G = Q8 is Quaternion group of order 8
The Hall-Senior number of this group is 5.
G has 2 minimal generators, rank 1 and exponent 4. The centre has rank 1.
The 3 maximal subgroups are: C4 (3x).
There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 1.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 3 generators:
There are 2 minimal relations:
This minimal generating set constitutes a Gröbner basis for the relations ideal.
Essential ideal: There are 2 minimal generators:
Nilradical: There are 2 minimal generators:
This cohomology ring was obtained from a calculation out to degree 4. The cohomology ring approximation is stable from degree 4 onwards, and Carlson's tests detect stability from degree 4 onwards.
This cohomology ring has dimension 1 and depth 1. Here is a homogeneous system of parameters:
The first term h1 forms 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 free of rank 3 as a module over the polynomial algebra on h1. These free generators are:
A basis for R/(h1) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 4.
(1 + 2t + 2t2 + t3) / (1 - t4)