David J. Green

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Cohomology of group number 863 of order 128

About the group

Ring generators

Ring relations

Completion information

Restriction maps

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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 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) · (t5  +  t2  +  1)

  (t  −  1)3 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. c_1_2, a Duflot regular element of degree 1
4. a_2_4, a nilpotent element of degree 2
5. a_2_5, a nilpotent element of degree 2
6. a_3_8, a nilpotent element of degree 3
7. a_3_9, a nilpotent element of degree 3
8. b_4_13, an element of degree 4
9. a_5_16, a nilpotent element of degree 5
10. a_5_18, a nilpotent element of degree 5
11. b_6_24, an element of degree 6
12. a_7_30, a nilpotent element of degree 7
13. c_8_37, a Duflot regular element of degree 8

About the group

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Ring relations

There are 47 minimal relations of maximal degree 14:

1. a_1_0²
2. a_1_0·a_1_1
3. a_2_4·a_1_0
4. a_2_5·a_1_1
5. a_2_5·a_1_0 + a_2_4·a_1_1
6. a_1_1⁴
7. a_2_5² + a_2_4·a_2_5 + a_2_4²
8. a_1_0·a_3_8
9. a_1_1·a_3_9 + a_2_5²
10. a_1_0·a_3_9 + a_2_4²
11. a_2_5·a_3_8
12. a_2_4·a_3_8
13. a_2_4·a_3_9 + a_1_1²·a_3_8
14. b_4_13·a_1_0
15. a_3_8·a_3_9
16. a_3_8² + b_4_13·a_1_1² + a_1_1³·a_3_8
17. a_2_4·b_4_13 + a_3_9² + a_1_1³·a_3_8
18. a_3_8² + a_1_1·a_5_16
19. a_1_0·a_5_16
20. a_1_0·a_5_18 + a_1_1³·a_3_8
21. a_2_4·a_5_16 + b_4_13·a_1_1³
22. a_2_5·a_5_18 + b_4_13·a_1_1³
23. a_2_5·a_5_16 + a_2_4·a_5_18 + b_4_13·a_1_1³
24. b_6_24·a_1_1 + b_4_13·a_3_8 + a_2_5·a_5_16 + b_4_13·a_1_1³
25. b_6_24·a_1_0
26. a_3_8·a_5_16 + b_4_13·a_1_1·a_3_8 + a_2_5·a_3_9²
27. a_2_5·a_3_9² + a_1_1³·a_5_18
28. a_2_5·b_6_24 + a_3_9·a_5_18 + a_3_9·a_5_16
29. a_2_4·b_6_24 + a_3_9·a_5_16
30. a_3_8·a_5_18 + a_1_1·a_7_30 + a_2_5·a_3_9²
31. a_1_0·a_7_30
32. b_6_24·a_3_8 + b_4_13²·a_1_1 + a_2_5·b_4_13·a_3_9 + b_4_13·a_1_1²·a_3_8
33. a_2_5·a_7_30 + a_2_5·b_4_13·a_3_9 + b_4_13·a_1_1²·a_3_8
34. b_6_24·a_3_9 + b_4_13·a_5_16 + b_4_13²·a_1_1 + a_2_5·b_4_13·a_3_9 + a_1_1²·a_7_30
35. a_2_5·b_4_13·a_3_9 + a_2_4·a_7_30
36. a_5_16² + b_4_13·a_3_9² + b_4_13²·a_1_1²
37. a_2_5·b_4_13² + a_5_16·a_5_18 + b_4_13·a_3_9² + b_4_13·a_1_1·a_5_18
38. a_2_5·b_4_13² + a_3_9·a_7_30
39. a_3_8·a_7_30 + b_4_13·a_1_1·a_5_18
40. a_2_5·b_4_13² + a_5_18² + b_4_13·a_1_1·a_5_18 + b_4_13²·a_1_1² + c_8_37·a_1_1²
41. b_6_24·a_5_16 + b_4_13²·a_3_9 + b_4_13²·a_3_8 + b_4_13·a_1_1²·a_5_18
42. b_6_24·a_5_18 + b_4_13·a_7_30 + b_4_13²·a_3_9 + a_3_9²·a_5_18 + b_4_13·a_1_1²·a_5_18
       + b_4_13²·a_1_1³ + a_2_4·c_8_37·a_1_1
43. a_5_16·a_7_30 + b_4_13·a_3_9·a_5_18 + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_1·a_7_30
44. a_5_18·a_7_30 + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_1·a_7_30 + b_4_13²·a_1_1·a_3_8
       + b_4_13·a_1_1³·a_5_18 + c_8_37·a_1_1·a_3_8
45. b_6_24² + b_4_13³ + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_1³·a_5_18 + a_2_4²·c_8_37
46. b_6_24·a_7_30 + b_4_13²·a_5_18 + b_4_13²·a_5_16 + b_4_13³·a_1_1 + a_3_9²·a_7_30
       + b_4_13²·a_1_1²·a_3_8
47. a_7_30² + b_4_13·a_3_9·a_7_30 + b_4_13²·a_3_9² + b_4_13²·a_1_1·a_5_18
       + b_4_13³·a_1_1² + b_4_13·a_1_1³·a_7_30 + b_4_13·c_8_37·a_1_1²
       + c_8_37·a_1_1³·a_3_8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Data used for Benson′s test

• Benson′s completion test succeeded in degree 14.
• 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_1_2, a Duflot regular element of degree 1
  2. c_8_37, a Duflot regular element of degree 8
  3. b_4_13, an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, 6, 10].
• 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 128

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. c_1_2 → c_1_0, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_2_5 → 0, an element of degree 2
6. a_3_8 → 0, an element of degree 3
7. a_3_9 → 0, an element of degree 3
8. b_4_13 → 0, an element of degree 4
9. a_5_16 → 0, an element of degree 5
10. a_5_18 → 0, an element of degree 5
11. b_6_24 → 0, an element of degree 6
12. a_7_30 → 0, an element of degree 7
13. c_8_37 → c_1_1⁸, an element of degree 8

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. c_1_2 → c_1_0, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_2_5 → 0, an element of degree 2
6. a_3_8 → 0, an element of degree 3
7. a_3_9 → 0, an element of degree 3
8. b_4_13 → c_1_2⁴, an element of degree 4
9. a_5_16 → 0, an element of degree 5
10. a_5_18 → 0, an element of degree 5
11. b_6_24 → c_1_2⁶, an element of degree 6
12. a_7_30 → 0, an element of degree 7
13. c_8_37 → c_1_1⁴·c_1_2⁴ + c_1_1⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128