Cohomology of group number 9 of order 64

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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
    ( − 2) · (t4  +  1/2·t3  +  1/2·t2  +  1/2·t  +  1/2)
    (t  +  1)2 · (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 14 minimal generators of maximal degree 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_2, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. a_3_1, a nilpotent element of degree 3
  6. a_3_2, a nilpotent element of degree 3
  7. a_3_3, a nilpotent element of degree 3
  8. a_4_4, a nilpotent element of degree 4
  9. a_4_5, a nilpotent element of degree 4
  10. b_4_3, an element of degree 4
  11. c_4_6, a Duflot regular element of degree 4
  12. c_4_7, a Duflot regular element of degree 4
  13. a_5_9, a nilpotent element of degree 5
  14. a_6_11, a nilpotent element of degree 6

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Ring relations

There are 65 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. a_2_2·a_1_1
  5. a_2_2·a_1_0
  6. b_2_1·a_1_1
  7. a_2_22
  8. a_1_1·a_3_1
  9. a_1_0·a_3_1
  10. a_1_1·a_3_2
  11. a_1_0·a_3_2
  12. a_1_0·a_3_3
  13. a_2_2·a_3_3 + a_2_2·a_3_2
  14. b_2_1·a_3_3 + b_2_1·a_3_2 + a_2_2·a_3_2 + a_2_2·a_3_1
  15. a_2_2·a_3_1 + a_1_12·a_3_3
  16. a_4_4·a_1_1 + a_2_2·a_3_1
  17. a_4_4·a_1_0 + a_2_2·a_3_1
  18. a_4_5·a_1_1
  19. a_4_5·a_1_0 + a_2_2·a_3_2 + a_2_2·a_3_1
  20. b_4_3·a_1_1
  21. b_4_3·a_1_0 + b_2_1·a_3_1
  22. a_3_12
  23. a_3_22
  24. a_3_1·a_3_2
  25. a_3_2·a_3_3
  26. a_3_1·a_3_3
  27. a_3_32 + c_4_6·a_1_12
  28. a_2_2·a_4_4
  29. a_2_2·a_4_5
  30. b_2_1·a_4_4 + a_2_2·b_4_3
  31. a_1_1·a_5_9 + c_4_7·a_1_12
  32. a_1_0·a_5_9
  33. a_4_4·a_3_1
  34. a_4_4·a_3_3 + a_4_4·a_3_2
  35. a_4_5·a_3_2 + a_4_4·a_3_2 + a_2_2·b_2_1·a_3_2
  36. a_4_5·a_3_1 + a_4_4·a_3_2
  37. a_4_5·a_3_3 + a_4_4·a_3_2 + a_2_2·b_2_1·a_3_2
  38. b_4_3·a_3_1 + b_2_12·a_3_1 + b_2_1·c_4_7·a_1_0
  39. b_4_3·a_3_3 + b_4_3·a_3_2 + a_4_4·a_3_2
  40. a_4_4·a_3_2 + a_2_2·a_5_9
  41. b_4_3·a_3_2 + b_2_1·a_5_9 + b_2_12·a_3_1 + a_2_2·b_2_1·a_3_2 + b_2_1·c_4_6·a_1_0
  42. a_6_11·a_1_1
  43. a_6_11·a_1_0 + a_4_4·a_3_2
  44. a_4_42
  45. a_4_4·a_4_5
  46. a_4_52
  47. b_4_32 + b_2_12·b_4_3 + b_2_12·c_4_7
  48. a_4_4·b_4_3 + a_2_2·b_2_1·b_4_3 + a_2_2·b_2_1·c_4_7
  49. a_3_2·a_5_9
  50. a_3_1·a_5_9
  51. a_3_3·a_5_9 + c_4_7·a_1_1·a_3_3
  52. a_2_2·a_6_11
  53. a_4_5·b_4_3 + b_2_1·a_6_11 + a_2_2·b_2_1·b_4_3
  54. b_4_3·a_5_9 + b_2_12·a_5_9 + a_2_2·b_2_1·a_5_9 + a_2_2·b_2_12·a_3_2
       + b_2_1·c_4_7·a_3_2 + b_2_1·c_4_6·a_3_1 + b_2_12·c_4_7·a_1_0 + b_2_12·c_4_6·a_1_0
  55. a_4_4·a_5_9 + a_2_2·b_2_1·a_5_9 + a_2_2·c_4_7·a_3_2 + c_4_7·a_1_12·a_3_3
       + c_4_6·a_1_12·a_3_3
  56. a_4_5·a_5_9 + a_2_2·b_2_1·a_5_9 + a_2_2·c_4_7·a_3_2 + a_2_2·c_4_6·a_3_2
  57. a_6_11·a_3_2 + a_2_2·b_2_1·a_5_9 + a_2_2·c_4_7·a_3_2 + c_4_6·a_1_12·a_3_3
  58. a_6_11·a_3_1 + a_2_2·b_2_1·a_5_9 + a_2_2·c_4_7·a_3_2 + c_4_7·a_1_12·a_3_3
  59. a_6_11·a_3_3 + a_2_2·b_2_1·a_5_9 + a_2_2·c_4_7·a_3_2 + c_4_6·a_1_12·a_3_3
  60. a_5_92 + c_4_72·a_1_12
  61. b_4_3·a_6_11 + b_2_12·a_6_11 + b_2_1·a_4_5·c_4_7 + a_2_2·b_2_12·c_4_7
  62. a_4_4·a_6_11
  63. a_4_5·a_6_11
  64. a_6_11·a_5_9 + a_2_2·c_4_7·a_5_9 + a_2_2·c_4_6·a_5_9
  65. a_6_112

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_6, a Duflot regular element of degree 4
    2. c_4_7, a Duflot regular element of degree 4
    3. b_2_1, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • 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_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_4_4 → 0, an element of degree 4
  9. a_4_5 → 0, an element of degree 4
  10. b_4_3 → 0, an element of degree 4
  11. c_4_6 → c_1_04, an element of degree 4
  12. c_4_7 → c_1_14 + c_1_04, an element of degree 4
  13. a_5_9 → 0, an element of degree 5
  14. a_6_11 → 0, an element of degree 6

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_2 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_4_4 → 0, an element of degree 4
  9. a_4_5 → 0, an element of degree 4
  10. b_4_3 → c_1_12·c_1_22 + c_1_02·c_1_22, an element of degree 4
  11. c_4_6 → c_1_12·c_1_22 + c_1_04, an element of degree 4
  12. c_4_7 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. a_5_9 → 0, an element of degree 5
  14. a_6_11 → 0, an element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009