Small group number 124 of order 64
G is the group 64gp124
The Hall-Senior number of this group is 58.
G has 3 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
32gp11 (2x), 32gp15, Ab(8,4), 32gp38 (2x), 32gp42.
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 5 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x in degree 2
- v in degree 4, a regular element
There are 3 minimal relations:
- y2.y3 =
y1.y2
- y12 =
0
- y3.x =
y2.x
+ y1.x
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 are 2 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 8. The cohomology ring approximation
is stable from degree 4 onwards, and
Carlson's tests detect stability from degree 6
onwards.
This cohomology ring has dimension 2 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
x
+ y32
+ y22
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 6.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y32
in degree 2
-
y22
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y23
in degree 3
-
y1.y32
in degree 3
-
y1.y22
in degree 3
-
y1.y23
in degree 4
Restriction to maximal subgroup number 1, which is 32gp38
- y1 restricts to
y1
- y2 restricts to
y3
+ y2
- y3 restricts to
y1
- x restricts to
y2.y3
- v restricts to
y23.y3
+ v
Restriction to maximal subgroup number 2, which is 32gp38
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
+ y1
- x restricts to
y2.y3
+ y22
- v restricts to
y23.y3
+ v
Restriction to maximal subgroup number 3, which is 32gp3
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
x1
- v restricts to
x22
+ x1.x2
Restriction to maximal subgroup number 4, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x restricts to
x2
- v restricts to
v
+ y1.w
Restriction to maximal subgroup number 5, which is 32gp42
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y3
- x restricts to
y22
- v restricts to
v
Restriction to maximal subgroup number 6, which is 32gp15
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- x restricts to
x
+ y1.y2
- v restricts to
x2
+ v
Restriction to maximal subgroup number 7, which is 32gp11
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x restricts to
x2
- v restricts to
v
+ y1.w
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- x restricts to
0
- v restricts to
y12.y22
+ y14
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x restricts to
y22
- v restricts to
y12.y22
+ y14
Restriction to maximal elementary abelian number 3, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- x restricts to
0
- v restricts to
y12.y22
+ y14
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x restricts to
0
- v restricts to
y4
(1 + 3t + 4t2
+ 3t3 + t4) /
(1 - t2) (1 - t4)
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