G = D8 is Dihedral group of order 8
The Hall-Senior number of this group is 4.
G has 2 minimal generators, rank 2 and exponent 4. The centre has rank 1.
The 3 maximal subgroups are: C4, V4 (2x).
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
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
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.
A basis for R/(h1, h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 4.
(1 + 2t + t2) / (1 - t2)2