David J. Green

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Cohomology of group number 12 of order 81

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General information on the group

• The group is also known as E27xC3, the Direct product E27 x C_3.
• The group has 3 minimal generators and exponent 3.
• 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 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  ( − 1) · (t2  +  1)2

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

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

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Ring generators

The cohomology ring has 11 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. b_2_2, an element of degree 2
5. b_2_3, an element of degree 2
6. b_2_4, an element of degree 2
7. b_2_5, an element of degree 2
8. c_2_6, a Duflot regular element of degree 2
9. a_3_11, a nilpotent element of degree 3
10. a_3_12, a nilpotent element of degree 3
11. c_6_36, a Duflot regular element of degree 6

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Ring relations

There are 5 "obvious" relations:
   a_1_0², a_1_1², a_1_2², a_3_11², a_3_12²

Apart from that, there are 17 minimal relations of maximal degree 6:

1. a_1_0·a_1_1
2. b_2_3·a_1_0 − b_2_2·a_1_1
3. b_2_4·a_1_1 + b_2_3·a_1_1 − b_2_2·a_1_1
4. b_2_4·a_1_0 − b_2_3·a_1_1 + b_2_2·a_1_1
5. b_2_5·a_1_0 − b_2_2·a_1_1
6.  − b_2_3² + b_2_2·b_2_4 + b_2_2·b_2_3
7.  − b_2_4² + b_2_3·b_2_5 + b_2_3·b_2_4 − b_2_3²
8.  − b_2_3·b_2_4 − b_2_3² + b_2_2·b_2_5
9.  − b_2_4² + b_2_3·b_2_4 + a_1_1·a_3_11
10.  − b_2_3·b_2_4 − b_2_3² + b_2_2·b_2_3 + a_1_0·a_3_11
11.  − b_2_4·b_2_5 + b_2_4² + a_1_1·a_3_12
12.  − b_2_4² − b_2_3² + b_2_2·b_2_3 + a_1_0·a_3_12
13. b_2_5·a_3_11 − b_2_3·a_3_11
14.  − b_2_4·a_3_11 + b_2_3·a_3_12 + b_2_3·a_3_11 + b_2_2·b_2_3·a_1_1 − b_2_2²·a_1_1
15.  − b_2_3·a_3_11 + b_2_2·a_3_12 − b_2_2·a_3_11 − b_2_2·b_2_3·a_1_1 + b_2_2²·a_1_1
16. b_2_4·a_3_12 + b_2_2·b_2_3·a_1_1 − b_2_2²·a_1_1
17. a_3_11·a_3_12 + b_2_2·a_1_1·a_3_12 + b_2_2·a_1_0·a_3_12 − b_2_2·a_1_0·a_3_11

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 7.
• However, the last relation was already found in degree 6 and the last generator in degree 6.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_6, a Duflot regular element of degree 2
  2. c_6_36, a Duflot regular element of degree 6
  3. b_2_5² − b_2_3·b_2_5 + b_2_2·b_2_5 − b_2_2·b_2_3 + b_2_2², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 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 → a_1_0, an element of degree 1
4. b_2_2 → 0, an element of degree 2
5. b_2_3 → 0, an element of degree 2
6. b_2_4 → 0, an element of degree 2
7. b_2_5 → 0, an element of degree 2
8. c_2_6 → c_2_1, an element of degree 2
9. a_3_11 → 0, an element of degree 3
10. a_3_12 → 0, an element of degree 3
11. c_6_36 →  − 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 → a_1_0, an element of degree 1
4. b_2_2 → c_2_5, an element of degree 2
5. b_2_3 →  − a_1_1·a_1_2, an element of degree 2
6. b_2_4 → a_1_1·a_1_2, an element of degree 2
7. b_2_5 → 0, an element of degree 2
8. c_2_6 → c_2_3, an element of degree 2
9. a_3_11 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
10. a_3_12 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
11. c_6_36 →  − c_2_5²·a_1_1·a_1_2 + 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 → a_1_0, an element of degree 1
4. b_2_2 → 0, an element of degree 2
5. b_2_3 → 0, an element of degree 2
6. b_2_4 → a_1_1·a_1_2, an element of degree 2
7. b_2_5 → c_2_5, an element of degree 2
8. c_2_6 → c_2_3, an element of degree 2
9. a_3_11 → 0, an element of degree 3
10. a_3_12 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
11. c_6_36 → 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 → a_1_0, an element of degree 1
4. b_2_2 → c_2_5, an element of degree 2
5. b_2_3 →  − a_1_1·a_1_2 + c_2_5, an element of degree 2
6. b_2_4 →  − a_1_1·a_1_2, an element of degree 2
7. b_2_5 → c_2_5, an element of degree 2
8. c_2_6 → c_2_3, an element of degree 2
9. a_3_11 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
10. a_3_12 → c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
11. c_6_36 → 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 → a_1_0, an element of degree 1
4. b_2_2 →  − c_2_5, an element of degree 2
5. b_2_3 → a_1_1·a_1_2 + c_2_5, an element of degree 2
6. b_2_4 → c_2_5, an element of degree 2
7. b_2_5 → c_2_5, an element of degree 2
8. c_2_6 → c_2_3, an element of degree 2
9. a_3_11 → c_2_5·a_1_2 + c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
10. a_3_12 →  − c_2_5·a_1_2, an element of degree 3
11. c_6_36 → c_2_5²·a_1_1·a_1_2 + c_2_5³ + c_2_4·c_2_5² − c_2_4³, an element of degree 6

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 81