Small group number 13 of order 81

G = M27xC3 is Direct product M27 x C_3

G has 3 minimal generators, rank 3 and exponent 9. The centre has rank 2.

The 13 maximal subgroups are: Ab(9,3) (3x), M27 (9x), V27.

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 8 generators:

There are 10 minimal relations:

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

Essential ideal: There are 4 minimal generators:

Nilradical: There are 5 minimal generators:


Completion information

This cohomology ring was obtained from a calculation out to degree 10. The cohomology ring approximation is stable from degree 10 onwards, and Carlson's tests detect stability from degree 10 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 free of rank 4 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, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 10.

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.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V27

Restriction to maximal subgroup number 2, which is 27gp2

Restriction to maximal subgroup number 3, which is 27gp2

Restriction to maximal subgroup number 4, which is 27gp2

Restriction to maximal subgroup number 5, which is 27gp4

Restriction to maximal subgroup number 6, which is 27gp4

Restriction to maximal subgroup number 7, which is 27gp4

Restriction to maximal subgroup number 8, which is 27gp4

Restriction to maximal subgroup number 9, which is 27gp4

Restriction to maximal subgroup number 10, which is 27gp4

Restriction to maximal subgroup number 11, which is 27gp4

Restriction to maximal subgroup number 12, which is 27gp4

Restriction to maximal subgroup number 13, which is 27gp4

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 V9


Poincaré series

(1 + 3t + 3t2 + t3) / (1 - t2)2 (1 - t6)


Back to the groups of order 81