Small group number 10 of order 32

G is the group 32gp10

The Hall-Senior number of this group is 28.

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

The 3 maximal subgroups are: Q8xC2, 16gp4, Ab(8,2).

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 2.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series


Ring structure

The cohomology ring has 6 generators:

There are 9 minimal relations:

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

Essential ideal: There is one minimal generator:

Nilradical: There are 4 minimal generators:


Completion information

This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 6 onwards, and Carlson's tests detect stability from degree 6 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 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 free of rank 1 as a module over the polynomial algebra on h1, h2. These free generators are:

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 6.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 16gp12

Restriction to maximal subgroup number 2, which is 16gp4

Restriction to maximal subgroup number 3, which is 16gp5

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V4

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is V4


Poincaré series

(1 + 2t + 2t2 + 2t3 + t4) / (1 - t2) (1 - t4)


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