Small group number 11 of order 16
G = D8xC2 is Direct product D8 x C_2
The Hall-Senior number of this group is 6.
G has 3 minimal generators, rank 3 and exponent 4.
The centre has rank 2.
The 7 maximal subgroups are:
Ab(4,2), D8 (4x), V8 (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 4 generators:
- y1 in degree 1
- y2 in degree 1
- y3 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 6. The cohomology ring approximation
is stable from degree 2 onwards, and
Carlson's tests detect stability from degree 6
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
y3
in degree 1
- h2 =
x
in degree 2
- h3 =
y22
+ y12
in degree 2
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
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 5.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
y12
in degree 2
Restriction to maximal subgroup number 1, which is V8
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y1
- x restricts to
y32
+ y2.y3
Restriction to maximal subgroup number 2, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
y32
+ y2.y3
Restriction to maximal subgroup number 3, which is 8gp2
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- x restricts to
x
Restriction to maximal subgroup number 4, which is 8gp3
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- x restricts to
x
Restriction to maximal subgroup number 5, which is 8gp3
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y1
- x restricts to
x
Restriction to maximal subgroup number 6, which is 8gp3
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y1
- x restricts to
x
Restriction to maximal subgroup number 7, which is 8gp3
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- x restricts to
x
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- x restricts to
y1.y2
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
y3
- y2 restricts to
0
- y3 restricts to
y2
- x restricts to
y1.y3
+ y12
Restriction to the greatest central elementary abelian, which is V4
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y1
- x restricts to
y22
(1 + 2t + t2) /
(1 - t) (1 - t2)2
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