David J. Green

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Cohomology of group number 56 of order 128

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General information on the group

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

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  − 1

  (t  +  1) · (t  −  1)3

• The a-invariants are -∞,-∞,-3,-3. 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. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_2_1, a nilpotent element of degree 2
4. b_2_2, an element of degree 2
5. c_2_3, a Duflot regular element of degree 2
6. a_3_4, a nilpotent element of degree 3
7. a_3_5, a nilpotent element of degree 3
8. c_4_8, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

1. a_1_0²
2. a_1_0·a_1_1
3. a_2_1·a_1_1 + a_1_1³
4. a_2_1·a_1_0 + a_1_1³
5. b_2_2·a_1_0
6. a_2_1²
7. a_1_1·a_3_4 + b_2_2·a_1_1²
8. a_1_0·a_3_4
9. a_1_0·a_3_5
10. a_2_1·a_3_5 + a_2_1·a_3_4 + c_2_3·a_1_1³
11. a_2_1·a_3_4 + a_1_1²·a_3_5 + c_2_3·a_1_1³
12. a_3_4²
13. a_2_1·b_2_2² + a_3_4·a_3_5 + b_2_2·a_1_1·a_3_5
14. a_2_1·b_2_2² + a_3_5² + b_2_2·a_1_1·a_3_5 + b_2_2²·a_1_1² + c_4_8·a_1_1²
       + c_2_3²·a_1_1²

About the group

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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_2_3, a Duflot regular element of degree 2
  2. c_4_8, a Duflot regular element of degree 4
  3. b_2_2, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 5].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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_2_1 → 0, an element of degree 2
4. b_2_2 → 0, an element of degree 2
5. c_2_3 → c_1_0², an element of degree 2
6. a_3_4 → 0, an element of degree 3
7. a_3_5 → 0, an element of degree 3
8. c_4_8 → c_1_1⁴, an element of degree 4

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 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. b_2_2 → c_1_2², an element of degree 2
5. c_2_3 → c_1_0², an element of degree 2
6. a_3_4 → 0, an element of degree 3
7. a_3_5 → 0, an element of degree 3
8. c_4_8 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2², an element of degree 4

About the group

Ring generators

Ring relations

Completion information

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Back to groups of order 128