Small group number 245 of order 64

G is the group 64gp245

The Hall-Senior number of this group is 187.

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

The 15 maximal subgroups are: 32gp32 (15x).

There is one conjugacy class of maximal elementary abelian subgroups. Each maximal elementary abelian has rank 2.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 26 generators:

There are 270 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 are 21 minimal generators:

Nilradical: There are 24 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 22. The cohomology ring approximation is stable from degree 22 onwards, and Carlson's tests detect stability from degree 22 onwards.

This cohomology ring has dimension 2 and depth 2. Here is a homogeneous system of parameters:

The first 2 terms h1, h2 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 free of rank 75 as a module over the polynomial algebra on h1, h2. These free generators are:

The essential ideal does NOT square to zero. Here are the relations which are not of the form Gi.Gj = 0 (i=j allowed):

Koszul information

A basis for R/(h1, h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 16.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 32gp32

Restriction to maximal subgroup number 2, which is 32gp32

Restriction to maximal subgroup number 3, which is 32gp32

Restriction to maximal subgroup number 4, which is 32gp32

Restriction to maximal subgroup number 5, which is 32gp32

Restriction to maximal subgroup number 6, which is 32gp32

Restriction to maximal subgroup number 7, which is 32gp32

Restriction to maximal subgroup number 8, which is 32gp32

Restriction to maximal subgroup number 9, which is 32gp32

Restriction to maximal subgroup number 10, which is 32gp32

Restriction to maximal subgroup number 11, which is 32gp32

Restriction to maximal subgroup number 12, which is 32gp32

Restriction to maximal subgroup number 13, which is 32gp32

Restriction to maximal subgroup number 14, which is 32gp32

Restriction to maximal subgroup number 15, which is 32gp32

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V4

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is V4

Poincaré series

(1 + 4t + 8t2 + 10t3 + 12t4 + 13t5 + 16t6 + 20t7 + 16t8 + 13t9 + 12t10 + 10t11 + 8t12 + 4t13 + t14) / (1 - t8)2

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