Small group number 46 of order 32
G is the group 32gp46
The Hall-Senior number of this group is 8.
G has 4 minimal generators, rank 4 and exponent 4.
The centre has rank 3.
The 15 maximal subgroups are:
Ab(4,2,2), D8xC2 (12x), V16 (2x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
4, 4.
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
- y2 in degree 1
- y3 in degree 1, a regular element
- y4 in degree 1, a regular element
- x in degree 2, a regular element
There is one minimal relation:
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 2 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 4 and depth 4.
Here is a homogeneous system of parameters:
- h1 =
y3
in degree 1
- h2 =
y4
in degree 1
- h3 =
x
in degree 2
- h4 =
y22
+ y12
in degree 2
The first
4 terms h1, h2, h3, h4 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
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 6.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y12
in degree 2
Restriction to maximal subgroup number 1, which is V16
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
y1
- x restricts to
y42
+ y3.y4
Restriction to maximal subgroup number 2, which is V16
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
y1
- x restricts to
y42
+ y3.y4
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
- x restricts to
x
Restriction to maximal subgroup number 4, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- y4 restricts to
y3
- x restricts to
x
Restriction to maximal subgroup number 5, which is 16gp11
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
- y4 restricts to
y3
- x restricts to
x
Restriction to maximal subgroup number 6, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- y4 restricts to
y3
- x restricts to
x
Restriction to maximal subgroup number 7, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- y4 restricts to
y2
+ y1
+ y3
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 8, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- x restricts to
x
Restriction to maximal subgroup number 9, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y2
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 10, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y1
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 11, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
+ y3
- y4 restricts to
y2
+ y1
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 12, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
- x restricts to
x
Restriction to maximal subgroup number 13, which is 16gp11
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
+ y3
- y4 restricts to
y3
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 14, which is 16gp11
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
+ y3
- y4 restricts to
y2
+ y3
- x restricts to
x
+ y2.y3
+ y1.y3
+ y32
Restriction to maximal subgroup number 15, which is 16gp11
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
+ y3
- y4 restricts to
y3
- x restricts to
x
Restriction to maximal elementary abelian number 1, which is V16
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- y3 restricts to
y4
- y4 restricts to
y3
- x restricts to
y1.y2
Restriction to maximal elementary abelian number 2, which is V16
- y1 restricts to
y4
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
y2
- x restricts to
y1.y4
+ y12
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
- x restricts to
y32
(1 + 2t + t2) /
(1 - t)2 (1 - t2)2
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