David J. Green

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Cohomology of group number 163 of order 64

About the group

Ring generators

Ring relations

Completion information

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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 3.
• Its center has rank 2.
• It has 2 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 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 2) · (t3  +  1/2·t  +  1/2)

  (t  +  1) · (t  −  1)3 · (t2  +  1)2

• 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 10 minimal generators of maximal degree 5:

1. a_1_2, a nilpotent element of degree 1
2. b_1_0, an element of degree 1
3. b_1_1, an element of degree 1
4. a_3_1, a nilpotent element of degree 3
5. b_3_2, an element of degree 3
6. b_3_4, an element of degree 3
7. b_3_5, an element of degree 3
8. c_4_8, a Duflot regular element of degree 4
9. c_4_9, a Duflot regular element of degree 4
10. b_5_13, an element of degree 5

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Ring relations

There are 27 minimal relations of maximal degree 10:

1. a_1_2·b_1_0
2. b_1_0·b_1_1 + a_1_2²
3. a_1_2·b_1_1²
4. a_1_2²·b_1_1
5. b_1_0·a_3_1
6. b_1_1·b_3_2 + b_1_1⁴ + a_1_2·a_3_1
7. a_1_2·b_3_2
8. a_1_2·b_3_4
9. b_1_0·b_3_5 + b_1_0·b_3_2 + a_1_2·a_3_1
10. b_1_1·b_3_4 + b_1_1⁴ + b_1_1·a_3_1 + a_1_2·b_3_5 + a_1_2·a_3_1
11. a_1_2·b_1_1·b_3_5
12. a_3_1·b_3_2 + b_1_1³·a_3_1
13. a_3_1·b_3_4 + b_1_1³·a_3_1
14. b_3_2·b_3_5 + b_3_2² + b_1_1³·b_3_5 + b_1_1⁶ + a_3_1²
15. b_3_4·b_3_5 + b_3_2·b_3_4 + b_1_1³·b_3_5 + b_1_1⁶ + a_3_1·b_3_5 + b_1_1³·a_3_1
       + a_3_1² + c_4_8·a_1_2·b_1_1
16. b_3_2² + b_1_1⁶ + c_4_8·b_1_0²
17. a_3_1² + c_4_8·a_1_2²
18. b_3_4² + b_1_1⁶ + c_4_9·b_1_0²
19. b_3_5² + b_3_2² + b_1_1³·b_3_5 + b_1_1⁶ + a_3_1² + c_4_8·b_1_1² + c_4_9·a_1_2²
20. b_1_1·b_5_13 + b_1_1³·b_3_5 + a_3_1·b_3_5 + a_3_1² + c_4_9·b_1_1² + c_4_9·a_1_2·b_1_1
21. b_3_5² + b_3_2·b_3_4 + b_3_2² + b_1_1³·b_3_5 + b_1_0·b_5_13 + b_1_0³·b_3_4
       + b_1_1³·a_3_1 + a_3_1² + c_4_8·b_1_1²
22. b_3_5² + b_3_2² + b_1_1³·b_3_5 + b_1_1⁶ + a_1_2·b_5_13 + c_4_8·b_1_1²
       + c_4_9·a_1_2·b_1_1
23. a_3_1·b_5_13 + b_1_1²·a_3_1·b_3_5 + c_4_9·b_1_1·a_3_1 + c_4_9·a_1_2·a_3_1
       + c_4_8·a_1_2·a_3_1
24. b_3_2·b_5_13 + b_1_1⁵·b_3_5 + b_1_0³·b_5_13 + b_1_0⁵·b_3_4 + b_1_1²·a_3_1·b_3_5
       + c_4_9·b_1_1⁴ + c_4_8·b_1_0·b_3_4 + c_4_9·a_1_2·a_3_1
25. b_3_4·b_5_13 + b_1_1⁵·b_3_5 + c_4_9·b_1_1⁴ + c_4_9·b_1_0·b_3_2 + c_4_9·b_1_0⁴
       + c_4_9·b_1_1·a_3_1 + c_4_9·a_1_2·b_3_5 + c_4_9·a_1_2·a_3_1
26. b_3_5·b_5_13 + b_1_1⁵·b_3_5 + b_1_0³·b_5_13 + b_1_0⁵·b_3_4 + b_1_1²·a_3_1·b_3_5
       + c_4_9·b_1_1·b_3_5 + c_4_8·b_1_1⁴ + c_4_8·b_1_0·b_3_4 + c_4_9·a_1_2·b_3_5
       + c_4_8·b_1_1·a_3_1 + c_4_8·a_1_2·a_3_1
27. b_5_13² + b_1_1⁷·b_3_5 + c_4_9·b_1_0⁶ + c_4_8·b_1_1⁶ + c_4_9²·b_1_1²
       + c_4_8·c_4_9·b_1_0² + c_4_9²·a_1_2² + c_4_8²·a_1_2²

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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_4_8, a Duflot regular element of degree 4
  2. c_4_9, a Duflot regular element of degree 4
  3. b_1_1 + b_1_0, an element of degree 1
• The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 6].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 64

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. a_3_1 → 0, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_4 → 0, an element of degree 3
7. b_3_5 → 0, an element of degree 3
8. c_4_8 → c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1⁴, an element of degree 4
10. b_5_13 → 0, an element of degree 5

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

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → c_1_2, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. a_3_1 → 0, an element of degree 3
5. b_3_2 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
6. b_3_4 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
7. b_3_5 → c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
8. c_4_8 → c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
10. b_5_13 → c_1_1·c_1_2⁴ + c_1_1²·c_1_2³ + c_1_0·c_1_1·c_1_2³ + c_1_0·c_1_1²·c_1_2²
       + c_1_0²·c_1_1·c_1_2² + c_1_0²·c_1_1²·c_1_2, an element of degree 5

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

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → c_1_2, an element of degree 1
4. a_3_1 → 0, an element of degree 3
5. b_3_2 → c_1_2³, an element of degree 3
6. b_3_4 → c_1_2³, an element of degree 3
7. b_3_5 → c_1_2³ + c_1_0·c_1_2² + c_1_0²·c_1_2, an element of degree 3
8. c_4_8 → c_1_0·c_1_2³ + c_1_0⁴, an element of degree 4
9. c_4_9 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
10. b_5_13 → c_1_2⁵ + c_1_1²·c_1_2³ + c_1_1⁴·c_1_2 + c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 64