Small group number 30 of order 32
G is the group 32gp30
The Hall-Senior number of this group is 38.
G has 3 minimal generators, rank 3 and exponent 4.
The centre has rank 2.
The 7 maximal subgroups are:
Ab(4,2,2), D8xC2, 16gp3 (3x), 16gp4 (2x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3.
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
- 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:
- y2.y3 =
y12
- y1.y3 =
0
- y12.y2 =
0
- y3.w1 =
0
- y2.w2 =
y22.x
+ y1.w1
- y1.w2 =
y1.y2.x
- w22 =
y32.v
+ y22.x2
- w1.w2 =
y2.x.w1
+ y1.y23.x
- w12 =
y24.x
+ y1.y22.w1
+ y12.v
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 12. The cohomology ring approximation
is stable from degree 6 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
x
in degree 2
- h2 =
v
in degree 4
- h3 =
y32
+ y22
in degree 2
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y12.
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) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
w2
in degree 3
-
w1
in degree 3
-
y3.w2
in degree 4
-
y2.w1
in degree 4
-
y1.w1
in degree 4
-
y1.y2.w1
in degree 5
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 6.
Restriction to maximal subgroup number 1, which is 16gp10
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x restricts to
y32
+ y1.y3
- w1 restricts to
y22.y3
+ y1.x
- w2 restricts to
y2.y32
- v restricts to
x2
+ y22.x
Restriction to maximal subgroup number 2, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- x restricts to
x3
- w1 restricts to
y1.x2
- w2 restricts to
y2.x1
+ y2.x2
- v restricts to
y22.x2
+ x22
Restriction to maximal subgroup number 3, which is 16gp4
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y2
- x restricts to
x2
+ y1.y2
- w1 restricts to
y2.x1
+ y1.x1
- w2 restricts to
y2.x2
+ y2.x1
- v restricts to
x12
+ y1.y2.x2
+ y1.y2.x1
Restriction to maximal subgroup number 4, which is 16gp11
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
y22
+ y1.y3
+ y32
- w1 restricts to
y23
+ y22.y3
- w2 restricts to
y23
+ y1.x
+ y12.y3
+ y2.y32
+ y1.y32
- v restricts to
x2
+ y22.y32
+ y12.y32
+ y34
Restriction to maximal subgroup number 5, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- x restricts to
y22
+ x2
- w1 restricts to
y2.x1
+ y23
+ y1.x3
- w2 restricts to
y23
+ y2.x2
+ y1.x3
+ y1.x2
- v restricts to
x32
Restriction to maximal subgroup number 6, which is 16gp3
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x restricts to
x1
+ x3
+ x2
- w1 restricts to
y1.x3
- w2 restricts to
y2.x3
+ y1.x2
- v restricts to
x32
Restriction to maximal subgroup number 7, which is 16gp4
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- x restricts to
x2
+ y1.y2
- w1 restricts to
y2.x2
+ y2.x1
- w2 restricts to
y2.x2
+ y2.x1
+ y1.x1
- v restricts to
x22
+ x12
+ y1.y2.x2
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y3
+ y2
+ y1
- x restricts to
y1.y3
+ y1.y2
- w1 restricts to
0
- w2 restricts to
y2.y32
+ y22.y3
+ y1.y22
+ y12.y2
- v restricts to
y22.y32
+ y12.y22
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- x restricts to
y22
- w1 restricts to
y2.y32
- w2 restricts to
y22.y3
- v restricts to
y12.y32
+ y14
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x restricts to
y22
- w1 restricts to
0
- w2 restricts to
0
- v restricts to
y24
+ y14
(1 + 3t + 3t2
+ 2t3 + 2t4 + t5) /
(1 - t2)2 (1 - t4)
Back to the groups of order 32