Small group number 88 of order 64
G is the group 64gp88
The Hall-Senior number of this group is 94.
G has 3 minimal generators, rank 4 and exponent 8.
The centre has rank 2.
The 7 maximal subgroups are:
32gp37 (2x), Ab(4,2,2,2), 32gp5 (4x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 4.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 9 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, a regular element
- w1 in degree 3, a nilpotent element
- w2 in degree 3
- v1 in degree 4, a nilpotent element
- v2 in degree 4, a regular element
There are 18 minimal relations:
- y1.y3 =
0
- y12 =
0
- y1.y22 =
0
- y1.x1 =
0
- y3.w1 =
y22.x1
- y1.w2 =
0
- x12 =
0
- y1.w1 =
0
- x1.w2 =
y3.v1
- x1.w1 =
0
- y1.v1 =
0
- w22 =
y22.y3.w2
+ y32.v2
+ y24.x2
- w1.w2 =
y22.v1
- w12 =
0
- x1.v1 =
0
- w2.v1 =
y22.y3.v1
+ y3.x1.v2
+ y22.x2.w1
- w1.v1 =
0
- v12 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 10. The cohomology ring approximation
is stable from degree 8 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 4 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x2
in degree 2
- h2 =
v2
in degree 4
- h3 =
y22
in degree 2
- h4 =
y32
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
2 terms h3, h4 are all
annihilated by the class
y1.
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
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y2.y3
in degree 2
-
x1
in degree 2
-
y1.y2
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y3.x1
in degree 3
-
y2.x1
in degree 3
-
y3.w2
in degree 4
-
y2.w2
in degree 4
-
v1
in degree 4
-
y2.w1
in degree 4
-
y2.y3.x1
in degree 4
-
y2.y3.w2
in degree 5
-
y3.v1
in degree 5
-
y2.v1
in degree 5
-
y2.y3.v1
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
in degree 1
-
y1.y2
in degree 2
-
w1
in degree 3
-
y2.w1
in degree 4
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 6.
-
y1
in degree 1
-
y1.y2
in degree 2
A basis for AnnR/(h1, h2)(h3)
/ h4 AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y1
in degree 1
-
y1.y2
in degree 2
Restriction to maximal subgroup number 1, which is 32gp45
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- x1 restricts to
y1.y3
- x2 restricts to
y42
+ y3.y4
- w1 restricts to
y1.y22
- w2 restricts to
y3.x
+ y22.y4
+ y1.y2.y3
- v1 restricts to
y1.y3.x
+ y1.y22.y4
- v2 restricts to
x2
+ y22.x
+ y1.y23
Restriction to maximal subgroup number 2, which is 32gp37
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x1 restricts to
y1.y2
+ y1.y3
- x2 restricts to
y22
+ y32
+ y1.y2
+ y1.y3
- w1 restricts to
w
- w2 restricts to
y23
+ y22.y3
- v1 restricts to
y2.w
+ y3.w
- v2 restricts to
v
Restriction to maximal subgroup number 3, which is 32gp37
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- x1 restricts to
y1.y2
+ y1.y3
- x2 restricts to
y22
+ y32
+ y1.y2
- w1 restricts to
w
- w2 restricts to
y23
+ y22.y3
+ w
- v1 restricts to
y2.w
+ y3.w
- v2 restricts to
v
Restriction to maximal subgroup number 4, which is 32gp5
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
+ y1
- x1 restricts to
x1
- x2 restricts to
x2
+ x1
- w1 restricts to
y1.x3
- w2 restricts to
y2.x3
+ y1.x3
- v1 restricts to
x1.x3
- v2 restricts to
x32
Restriction to maximal subgroup number 5, which is 32gp5
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x1 restricts to
x1
- x2 restricts to
x2
+ x1
- w1 restricts to
y1.x3
- w2 restricts to
y23
+ y2.x3
+ y1.x3
- v1 restricts to
y22.x1
+ x1.x3
- v2 restricts to
y24
+ x32
Restriction to maximal subgroup number 6, which is 32gp5
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- x1 restricts to
x1
- x2 restricts to
x2
- w1 restricts to
y2.x1
+ y1.x3
+ y1.x2
- w2 restricts to
y23
+ y2.x3
+ y2.x2
+ y2.x1
- v1 restricts to
y22.x1
+ x1.x3
+ x1.x2
- v2 restricts to
y22.x3
+ x32
+ x22
+ y22.x1
Restriction to maximal subgroup number 7, which is 32gp5
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- x1 restricts to
x1
- x2 restricts to
x2
+ x1
- w1 restricts to
y2.x1
+ y1.x3
+ y1.x2
- w2 restricts to
y2.x3
+ y2.x2
+ y1.x3
+ y1.x2
- v1 restricts to
x1.x3
+ x1.x2
- v2 restricts to
y22.x3
+ x32
+ x22
+ y22.x1
Restriction to maximal elementary abelian number 1, which is V16
- y1 restricts to
0
- y2 restricts to
y4
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
y1.y3
+ y12
- w1 restricts to
0
- w2 restricts to
y22.y3
+ y1.y42
- v1 restricts to
0
- v2 restricts to
y22.y42
+ y24
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
y12
- w1 restricts to
0
- w2 restricts to
0
- v1 restricts to
0
- v2 restricts to
y24
(1 + 3t + 3t2
+ 2t3 + 3t4 + 3t5
+ t6) /
(1 - t2)3 (1 - t4)
Back to the groups of order 64