Small group number 47 of order 32
G is the group 32gp47
The Hall-Senior number of this group is 9.
G has 4 minimal generators, rank 3 and exponent 4.
The centre has rank 3.
The 15 maximal subgroups are:
Ab(4,2,2) (3x), Q8xC2 (12x).
There is one conjugacy class of maximal elementary abelian
subgroups. Each maximal elementary abelian has rank 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 5 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1, a nilpotent element
- y3 in degree 1, a regular element
- y4 in degree 1, a regular element
- v in degree 4, a regular element
There are 2 minimal relations:
- y22 =
y1.y2
+ y12
- y13 =
0
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
There are 3 minimal generators:
-
y12.y2.y3.y42
+ y12.y2.y32.y4
-
y1.y2.y32.y44
+ y1.y2.y34.y42
-
y12.y32.y44
+ y12.y34.y42
Nilradical:
There are 2 minimal generators:
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 4 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
y3
in degree 1
- h2 =
y4
in degree 1
- h3 =
v
in degree 4
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
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
free of rank 3 as a module over the polynomial algebra
on h1, h2, h3.
These free generators are:
- G1 =
y12.y2.y3.y42
+ y12.y2.y32.y4
in degree 6
- G2 =
y1.y2.y32.y44
+ y1.y2.y34.y42
in degree 8
- G3 =
y12.y32.y44
+ y12.y34.y42
in degree 8
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 6.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y1.y2
in degree 2
-
y12
in degree 2
-
y12.y2
in degree 3
Restriction to maximal subgroup number 1, which is 16gp10
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
x2
Restriction to maximal subgroup number 2, which is 16gp10
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
x2
Restriction to maximal subgroup number 3, which is 16gp10
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
x2
Restriction to maximal subgroup number 4, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 5, which is 16gp12
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 6, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 7, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 8, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- v restricts to
v
Restriction to maximal subgroup number 9, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y2
+ y1
- v restricts to
v
+ y34
Restriction to maximal subgroup number 10, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
y1
- v restricts to
v
Restriction to maximal subgroup number 11, which is 16gp12
- y1 restricts to
y2
+ y1
- y2 restricts to
y2
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
y1
- v restricts to
v
Restriction to maximal subgroup number 12, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 13, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
+ y2
- y4 restricts to
y3
+ y2
+ y1
- v restricts to
v
+ y34
Restriction to maximal subgroup number 14, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
+ y2
+ y1
- v restricts to
v
+ y34
Restriction to maximal subgroup number 15, which is 16gp12
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
y3
+ y1
- v restricts to
v
+ y34
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
y14
Restriction to the greatest central elementary abelian, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
y1
- v restricts to
y34
(1 + 2t + 2t2
+ t3) /
(1 - t)2 (1 - t4)
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