Small group number 202 of order 64
G is the group 64gp202
The Hall-Senior number of this group is 68.
G has 4 minimal generators, rank 5 and exponent 4.
The centre has rank 3.
The 15 maximal subgroups are:
32gp22 (3x), 32gp27 (8x), 32gp46 (3x), V32.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 5.
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
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1, a regular element
- x1 in degree 2
- x2 in degree 2, a regular element
- x3 in degree 2, a regular element
There are 4 minimal relations:
- y2.y3 =
0
- y1.y3 =
0
- y3.x1 =
0
- x12 =
y1.y2.x1
+ y22.x3
+ y12.x2
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 4 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 5 and depth 4.
Here is a homogeneous system of parameters:
- h1 =
y4
in degree 1
- h2 =
x2
in degree 2
- h3 =
x3
in degree 2
- h4 =
y32
+ y12
in degree 2
- h5 =
y22
in degree 2
The first
4 terms h1, h2, h3, h4 form
a regular sequence of maximum length.
The remaining
term h5 is
annihilated by the class
y3.
The first
3 terms h1, h2, h3 form
a complete Duflot regular sequence.
That is, their restrictions to the greatest central elementary abelian
subgroup form 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, h4, h5) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 9.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
x1
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y2.x1
in degree 3
-
y1.x1
in degree 3
-
y1.y2.x1
in degree 4
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 7.
-
y3
in degree 1
-
y12
in degree 2
Restriction to maximal subgroup number 1, which is V32
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- y4 restricts to
y1
- x1 restricts to
y3.y4
+ y2.y5
- x2 restricts to
y52
+ y3.y5
- x3 restricts to
y42
+ y2.y4
Restriction to maximal subgroup number 2, which is 32gp46
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- y4 restricts to
y2
+ y1
+ y3
- x1 restricts to
y2.y4
+ y2.y3
- x2 restricts to
y1.y4
+ y1.y3
+ y42
+ y32
- x3 restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 3, which is 32gp22
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
- y4 restricts to
y2
+ y3
- x1 restricts to
x1
+ y22
+ y2.y3
+ y1.y3
- x2 restricts to
y22
+ x2
+ y32
- x3 restricts to
x3
+ y2.y3
+ y32
+ y1.y3
Restriction to maximal subgroup number 4, which is 32gp46
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y1
+ y4
+ y3
- x1 restricts to
y2.y4
- x2 restricts to
x
- x3 restricts to
y1.y4
+ y42
Restriction to maximal subgroup number 5, which is 32gp22
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y3
+ y1
- x1 restricts to
x1
+ y22
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
y22
+ x2
Restriction to maximal subgroup number 6, which is 32gp46
- y1 restricts to
y2
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y2
+ y1
+ y3
- x1 restricts to
y22
+ y2.y4
+ y2.y3
- x2 restricts to
x
- x3 restricts to
x
+ y2.y4
+ y2.y3
+ y1.y4
+ y1.y3
+ y42
+ y32
Restriction to maximal subgroup number 7, which is 32gp22
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y2
+ y3
- x1 restricts to
x1
+ y22
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
x3
Restriction to maximal subgroup number 8, which is 32gp27
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
0
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 9, which is 32gp27
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
- x1 restricts to
x1
+ y1.y2
- x2 restricts to
x1
+ y1.y2
+ x3
+ x2
- x3 restricts to
x3
Restriction to maximal subgroup number 10, which is 32gp27
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x3
- x3 restricts to
x2
Restriction to maximal subgroup number 11, which is 32gp27
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
- x1 restricts to
x1
+ y1.y2
+ y12
- x2 restricts to
y1.y2
+ y12
+ x3
- x3 restricts to
x1
+ x3
+ x2
Restriction to maximal subgroup number 12, which is 32gp27
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
- x3 restricts to
x3
Restriction to maximal subgroup number 13, which is 32gp27
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
- x1 restricts to
x1
+ y22
+ y12
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
y1.y2
+ y12
+ x3
Restriction to maximal subgroup number 14, which is 32gp27
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x1
+ x3
+ x2
- x3 restricts to
x2
Restriction to maximal subgroup number 15, which is 32gp27
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
- x1 restricts to
x1
+ y22
- x2 restricts to
x1
+ y1.y2
+ x3
+ x2
- x3 restricts to
y22
+ y1.y2
+ x2
Restriction to maximal elementary abelian number 1, which is V16
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y4
- y4 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y2.y4
+ y22
- x3 restricts to
y3.y4
+ y32
Restriction to maximal elementary abelian number 2, which is V32
- y1 restricts to
y5
- y2 restricts to
y4
- y3 restricts to
0
- y4 restricts to
y1
- x1 restricts to
y3.y4
+ y2.y5
- x2 restricts to
y2.y4
+ y22
- x3 restricts to
y3.y5
+ y32
Restriction to the greatest central elementary abelian, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
y1
- x1 restricts to
0
- x2 restricts to
y22
- x3 restricts to
y32
(1 + 3t + 3t2
+ t3) /
(1 - t) (1 - t2)4
Back to the groups of order 64