Small group number 8 of order 81

G is the group 81gp8

G has 2 minimal generators, rank 2 and exponent 9. The centre has rank 1.

The 4 maximal subgroups are: Ab(9,3), E27, M27 (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


Ring structure

The cohomology ring has 13 generators:

There are 65 minimal relations:

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

Essential ideal: There is one minimal generator:

Nilradical: There are 9 minimal generators:


Completion information

This cohomology ring was obtained from a calculation out to degree 14. The cohomology ring approximation is stable from degree 14 onwards, and Carlson's tests detect stability from degree 14 onwards.

This cohomology ring has dimension 2 and depth 1. Here is a homogeneous system of parameters:

The first term h1 forms a regular sequence of maximum length. The remaining term h2 is annihilated by the class x1.

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 1 as a module over the polynomial algebra on h1. 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 10.

A basis for AnnR/(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 27gp2

Restriction to maximal subgroup number 2, which is 27gp3

Restriction to maximal subgroup number 3, which is 27gp4

Restriction to maximal subgroup number 4, which is 27gp4

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V9

Restriction to maximal elementary abelian number 2, which is V9

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C3


Poincaré series

(1 + 2t + 3t2 + 3t3 + 2t4 + 2t5 + t6 + t7 + t8) / (1 - t4) (1 - t6)


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