Small group number 12 of order 81

G = E27xC3 is Direct product E27 x C_3

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

The 13 maximal subgroups are: E27 (9x), V27 (4x).

There are 4 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3.

This cohomology ring calculation is complete.

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


Ring structure

The cohomology ring has 11 generators:

There are 22 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 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 6 onwards, and Carlson's tests detect stability from degree 10 onwards.

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

The first 3 terms h1, h2, h3 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 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 10.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V27

Restriction to maximal subgroup number 2, which is V27

Restriction to maximal subgroup number 3, which is V27

Restriction to maximal subgroup number 4, which is V27

Restriction to maximal subgroup number 5, which is 27gp3

Restriction to maximal subgroup number 6, which is 27gp3

Restriction to maximal subgroup number 7, which is 27gp3

Restriction to maximal subgroup number 8, which is 27gp3

Restriction to maximal subgroup number 9, which is 27gp3

Restriction to maximal subgroup number 10, which is 27gp3

Restriction to maximal subgroup number 11, which is 27gp3

Restriction to maximal subgroup number 12, which is 27gp3

Restriction to maximal subgroup number 13, which is 27gp3

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V27

Restriction to maximal elementary abelian number 2, which is V27

Restriction to maximal elementary abelian number 3, which is V27

Restriction to maximal elementary abelian number 4, 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 + 5t2 + 7t3 + 7t4 + 5t5 + 3t6 + t7) / (1 - t2)2 (1 - t6)


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