David J. Green

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Cohomology of group number 67 of order 243

About the group

Ring generators

Ring relations

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General information on the group

• The group has 5 minimal generators and exponent 3.
• It is abelian, isomorphic to C₃×C₃×C₃×C₃×C₃.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 5 and depth 5.
• The Poincaré series is

  − 1

  (t  −  1)5

• The a-invariants are -∞,-∞,-∞,-∞,-∞,-5.

About the group

Ring generators

Ring relations

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 2:

1. a_1_0, an element of degree 1
2. a_1_1, an element of degree 1
3. a_1_2, an element of degree 1
4. a_1_3, an element of degree 1
5. a_1_4, an element of degree 1
6. c_2_10, an element of degree 2
7. c_2_11, an element of degree 2
8. c_2_12, an element of degree 2
9. c_2_13, an element of degree 2
10. c_2_14, an element of degree 2

About the group

Ring generators

Ring relations

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Ring relations

There are 5 "obvious" relations:
   a_1_0², a_1_1², a_1_2², a_1_3², a_1_4²

Apart from that, there are no relations.

About the group

Ring generators

Ring relations

Back to groups of order 243