Small group number 215 of order 64
G is the group 64gp215
The Hall-Senior number of this group is 169.
G has 4 minimal generators, rank 4 and exponent 4.
The centre has rank 2.
The 15 maximal subgroups are:
32gp22, 32gp27 (4x), 32gp28 (4x), 32gp30 (4x), 32gp46 (2x).
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 4, 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 8 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- x in degree 2, a regular element
- w1 in degree 3
- w2 in degree 3
- v in degree 4, a regular element
There are 9 minimal relations:
- y3.y4 =
y1.y4
+ y1.y2
- y2.y4 =
0
- y1.y2.y3 =
0
- y4.w1 =
0
- y3.w2 =
y1.w2
+ y1.w1
- y2.w2 =
0
- w22 =
y1.y42.w2
+ y1.y45
+ y12.y4.w2
+ y42.v
+ y12.y42.x
+ y12.y32.x
- w1.w2 =
y1.y33.x
+ y12.y32.x
- w12 =
y2.y32.w1
+ y22.y3.w1
+ y23.w1
+ y34.x
+ y22.v
+ y12.y32.x
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 10. The cohomology ring approximation
is stable from degree 6 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 4 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x
in degree 2
- h2 =
v
in degree 4
- h3 =
y42
+ y22
+ y1.y3
+ y12
in degree 2
- h4 =
y32
in degree 2
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
The remaining
term h4 is
annihilated by the class
y1.y2.
The first
2 terms h1, h2 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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y4
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y4.w2
in degree 4
-
y3.w1
in degree 4
-
y2.w1
in degree 4
-
y1.w2
in degree 4
-
y1.w1
in degree 4
-
y2.y3.w1
in degree 5
-
y1.y4.w2
in degree 5
-
y12.w2
in degree 5
-
y12.w1
in degree 5
-
y13.w1
in degree 6
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y1.y2
in degree 2
-
y12.y2
in degree 3
Restriction to maximal subgroup number 1, which is 32gp46
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y2
+ y3
- y4 restricts to
0
- x restricts to
y1.y4
+ y1.y3
+ y42
+ y32
- w1 restricts to
y1.x
+ y2.y3.y4
+ y2.y32
+ y1.y3.y4
+ y1.y32
+ y32.y4
+ y33
- w2 restricts to
y22.y4
+ y22.y3
+ y2.y3.y4
+ y2.y32
- v restricts to
y12.x
+ x2
+ y2.y3.x
+ y22.y3.y4
+ y22.y32
+ y1.y3.x
+ y12.y3.y4
+ y12.y32
+ y32.x
+ y2.y3.y42
+ y2.y32.y4
+ y32.y42
+ y34
Restriction to maximal subgroup number 2, which is 32gp27
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- x restricts to
x2
- w1 restricts to
y2.x1
+ y1.x1
+ y1.x3
- w2 restricts to
y3.x3
- v restricts to
y1.y2.x1
+ y12.x1
+ y22.x2
+ y1.y2.x3
+ x32
Restriction to maximal subgroup number 3, which is 32gp27
- y1 restricts to
y3
- y2 restricts to
y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
- x restricts to
x3
- w1 restricts to
y2.x1
+ y1.x1
+ y1.y22
+ y12.y2
+ y2.x3
+ y2.x2
- w2 restricts to
y33
+ y3.x3
+ y3.x2
- v restricts to
y22.x1
+ y1.y2.x1
+ y1.y23
+ y12.y22
+ y22.x3
+ y1.y2.x3
+ y1.y2.x2
+ x32
+ x22
Restriction to maximal subgroup number 4, which is 32gp46
- y1 restricts to
y1
+ y3
- y2 restricts to
0
- y3 restricts to
y2
+ y1
+ y3
- y4 restricts to
y1
- x restricts to
y22
+ y1.y4
+ y1.y3
+ y42
+ y32
- w1 restricts to
y23
+ y22.y4
+ y2.y3.y4
+ y2.y32
- w2 restricts to
y13
+ y22.y3
+ y1.x
+ y12.y4
+ y12.y3
+ y2.y3.y4
+ y1.y42
+ y1.y3.y4
+ y1.y32
+ y32.y4
+ y33
- v restricts to
y24
+ x2
+ y2.y3.x
+ y22.y32
+ y1.y3.x
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y32.x
+ y1.y3.y42
+ y1.y32.y4
+ y1.y33
+ y44
+ y32.y42
Restriction to maximal subgroup number 5, which is 32gp22
- y1 restricts to
y3
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- x restricts to
y22
+ x2
+ y32
- w1 restricts to
y2.x1
+ y23
+ y3.x1
+ y2.y32
+ y1.x3
+ y1.x2
+ y1.y32
- w2 restricts to
y3.x1
+ y22.y3
+ y2.y32
+ y1.x3
+ y1.x2
+ y1.y32
- v restricts to
y24
+ x32
+ x22
+ y2.y3.x3
+ y22.y32
+ y32.x3
+ y32.x2
+ y1.y3.x3
Restriction to maximal subgroup number 6, which is 32gp30
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- x restricts to
y22
+ x
+ y1.y2
+ y12
- w1 restricts to
w1
+ y23
+ y1.x
- w2 restricts to
w2
+ y3.x
+ y2.x
+ y1.y22
- v restricts to
y24
+ v
+ x2
+ y1.w1
+ y1.y2.x
Restriction to maximal subgroup number 7, which is 32gp30
- y1 restricts to
y3
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y3
- x restricts to
y22
+ x
+ y1.y2
+ y12
- w1 restricts to
w1
+ y23
+ y1.y22
- w2 restricts to
w2
+ y33
+ y2.x
+ y1.y22
- v restricts to
y24
+ v
+ y22.x
+ y1.y2.x
Restriction to maximal subgroup number 8, which is 32gp28
- y1 restricts to
y2
- y2 restricts to
y3
+ y1
- y3 restricts to
0
- y4 restricts to
y1
- x restricts to
y22
+ y12
+ x2
- w1 restricts to
y3.x1
+ y1.x1
- w2 restricts to
y1.x1
- v restricts to
y32.x1
+ y12.x1
+ x12
Restriction to maximal subgroup number 9, which is 32gp28
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y1
- y4 restricts to
y1
- x restricts to
y22
+ y12
+ x2
- w1 restricts to
y1.y22
+ y3.x2
+ y3.x1
+ y1.x2
+ y1.x1
- w2 restricts to
y1.y22
+ y1.x2
+ y1.x1
- v restricts to
y14
+ y32.x2
+ y32.x1
+ y22.x2
+ y1.y2.x1
+ y12.x2
+ y12.x1
+ x22
+ x12
Restriction to maximal subgroup number 10, which is 32gp27
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
- y3 restricts to
y2
+ y1
- y4 restricts to
y2
- x restricts to
x3
- w1 restricts to
y3.x3
+ y3.x2
- w2 restricts to
y2.x1
+ y23
+ y1.x1
+ y1.y22
+ y2.x3
+ y2.x2
- v restricts to
y12.y22
+ y13.y2
+ y32.x3
+ y32.x2
+ y22.x2
+ y1.y2.x3
+ y1.y2.x2
+ y12.x3
+ x32
+ x22
Restriction to maximal subgroup number 11, which is 32gp30
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- x restricts to
x
- w1 restricts to
w2
+ y2.x
- w2 restricts to
w1
+ y1.y22
- v restricts to
y3.w2
+ v
+ y1.w1
+ y1.y23
+ y1.y2.x
+ y12.x
Restriction to maximal subgroup number 12, which is 32gp28
- y1 restricts to
y2
- y2 restricts to
y3
+ y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- x restricts to
y22
+ y12
+ x2
- w1 restricts to
y1.y22
+ y3.x2
+ y3.x1
+ y1.x2
+ y1.x1
- w2 restricts to
y1.y22
+ y13
+ y1.x2
+ y1.x1
- v restricts to
y14
+ y32.x1
+ y22.x2
+ y1.y2.x1
+ y12.x1
+ x22
+ x12
Restriction to maximal subgroup number 13, which is 32gp28
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y3
- y4 restricts to
y1
- x restricts to
y22
+ y12
+ x2
- w1 restricts to
y3.x1
+ y1.x1
- w2 restricts to
y13
+ y1.x1
- v restricts to
y32.x2
+ y32.x1
+ y12.x2
+ y12.x1
+ x12
Restriction to maximal subgroup number 14, which is 32gp27
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y3
+ y2
- y4 restricts to
y1
- x restricts to
y22
+ y1.y2
+ x2
- w1 restricts to
y3.x3
+ y3.x2
- w2 restricts to
y2.x1
+ y23
+ y1.x1
+ y1.x3
+ y1.x2
- v restricts to
y1.y23
+ y13.y2
+ y32.x3
+ y22.x2
+ y1.y2.x3
+ y1.y2.x2
+ y12.x3
+ x32
+ x22
Restriction to maximal subgroup number 15, which is 32gp30
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
y1
- x restricts to
y22
+ x
+ y1.y2
+ y12
- w1 restricts to
w2
+ y3.x
+ y2.x
+ y1.y22
- w2 restricts to
w1
+ y23
+ y1.y22
+ y1.x
- v restricts to
y3.w2
+ v
+ y22.x
+ x2
+ y1.y23
+ y1.y2.x
Restriction to maximal elementary abelian number 1, which is V16
- y1 restricts to
y4
- y2 restricts to
0
- y3 restricts to
y4
- y4 restricts to
y3
- x restricts to
y2.y3
+ y22
- w1 restricts to
0
- w2 restricts to
y32.y4
+ y2.y42
+ y2.y3.y4
+ y1.y32
+ y12.y3
- v restricts to
y3.y43
+ y33.y4
+ y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to maximal elementary abelian number 2, which is V16
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y4
- y4 restricts to
0
- x restricts to
y22
- w1 restricts to
y2.y42
+ y2.y3.y4
- w2 restricts to
y2.y3.y4
- v restricts to
y2.y3.y42
+ y2.y32.y4
+ y22.y42
+ y22.y3.y4
+ y1.y3.y42
+ y1.y32.y4
+ y12.y42
+ y12.y3.y4
+ y12.y32
+ y14
Restriction to maximal elementary abelian number 3, which is V16
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y4
- y4 restricts to
0
- x restricts to
y2.y3
+ y22
- w1 restricts to
y2.y42
+ y2.y3.y4
+ y1.y32
+ y12.y3
- w2 restricts to
0
- v restricts to
y2.y32.y4
+ y22.y42
+ y1.y3.y42
+ y1.y32.y4
+ y1.y33
+ y12.y42
+ y12.y3.y4
+ y14
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- x restricts to
y22
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y14
(1 + 4t + 6t2
+ 5t3 + 4t4 + 3t5
+ t6) /
(1 - t2)3 (1 - t4)
Back to the groups of order 64