.w2
+ x1.w2
+ x1.w1
+ y2.x22
+ y1.x32
+ y1.x12
- u3 restricts to
y2.v
+ y1.x32
+ y1.x12
- r restricts to
x34
+ x24
+ x12.x22
+ x14
+ y24.x22
+ y22.x2.v
+ y24.v
+ v2
Restriction to maximal subgroup number 6, which is 32gp30
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y3
- x1 restricts to
x
- x2 restricts to
y22
+ x
+ y1.y2
+ y12
- w restricts to
w1
+ y2.x
+ y1.y22
- u1 restricts to
y22.w1
+ x.w2
+ y3.v
+ y33.x
+ y23.x
+ y3.x2
+ y2.x2
+ y1.y2.w1
+ y1.y24
+ y1.y22.x
+ y1.x2
- u2 restricts to
y22.w1
+ x.w2
+ y3.v
+ y3.x2
+ y1.y24
+ y1.v
+ y1.y22.x
+ y1.x2
- u3 restricts to
y33.x
+ y3.x2
+ y1.y2.w1
+ y1.y22.x
- r restricts to
y28
+ y33.x.w2
+ y34.v
+ y36.x
+ v2
+ y34.x2
+ y22.x.v
+ y24.x2
+ x2.v
+ y22.x3
+ x4
+ y1.w1.v
+ y1.y22.x.w1
+ y1.y25.x
+ y1.x2.w1
+ y1.y23.x2
+ y1.y2.x3
Restriction to maximal subgroup number 7, which is 32gp6
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- x1 restricts to
x3
+ x2
- x2 restricts to
x2
- w restricts to
w2
+ y1.x3
- u1 restricts to
x3.w2
+ y2.x22
+ y23.x2
+ y2.v
+ y1.x32
+ y1.x12
+ y1.v
- u2 restricts to
x3.w2
+ x1.w2
+ x1.w1
+ y2.x22
+ y2.v
- u3 restricts to
y2.x22
+ y23.x2
+ y1.x12
+ y1.v
- r restricts to
x24
+ y24.x22
+ y26.x2
+ y24.v
+ v2
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- w restricts to
0
- u1 restricts to
y1.y2.y33
+ y1.y22.y32
+ y12.y33
+ y12.y2.y32
+ y12.y22.y3
+ y14.y3
- u2 restricts to
y1.y2.y33
+ y1.y22.y32
+ y12.y33
+ y12.y2.y32
+ y12.y22.y3
+ y14.y3
- u3 restricts to
y1.y2.y33
+ y1.y23.y3
+ y12.y33
+ y12.y23
+ y14.y3
+ y14.y2
- r restricts to
y1.y2.y36
+ y1.y26.y3
+ y12.y36
+ y12.y23.y33
+ y12.y26
+ y14.y2.y33
+ y14.y23.y3
+ y18
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
y32
+ y2.y3
- w restricts to
0
- u1 restricts to
0
- u2 restricts to
y2.y34
+ y23.y32
- u3 restricts to
y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- r restricts to
y38
+ y26.y32
+ y1.y23.y34
+ y1.y26.y3
+ y12.y26
+ y14.y34
+ y14.y23.y3
+ y18
Restriction to maximal elementary abelian number 3, 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
- w 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
y2.y34
+ y23.y32
+ y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u3 restricts to
y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- r restricts to
y24.y34
+ y26.y32
+ 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 4, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
y32
- x2 restricts to
y22
- w restricts to
y2.y32
+ y22.y3
- u1 restricts to
y2.y34
+ y22.y33
+ y23.y32
+ y24.y3
- u2 restricts to
y2.y34
+ y24.y3
- u3 restricts to
0
- r restricts to
y22.y36
+ y24.y34
+ y28
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y22.y32
+ y14.y24
+ y18
Restriction to maximal elementary abelian number 5, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y2
- x1 restricts to
y32
+ y2.y3
- x2 restricts to
y32
+ y2.y3
- w restricts to
0
- u1 restricts to
y2.y34
+ y24.y3
+ y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u2 restricts to
y2.y34
+ y23.y32
+ y1.y22.y32
+ y1.y23.y3
+ y12.y2.y32
+ y12.y22.y3
+ y12.y23
+ y14.y2
- u3 restricts to
y2.y34
+ y24.y3
- r restricts to
y38
+ y27.y3
+ y1.y25.y32
+ y1.y26.y3
+ y12.y22.y34
+ y12.y25.y3
+ 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
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- w restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- u3 restricts to
0
- r restricts to
y8
(1 + 3t + 5t2
+ 7t3 + 7t4 + 7t5
+ 7t6 + 6t7 + 6t8
+ 4t9 + 2t10 + t11) /
(1 - t2) (1 - t4) (1 - t8)
Back to the groups of order 64