Small group number 137 of order 64
G is the group 64gp137
The Hall-Senior number of this group is 264.
G has 3 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
32gp11 (2x), 32gp35, 32gp44 (2x), E32-, 32gp8.
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 7 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2, a nilpotent element
- x2 in degree 2
- u in degree 5, a nilpotent element
- r in degree 8, a regular element
There are 10 minimal relations:
- y1.y3 =
0
- y1.y2 =
0
- y3.x2 =
y22.y3
+ y23
+ y13
- y1.x1 =
y13
- y22.y32 =
y24
+ y2.y3.x1
+ y22.x1
+ x12
- y23.x2 =
y24.y3
+ y25
+ y22.y3.x1
+ y23.x1
+ y2.x12
+ y13.x2
- y22.x22 =
x1.x22
+ y3.u
+ y2.u
+ y2.y33.x1
+ y24.x1
+ y12.x22
- y25.y3 =
y26
+ y3.u
+ y2.y33.x1
+ y23.y3.x1
+ y24.x1
+ y2.y3.x12
- y24.y3.x1 =
y25.x1
+ x1.u
+ y2.y32.x12
+ y12.u
+ y13.x22
- x1.x24 =
u2
+ y23.x1.u
+ y26.x12
+ y1.x22.u
+ y12.r
+ y13.x2.u
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y14 =
0
- y22.x1.x2 =
y23.y3.x1
+ y24.x1
+ x12.x2
- x13 =
0
- y32.u =
y2.y3.u
+ y2.y34.x1
+ y25.x1
+ y22.y3.x12
- y2.x1.x22 =
y2.y3.u
+ y22.u
+ y2.y32.x12
+ y22.y3.x12
+ y23.x12
+ y13.x22
- y2.x12.x2 =
y22.y3.x12
+ y23.x12
- y22.y3.u =
y23.u
+ y24.x12
+ y13.u
- x12.x22 =
y13.u
- y3.x1.u =
y2.x1.u
+ y2.y33.x12
+ y23.y3.x12
- x1.x2.u =
y12.x2.u
+ y13.x23
- y3.u2 =
y24.x1.u
+ y27.x12
+ y22.x12.u
- y1.u2 =
y12.x22.u
+ y13.x24
+ y13.r
- y22.u2 =
y25.x1.u
+ y28.x12
+ y13.x22.u
- x1.u2 =
y23.x12.u
- u3 =
y12.u.r
+ y13.x22.r
Essential ideal:
Zero ideal
Nilradical:
There are 5 minimal generators:
-
y1
-
y2.y3
+ y22
-
x1
-
y2.x2
-
u
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 10 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
x2
+ y32
in degree 2
The first
2 terms h1, h2 form
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.
The ideal of essential classes is
the zero ideal.
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 10.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
x1
in degree 2
-
y12
in degree 2
-
y2.y32
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y3.x1
in degree 3
-
y2.x1
in degree 3
-
y13
in degree 3
-
y23.y3
in degree 4
-
y24
in degree 4
-
y32.x1
in degree 4
-
y2.y3.x1
in degree 4
-
y22.x1
in degree 4
-
x12
in degree 4
-
u
in degree 5
-
y2.y32.x1
in degree 5
-
y22.y3.x1
in degree 5
-
y23.x1
in degree 5
-
y3.x12
in degree 5
-
y2.x12
in degree 5
-
y24.x1
in degree 6
-
y32.x12
in degree 6
-
y2.y3.x12
in degree 6
-
y22.x12
in degree 6
-
y1.u
in degree 6
-
y22.y3.x12
in degree 7
-
y23.x12
in degree 7
-
y12.u
in degree 7
-
y13.u
in degree 8
Restriction to maximal subgroup number 1, which is 32gp50
- y1 restricts to
0
- y2 restricts to
y4
+ y1
- y3 restricts to
y4
+ y3
+ y2
- x1 restricts to
y2.y4
+ y2.y3
+ y1.y4
+ y1.y3
+ y1.y2
+ y12
- x2 restricts to
y2.y4
+ y22
+ y1.y2
- u restricts to
y24.y3
+ y25
+ y1.y44
+ y1.y2.y43
+ y1.y2.y33
+ y1.y23.y3
+ y12.y43
+ y12.y33
+ y12.y2.y42
+ y12.y23
+ y13.y2.y3
+ y14.y4
+ y15
- r restricts to
y27.y3
+ y28
+ y1.y2.y46
+ y12.y2.y45
+ y12.y25.y3
+ y15.y23
+ y16.y2.y3
+ y17.y4
+ y17.y2
+ r
Restriction to maximal subgroup number 2, which is 32gp35
- y1 restricts to
y3
+ y1
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y2.y3
+ y1.y3
+ y12
- x2 restricts to
x
+ y1.y3
+ y12
- u restricts to
y3.v
+ y1.v
+ y2.x2
+ y1.x2
+ y1.y2.y3.x
+ y12.y2.x
- r restricts to
v2
+ x2.v
+ y12.x.v
+ y1.y3.x3
+ y12.x3
+ y12.y2.y3.x2
Restriction to maximal subgroup number 3, which is 32gp8
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x1 restricts to
x1
+ y22
- x2 restricts to
x2
+ y22
- u restricts to
u1
+ x2.w
- r restricts to
x24
+ r
+ y2.x22.w
Restriction to maximal subgroup number 4, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- x1 restricts to
y2.y3
+ y12
- x2 restricts to
y22
+ y1.y2
- u restricts to
u1
+ y1.y24
- r restricts to
r
+ y23.u1
+ y12.y26
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
x1
- x2 restricts to
x2
- u restricts to
x2.w
+ y1.v
- r restricts to
v2
Restriction to maximal subgroup number 6, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y3
- x1 restricts to
y2.y3
+ y12
- x2 restricts to
y22
+ y1.y2
+ y12
- u restricts to
y2.y34
+ u1
- r restricts to
y2.y37
+ r
+ y23.u1
+ y12.y26
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
x1
- x2 restricts to
x2
- u restricts to
x2.w
+ y23.x1
+ y1.v
- r restricts to
v2
+ y26.x1
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- u restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y22
- u restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 3, which is V4
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y2
- x1 restricts to
0
- x2 restricts to
0
- u 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
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- u restricts to
0
- r restricts to
y8
(1 + 3t + 5t2
+ 6t3 + 6t4 + 6t5
+ 5t6 + 3t7 + t8) /
(1 - t2) (1 - t8)
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