G is the group 16gp3
The Hall-Senior number of this group is 9.
G has 2 minimal generators, rank 3 and exponent 4. The centre has rank 2.
The 3 maximal subgroups are: Ab(4,2) (2x), V8.
There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 3.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 5 generators:
There are 4 minimal relations:
This minimal generating set constitutes a Gröbner basis for the relations ideal.
Essential ideal: Zero ideal
Nilradical: There is one minimal generator:
This cohomology ring was obtained from a calculation out to degree 6. The cohomology ring approximation is stable from degree 4 onwards, and Carlson's tests detect stability from degree 6 onwards.
This cohomology ring has dimension 3 and depth 2. Here is a homogeneous system of parameters:
The first 2 terms h1, h2 form a regular sequence of maximum length. The remaining term h3 is annihilated by the class y1.
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 6.
A basis for AnnR/(h1, h2)(h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 4.
(1 + 2t + t2) / (1 - t2)3