David J. Green

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Cohomology of group number 64 of order 243

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General information on the group

• The group is also known as (E27*C3)xC3, the Direct product E27*C3 x C_3.
• The group has 4 minimal generators and exponent 9.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t4  +  t2  +  t  +  1)

  (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)

• The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 6:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. a_1_3, a nilpotent element of degree 1
5. b_2_6, an element of degree 2
6. b_2_7, an element of degree 2
7. c_2_8, a Duflot regular element of degree 2
8. b_4_17, an element of degree 4
9. c_6_38, a Duflot regular element of degree 6

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Ring relations

There are 4 "obvious" relations:
   a_1_0², a_1_1², a_1_2², a_1_3²

Apart from that, there are 6 minimal relations of maximal degree 8:

1. b_2_7·a_1_0 − b_2_6·a_1_1
2. b_4_17·a_1_1 − b_2_6²·a_1_1
3. b_4_17·a_1_0 − b_2_6·b_2_7·a_1_1
4. b_2_7·b_4_17 − b_2_6²·b_2_7
5. b_2_6·b_4_17 − b_2_6·b_2_7²
6. b_4_17² − b_2_6²·b_2_7²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_8, a Duflot regular element of degree 2
  2. c_6_38, a Duflot regular element of degree 6
  3. b_4_17 − b_2_7² − b_2_6², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 9].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
• We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 1 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_1_3 → a_1_0, an element of degree 1
5. b_2_6 → 0, an element of degree 2
6. b_2_7 → 0, an element of degree 2
7. c_2_8 → c_2_1, an element of degree 2
8. b_4_17 → 0, an element of degree 4
9. c_6_38 → c_2_2³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → a_1_2, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_1_3 → a_1_0, an element of degree 1
5. b_2_6 → c_2_5, an element of degree 2
6. b_2_7 → 0, an element of degree 2
7. c_2_8 → c_2_3, an element of degree 2
8. b_4_17 → 0, an element of degree 4
9. c_6_38 →  − c_2_4·c_2_5² + c_2_4³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_2, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_1_3 → a_1_0, an element of degree 1
5. b_2_6 → 0, an element of degree 2
6. b_2_7 → c_2_5, an element of degree 2
7. c_2_8 → c_2_3, an element of degree 2
8. b_4_17 → 0, an element of degree 4
9. c_6_38 →  − c_2_4·c_2_5² + c_2_4³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → a_1_2, an element of degree 1
2. a_1_1 → a_1_2, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_1_3 → a_1_0, an element of degree 1
5. b_2_6 → c_2_5, an element of degree 2
6. b_2_7 → c_2_5, an element of degree 2
7. c_2_8 → c_2_3, an element of degree 2
8. b_4_17 → c_2_5², an element of degree 4
9. c_6_38 →  − c_2_4·c_2_5² + c_2_4³, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 →  − a_1_2, an element of degree 1
2. a_1_1 → a_1_2, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. a_1_3 → a_1_0, an element of degree 1
5. b_2_6 →  − c_2_5, an element of degree 2
6. b_2_7 → c_2_5, an element of degree 2
7. c_2_8 → c_2_3, an element of degree 2
8. b_4_17 → c_2_5², an element of degree 4
9. c_6_38 →  − c_2_4·c_2_5² + c_2_4³, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243