David J. Green

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Cohomology of group number 42 of order 64

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Ring generators

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General information on the group

• The group has 2 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 1.
• 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 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t2  −  t  +  1)

  (t  −  1)3 · (t2  +  1)

• The a-invariants are -∞,-3,-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. b_1_1, an element of degree 1
3. b_2_1, an element of degree 2
4. b_2_2, an element of degree 2
5. b_3_3, an element of degree 3
6. b_3_4, an element of degree 3
7. b_4_6, an element of degree 4
8. c_4_7, a Duflot regular element of degree 4

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Ring relations

There are 18 minimal relations of maximal degree 8:

1. a_1_0²
2. a_1_0·b_1_1
3. b_2_1·a_1_0
4. b_2_2·a_1_0
5. b_2_2·b_1_1² + b_2_1²
6. a_1_0·b_3_3
7. b_1_1·b_3_3 + b_2_1·b_2_2
8. a_1_0·b_3_4
9. b_2_2²·b_1_1 + b_2_1·b_3_3
10. b_4_6·a_1_0
11. b_4_6·b_1_1 + b_2_2²·b_1_1 + b_2_1·b_3_4 + b_2_1·b_2_2·b_1_1
12. b_3_3² + b_2_2³
13. b_3_4² + b_2_2·b_1_1·b_3_4 + c_4_7·b_1_1²
14. b_2_2·b_1_1·b_3_4 + b_2_1·b_4_6 + b_2_1·b_2_2² + b_2_1²·b_2_2
15. b_3_3·b_3_4 + b_2_2·b_4_6 + b_2_2³ + b_2_1·b_2_2²
16. b_4_6·b_3_3 + b_2_2²·b_3_4 + b_2_2²·b_3_3 + b_2_1·b_2_2·b_3_3
17. b_4_6·b_3_4 + b_2_2²·b_3_4 + b_2_1·c_4_7·b_1_1
18. b_4_6² + b_2_2⁴ + b_2_1·b_2_2·b_4_6 + b_2_1·b_2_2³ + b_2_1²·c_4_7

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_7, a Duflot regular element of degree 4
  2. b_1_1² + b_2_2 + b_2_1, an element of degree 2
  3. b_1_1, an element of degree 1
• The Raw Filter Degree Type of that HSOP is [-1, 1, 3, 4].
• 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 1

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_2_1 → 0, an element of degree 2
4. b_2_2 → 0, an element of degree 2
5. b_3_3 → 0, an element of degree 3
6. b_3_4 → 0, an element of degree 3
7. b_4_6 → 0, an element of degree 4
8. c_4_7 → c_1_0⁴, 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. b_1_1 → c_1_1, an element of degree 1
3. b_2_1 → c_1_1·c_1_2, an element of degree 2
4. b_2_2 → c_1_2², an element of degree 2
5. b_3_3 → c_1_2³, an element of degree 3
6. b_3_4 → c_1_1²·c_1_2 + c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
7. b_4_6 → c_1_2⁴ + c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_1·c_1_2, an element of degree 4
8. c_4_7 → c_1_1·c_1_2³ + c_1_1²·c_1_2² + c_1_0·c_1_1·c_1_2² + c_1_0²·c_1_2²
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4

About the group

Ring generators

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