Small group number 18 of order 32

G = D32 is Dihedral group of order 32

The Hall-Senior number of this group is 49.

G has 2 minimal generators, rank 2 and exponent 16. The centre has rank 1.

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.


Completion information

This cohomology ring was obtained from a calculation out to degree 8. The cohomology ring approximation is stable from degree 2 onwards, and Benson's tests detect stability from degree 3 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.

Data for Benson's test:


Koszul information

A basis for R/(h1, h2) is as follows.


Restriction information