David J. Green

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Cohomology of group number 6661 of order 256

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General information on the group

• The group is also known as Syl2(S4(7)), the Sylow 2-group of Symplectic group S4(7).
• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 1.
• It has 6 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 3, 3, 4 and 4, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1) · (t3  −  t2  −  1)

  (t  +  1) · (t  −  1)4 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-4,-4. 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_2_4, an element of degree 2
5. b_2_5, an element of degree 2
6. b_2_6, an element of degree 2
7. b_3_11, an element of degree 3
8. c_4_18, a Duflot regular element of degree 4

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Ring relations

There are 9 minimal relations of maximal degree 6:

1. b_1_0·b_1_1
2. b_1_0·b_1_2
3. b_2_4·b_1_2
4. b_2_5·b_1_1
5. b_2_6·b_1_2 + b_2_6·b_1_1
6. b_2_4·b_2_5
7. b_1_2·b_3_11
8. b_1_1·b_3_11
9. b_3_11² + b_2_6·b_1_0·b_3_11 + b_2_5·b_2_6² + b_2_4·b_2_6² + c_4_18·b_1_0²

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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 4.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_18, a Duflot regular element of degree 4
  2. b_1_2⁴ + b_1_1²·b_1_2² + b_1_1⁴ + b_1_0·b_3_11 + b_1_0⁴ + b_2_6·b_1_0² + b_2_6²
        + b_2_5² + b_2_4², an element of degree 4
  3. b_1_1²·b_1_2⁴ + b_1_1⁴·b_1_2² + b_1_0³·b_3_11 + b_2_6·b_1_0⁴ + b_2_6²·b_1_1²
        + b_2_6²·b_1_0² + b_2_5·b_1_0·b_3_11 + b_2_5·b_2_6·b_1_0² + b_2_5·b_2_6²
        + b_2_5²·b_1_2² + b_2_5²·b_1_0² + b_2_4·b_1_0·b_3_11 + b_2_4·b_2_6·b_1_0²
        + b_2_4·b_2_6² + b_2_4²·b_1_1² + b_2_4²·b_1_0², an element of degree 6
  4. b_1_0, an element of degree 1
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 10, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
• We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 3 elements of degree 2.

About the group

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

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_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_3_11 → 0, an element of degree 3
8. c_4_18 → c_1_0⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_1, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_3_11 → 0, an element of degree 3
8. c_4_18 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_1, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_4 → c_1_2² + c_1_1·c_1_2, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_3_11 → 0, an element of degree 3
8. c_4_18 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4

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

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_1, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. b_2_5 → c_1_2² + c_1_1·c_1_2, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. b_3_11 → 0, an element of degree 3
8. c_4_18 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4

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

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_2, an element of degree 1
4. b_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. b_3_11 → 0, an element of degree 3
8. c_4_18 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + 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_1, 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_2_4 → c_1_2² + c_1_1·c_1_2, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_0·c_1_1, an element of degree 2
7. b_3_11 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3 + c_1_0·c_1_1²
      + c_1_0²·c_1_1, an element of degree 3
8. c_4_18 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0·c_1_1·c_1_2·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_0²·c_1_1·c_1_3
      + c_1_0²·c_1_1·c_1_2 + c_1_0³·c_1_1 + 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_1, 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_2_4 → 0, an element of degree 2
5. b_2_5 → c_1_2² + c_1_1·c_1_2, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_0·c_1_1, an element of degree 2
7. b_3_11 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3 + c_1_0·c_1_1²
      + c_1_0²·c_1_1, an element of degree 3
8. c_4_18 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0·c_1_1·c_1_2·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_0²·c_1_1·c_1_3
      + c_1_0²·c_1_1·c_1_2 + c_1_0³·c_1_1 + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 256