David J. Green

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Cohomology of group number 2216 of order 128

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General information on the group

• The group has 5 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 5.
• Its center has rank 2.
• It has 6 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4, 4, 4, 4, 5 and 5, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 5 and depth 4.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  − 1

  (t  −  1)5 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 4:

1. b_1_0, an element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. b_1_4, an element of degree 1
6. c_2_13, a Duflot regular element of degree 2
7. b_3_29, an element of degree 3
8. c_4_55, a Duflot regular element of degree 4

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Ring relations

There are 5 minimal relations of maximal degree 6:

1. b_1_0·b_1_2
2. b_1_1·b_1_3 + b_1_0·b_1_4 + b_1_0·b_1_1
3. b_1_0·b_1_4² + b_1_0·b_1_3·b_1_4 + b_1_0·b_1_1·b_1_4
4. b_1_0·b_3_29
5. b_3_29² + b_1_2·b_1_4²·b_3_29 + b_1_2·b_1_3·b_1_4·b_3_29 + b_1_1·b_1_2·b_1_4·b_3_29
      + c_4_55·b_1_2² + c_2_13·b_1_4⁴ + c_2_13·b_1_3²·b_1_4² + c_2_13·b_1_1²·b_1_4²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 13.
• However, the last relation was already found in degree 6 and the last generator in degree 4.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_13, a Duflot regular element of degree 2
  2. c_4_55, a Duflot regular element of degree 4
  3. b_1_4⁴ + b_1_3²·b_1_4² + b_1_3⁴ + b_1_2·b_1_3·b_1_4² + b_1_2·b_1_3²·b_1_4
        + b_1_2²·b_1_4² + b_1_2²·b_1_3·b_1_4 + b_1_2²·b_1_3² + b_1_2⁴
        + b_1_1·b_1_2·b_1_4² + b_1_1·b_1_2²·b_1_4 + b_1_1²·b_1_4² + b_1_1²·b_1_2·b_1_4
        + b_1_1²·b_1_2² + b_1_1⁴ + b_1_0²·b_1_3² + b_1_0²·b_1_1² + b_1_0⁴, an element of degree 4
  4. b_1_3²·b_1_4⁴ + b_1_3⁴·b_1_4² + b_1_2·b_1_3·b_1_4⁴ + b_1_2·b_1_3⁴·b_1_4
        + b_1_2²·b_1_4⁴ + b_1_2²·b_1_3²·b_1_4² + b_1_2²·b_1_3⁴ + b_1_2⁴·b_1_4²
        + b_1_2⁴·b_1_3·b_1_4 + b_1_2⁴·b_1_3² + b_1_1·b_1_2·b_1_4⁴ + b_1_1·b_1_2⁴·b_1_4
        + b_1_1²·b_1_4⁴ + b_1_1²·b_1_2²·b_1_4² + b_1_1²·b_1_2⁴ + b_1_1⁴·b_1_4²
        + b_1_1⁴·b_1_2·b_1_4 + b_1_1⁴·b_1_2² + b_1_0²·b_1_3⁴ + b_1_0²·b_1_1⁴
        + b_1_0⁴·b_1_3² + b_1_0⁴·b_1_1², an element of degree 6
  5. b_1_4², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 11, 13].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

About the group

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Restriction maps

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. c_2_13 → c_1_1², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. c_4_55 → c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_2, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. c_2_13 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. c_4_55 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_2, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_1_4 → c_1_3, an element of degree 1
6. c_2_13 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. c_4_55 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → 0, an element of degree 1
6. c_2_13 → c_1_1·c_1_3 + c_1_1², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. c_4_55 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_3 + c_1_2, an element of degree 1
5. b_1_4 → c_1_2, an element of degree 1
6. c_2_13 → c_1_1·c_1_3 + c_1_1², an element of degree 2
7. b_3_29 → 0, an element of degree 3
8. c_4_55 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_1_4 → c_1_4, an element of degree 1
6. c_2_13 → c_1_1·c_1_3 + c_1_1², an element of degree 2
7. b_3_29 → c_1_1·c_1_4² + c_1_1·c_1_2·c_1_4 + c_1_0·c_1_2·c_1_3 + c_1_0²·c_1_3, an element of degree 3
8. c_4_55 → c_1_0·c_1_2·c_1_4² + c_1_0·c_1_2²·c_1_4 + c_1_0²·c_1_4² + c_1_0²·c_1_2·c_1_4
      + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. b_1_4 → c_1_4, an element of degree 1
6. c_2_13 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. b_3_29 → c_1_1·c_1_4² + c_1_1·c_1_3·c_1_4 + c_1_0·c_1_2·c_1_3 + c_1_0²·c_1_2, an element of degree 3
8. c_4_55 → c_1_0·c_1_3·c_1_4² + c_1_0·c_1_3²·c_1_4 + c_1_0²·c_1_4² + c_1_0²·c_1_3·c_1_4
      + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128