David J. Green

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Cohomology of group number 137 of order 64

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Ring generators

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  (t2  −  t  +  1) · (t2  +  t  +  1)2

  (t  −  1)2 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-2. 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 8:

1. a_1_0, a nilpotent element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. a_2_3, a nilpotent element of degree 2
5. b_2_5, an element of degree 2
6. a_5_7, a nilpotent element of degree 5
7. c_8_18, a Duflot regular element of degree 8

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Ring relations

There are 10 minimal relations of maximal degree 10:

1. a_1_0·b_1_1
2. a_1_0·b_1_2
3. a_2_3·a_1_0 + a_1_0³
4. b_1_2³ + b_1_1·b_1_2² + b_2_5·b_1_1 + a_1_0³
5. b_2_5·b_1_1·b_1_2 + b_2_5·b_1_1² + a_2_3·b_1_2² + a_2_3·b_1_1·b_1_2 + a_2_3²
6. b_2_5²·b_1_1 + a_2_3·b_2_5·b_1_1 + a_2_3²·b_1_2
7. b_2_5·b_1_1⁴ + b_2_5²·b_1_2² + b_1_2·a_5_7 + a_2_3·b_1_1³·b_1_2
      + a_2_3·b_2_5·b_1_1² + a_2_3·b_2_5² + b_2_5²·a_1_0² + a_2_3²·b_1_1·b_1_2
      + a_2_3²·b_1_1²
8. b_2_5·b_1_1⁴ + b_1_1·a_5_7 + a_2_3·b_1_1²·b_1_2² + a_2_3·b_2_5·b_1_1²
      + a_2_3²·b_1_2² + a_2_3²·b_1_1²
9. a_2_3·b_2_5·b_1_1³ + a_2_3·a_5_7 + a_2_3²·b_1_1·b_1_2² + a_2_3²·b_2_5·b_1_1
      + a_1_0²·a_5_7 + b_2_5²·a_1_0³
10. a_2_3·b_2_5⁴ + a_5_7² + b_2_5²·a_1_0·a_5_7 + a_2_3·b_1_1³·a_5_7
       + a_2_3²·b_1_1⁵·b_1_2 + a_2_3²·b_1_1·a_5_7 + b_2_5·a_1_0³·a_5_7 + c_8_18·a_1_0²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• 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_8_18, a Duflot regular element of degree 8
  2. b_1_1² + b_2_5, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 8].
• The filter degree type of any filter regular HSOP is [-1, -2, -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. a_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. a_2_3 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. a_5_7 → 0, an element of degree 5
7. c_8_18 → c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_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. a_2_3 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. a_5_7 → 0, an element of degree 5
7. c_8_18 → c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_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. a_2_3 → 0, an element of degree 2
5. b_2_5 → c_1_1², an element of degree 2
6. a_5_7 → 0, an element of degree 5
7. c_8_18 → c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_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_1, an element of degree 1
4. a_2_3 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. a_5_7 → 0, an element of degree 5
7. c_8_18 → c_1_1⁸ + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 64