Small group number 173 of order 64

G is the group 64gp173

The Hall-Senior number of this group is 189.

G has 3 minimal generators, rank 3 and exponent 8. The centre has rank 2.

The 7 maximal subgroups are: Ab(8,4), 32gp34, 32gp35, 32gp40 (4x).

There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3.

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 5 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relation:

Essential ideal: Zero ideal

Nilradical: There is one minimal generator:


Completion information

This cohomology ring was obtained from a calculation out to degree 10. The cohomology ring approximation is stable from degree 6 onwards, and Carlson's tests detect stability from degree 8 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.


Koszul information

A basis for R/(h1, h2, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 8.

A basis for AnnR/(h1, h2)(h3) 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 32gp3

Restriction to maximal subgroup number 2, which is 32gp34

Restriction to maximal subgroup number 3, which is 32gp35

Restriction to maximal subgroup number 4, which is 32gp40

Restriction to maximal subgroup number 5, which is 32gp40

Restriction to maximal subgroup number 6, which is 32gp40

Restriction to maximal subgroup number 7, which is 32gp40

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V8

Restriction to maximal elementary abelian number 2, which is V8

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is V4


Poincaré series

(1 + 3t + 3t2 + t3) / (1 - t2)2 (1 - t4)


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