Small group number 44 of order 32
G is the group 32gp44
The Hall-Senior number of this group is 45.
G has 3 minimal generators, rank 2 and exponent 8.
The centre has rank 1.
The 7 maximal subgroups are:
Q8xC2, 16gp13, 16gp6, SD16 (2x), 16gp9 (2x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 2.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 6 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- u1 in degree 5, a nilpotent element
- u2 in degree 5
- r in degree 8, a regular element
There are 9 minimal relations:
- y1.y3 =
0
- y22.y3 =
y13
- y13.y22 =
0
- y3.u1 =
0
- y1.u2 =
0
- y22.u2 =
y12.u1
- u22 =
y32.r
- u1.u2 =
0
- u12 =
y1.y24.u1
+ y12.y28
+ y12.r
+ y12.y23.u1
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
Essential ideal:
There is one minimal generator:
Nilradical:
There are 4 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 10 onwards, and
Carlson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 2 and depth 1.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
y32
+ y22
in degree 2
The first
term h1 forms
a regular sequence of maximum length.
The remaining
term h2 is
annihilated by the class
y13.
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
free of rank 1 as a module over the polynomial algebra
on h1.
These free generators are:
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 10.
-
1
in degree 0
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y23
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y13.y2
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y3.u2
in degree 6
-
y2.u2
in degree 6
-
y2.u1
in degree 6
-
y1.u1
in degree 6
-
y2.y3.u2
in degree 7
-
y1.y2.u1
in degree 7
-
y12.u1
in degree 7
-
y12.y2.u1
in degree 8
A basis for AnnR/(h1)(h2) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 8.
-
y13
in degree 3
-
y13.y2
in degree 4
Restriction to maximal subgroup number 1, which is 16gp12
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- u1 restricts to
y2.v
+ y1.y34
- u2 restricts to
y12.y2.y32
- r restricts to
v2
+ y34.v
+ y2.y33.v
+ y1.y37
Restriction to maximal subgroup number 2, which is 16gp13
- y1 restricts to
0
- y2 restricts to
y3
+ y2
- y3 restricts to
y1
- u1 restricts to
y2.y34
+ y13.y2.y3
- u2 restricts to
y14.y3
+ y1.v
- r restricts to
y2.y37
+ y16.y2.y3
+ y17.y3
+ v2
Restriction to maximal subgroup number 3, which is 16gp6
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- u1 restricts to
y22.w
+ y1.v
- u2 restricts to
y1.v
- r restricts to
v2
+ y25.w
Restriction to maximal subgroup number 4, which is 16gp8
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- u1 restricts to
y1.v
- u2 restricts to
y2.v
- r restricts to
v2
Restriction to maximal subgroup number 5, which is 16gp9
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- u1 restricts to
y1.v
- u2 restricts to
y2.v
- r restricts to
v2
Restriction to maximal subgroup number 6, which is 16gp8
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- u1 restricts to
y1.v
- u2 restricts to
y2.v
- r restricts to
v2
Restriction to maximal subgroup number 7, which is 16gp9
- y1 restricts to
y2
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- u1 restricts to
y2.v
- u2 restricts to
y1.v
- r restricts to
v2
Restriction to maximal elementary abelian number 1, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- u1 restricts to
0
- u2 restricts to
y12.y23
+ y14.y2
- r restricts to
y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V4
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y14.y24
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y8
(1 + 3t + 4t2
+ 3t3 + t4 + t5
+ 3t6 + 3t7 + t8) /
(1 - t2) (1 - t8)
Back to the groups of order 32