Small group number 7 of order 81

G = Syl3(A9) is Sylow 3-subgroup of A_9

G has 2 minimal generators, rank 3 and exponent 9. The centre has rank 1.

The 4 maximal subgroups are: E27, M27 (2x), V27.

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

This cohomology ring calculation is complete.

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


Ring structure

The cohomology ring has 16 generators:

There are 88 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 11 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 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 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 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 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.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V27

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 V27

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C3


Poincaré series

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


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