Small group number 88 of order 64

G is the group 64gp88

The Hall-Senior number of this group is 94.

G has 3 minimal generators, rank 4 and exponent 8. The centre has rank 2.

The 7 maximal subgroups are: 32gp37 (2x), Ab(4,2,2,2), 32gp5 (4x).

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 4.

This cohomology ring calculation is complete.

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


Ring structure

The cohomology ring has 9 generators:

There are 18 minimal relations:

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

Essential ideal: Zero ideal

Nilradical: There are 4 minimal generators:


Completion information

This cohomology ring was obtained from a calculation out to degree 10. The cohomology ring approximation is stable from degree 8 onwards, and Carlson's tests detect stability from degree 10 onwards.

This cohomology ring has dimension 4 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 2 terms h3, h4 are all 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 the zero ideal. The essential ideal squares to zero.


Koszul information

A basis for R/(h1, h2, h3, h4) 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)(h4) 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, h4) 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) / h4 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 32gp45

Restriction to maximal subgroup number 2, which is 32gp37

Restriction to maximal subgroup number 3, which is 32gp37

Restriction to maximal subgroup number 4, which is 32gp5

Restriction to maximal subgroup number 5, which is 32gp5

Restriction to maximal subgroup number 6, which is 32gp5

Restriction to maximal subgroup number 7, which is 32gp5

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V16

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is V4


Poincaré series

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


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