David J. Green

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Cohomology of group number 134 of order 64

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Ring generators

Ring relations

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General information on the group

• The group is also known as Syl2(M12), the Sylow 2-subgroup of Mathieu Group M_12.
• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 1.
• It has 5 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 1.
• The Poincaré series is

  − 1

  (t  −  1)3

• 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 7 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_3_9, an element of degree 3
7. c_4_14, 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_0
4. b_2_5·b_1_1
5. b_2_4·b_1_2² + b_2_4·b_1_1·b_1_2 + b_2_4²
6. b_1_2·b_3_9 + b_2_4·b_2_5
7. b_1_1·b_3_9
8. b_2_4·b_3_9 + b_2_4·b_2_5·b_1_2
9. b_3_9² + b_1_0³·b_3_9 + b_2_5·b_1_0·b_3_9 + b_2_4·b_2_5² + c_4_14·b_1_0²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• 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_4_14, a Duflot regular element of degree 4
  2. b_1_2² + b_1_1·b_1_2 + b_1_1² + b_1_0² + b_2_5, an element of degree 2
  3. b_1_1·b_1_2² + b_1_1²·b_1_2 + b_2_5·b_1_2 + b_2_5·b_1_0, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6].
• 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 term of the above HSOP, together with 2 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_3_9 → 0, an element of degree 3
7. c_4_14 → c_1_0⁴, an element of degree 4

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

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_3_9 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
7. c_4_14 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0·c_1_1³ + c_1_0²·c_1_2²
      + c_1_0²·c_1_1·c_1_2 + 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_3_9 → 0, an element of degree 3
7. c_4_14 → 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 → c_1_2, 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_3_9 → 0, an element of degree 3
7. c_4_14 → c_1_2⁴ + c_1_1·c_1_2³ + 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_3_9 → 0, an element of degree 3
7. c_4_14 → 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_2, an element of degree 1
4. b_2_4 → c_1_2², an element of degree 2
5. b_2_5 → c_1_1·c_1_2 + c_1_1², an element of degree 2
6. b_3_9 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
7. c_4_14 → c_1_2⁴ + 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

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 64