Small group number 16 of order 243

G is the group 243gp16

G has 2 minimal generators, rank 3 and exponent 27. The centre has rank 1.

The 4 maximal subgroups are: Ab(9,3,3), 81gp6 (3x).

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

Ring structure

The cohomology ring has 17 generators:

There are 103 minimal relations:

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

Essential ideal: There is one minimal generator:

Nilradical: There are 13 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 16. The cohomology ring approximation is stable from degree 16 onwards, and Carlson's tests detect stability from degree 16 onwards.

This cohomology ring has dimension 3 and depth 1. Here is a homogeneous system of parameters:

The first term h1 forms a regular sequence of maximum length. The remaining 2 terms h2, h3 are all annihilated by the class y1.

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, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 12.

A basis for AnnR/(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.

A basis for AnnR/(h1)(h2) / h3 AnnR/(h1)(h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 10.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 81gp11

Restriction to maximal subgroup number 2, which is 81gp6

Restriction to maximal subgroup number 3, which is 81gp6

Restriction to maximal subgroup number 4, which is 81gp6

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V27

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C3

Poincaré series

(1 + 2t + 2t2 + 2t3 + t4 + t5 + 2t6 + 2t7 + 2t8 + t9) / (1 - t2) (1 - t4) (1 - t6)

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