Small group number 124 of order 64

G is the group 64gp124

The Hall-Senior number of this group is 58.

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

The 7 maximal subgroups are: 32gp11 (2x), 32gp15, Ab(8,4), 32gp38 (2x), 32gp42.

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

This cohomology ring calculation is complete.

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


Ring structure

The cohomology ring has 5 generators:

There are 3 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 2 minimal generators:


Completion information

This cohomology ring was obtained from a calculation out to degree 8. The cohomology ring approximation is stable from degree 4 onwards, and Carlson's tests detect stability from degree 6 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 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) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 6.


Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 32gp38

Restriction to maximal subgroup number 2, which is 32gp38

Restriction to maximal subgroup number 3, which is 32gp3

Restriction to maximal subgroup number 4, which is 32gp11

Restriction to maximal subgroup number 5, which is 32gp42

Restriction to maximal subgroup number 6, which is 32gp15

Restriction to maximal subgroup number 7, which is 32gp11

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V4

Restriction to maximal elementary abelian number 2, which is V4

Restriction to maximal elementary abelian number 3, which is V4

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C2


Poincaré series

(1 + 3t + 4t2 + 3t3 + t4) / (1 - t2) (1 - t4)


Back to the groups of order 64