Small group number 7 of order 16

G = D16 is Dihedral group of order 16

The Hall-Senior number of this group is 12.

G has 2 minimal generators, rank 2 and exponent 8. The centre has rank 1.

The 3 maximal subgroups are: D8 (2x), C8.

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


Ring structure

The cohomology ring has 3 generators:

There is one minimal relation:

This minimal generating set constitutes a Gröbner basis for the relations ideal.

Essential ideal: Zero ideal

Nilradical: Zero ideal


Completion information

This cohomology ring was obtained from a calculation out to degree 4. The cohomology ring approximation is stable from degree 2 onwards, and Carlson's tests detect stability from degree 4 onwards.

This cohomology ring has dimension 2 and depth 2. Here is a homogeneous system of parameters:

The first 2 terms h1, h2 form a regular sequence of maximum length.

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 the zero ideal. The essential ideal squares to zero.


Koszul information

A basis for R/(h1, h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 4.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 8gp3

Restriction to maximal subgroup number 2, which is 8gp3

Restriction to maximal subgroup number 3, which is C8

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V4

Restriction to maximal elementary abelian number 2, which is V4

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C2


Poincaré series

(1 + 2t + t2) / (1 - t2)2


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