Small group number 33 of order 64

G is the group 64gp33

The Hall-Senior number of this group is 251.

G has 2 minimal generators, rank 3 and exponent 8. The centre has rank 1.

The 3 maximal subgroups are: 32gp30, 32gp6, 32gp7.

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

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 19 generators:

There are 134 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

Essential ideal: There is one minimal generator:

Nilradical: There are 3 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 18 onwards, and Carlson's tests detect stability from degree 18 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 y22.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, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 14.

A basis for AnnR/(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.

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 12.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 32gp30

Restriction to maximal subgroup number 2, which is 32gp6

Restriction to maximal subgroup number 3, which is 32gp7

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V8

Restriction to maximal elementary abelian number 2, which is V8

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C2

Poincaré series

(1 + 2t + 2t2 + 3t3 + 3t4 + 2t5 + 3t6 + 3t7 + 2t8 + 2t9 + t10) / (1 - t2) (1 - t4) (1 - t8)

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