David J. Green

Cohomology
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Cohomology of group number 83 of order 64

About the group

Ring generators

Ring relations

Back to groups of order 64

General information on the group

• The group has 3 minimal generators and exponent 8.
• It is abelian, isomorphic to C₂×C₄×C₈.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The Poincaré series is

  − 1

  (t  −  1)3

• The a-invariants are -∞,-∞,-∞,-3.

About the group

Ring generators

Ring relations

Back to groups of order 64

Ring generators

The cohomology ring has 5 minimal generators of maximal degree 2:

1. c_1_0, an element of degree 1
2. c_1_1, an element of degree 1
3. c_1_2, an element of degree 1
4. c_2_4, an element of degree 2
5. c_2_5, an element of degree 2

About the group

Ring generators

Ring relations

Back to groups of order 64

Ring relations

There are 2 minimal relations of maximal degree 2:

1. c_1_0²
2. c_1_1²

About the group

Ring generators

Ring relations

Back to groups of order 64