Cohomology of group number 12 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 4.
  • Its center has rank 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t3  +  t2  +  1
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 15 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. c_2_3, a Duflot regular element of degree 2
  7. c_2_4, a Duflot regular element of degree 2
  8. a_3_5, a nilpotent element of degree 3
  9. a_3_6, a nilpotent element of degree 3
  10. a_3_7, a nilpotent element of degree 3
  11. b_3_8, an element of degree 3
  12. a_4_5, a nilpotent element of degree 4
  13. a_4_13, a nilpotent element of degree 4
  14. c_4_14, a Duflot regular element of degree 4
  15. a_5_22, a nilpotent element of degree 5

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Ring relations

There are 65 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. a_2_1·a_1_1
  7. a_2_1·a_1_0
  8. b_2_2·a_1_0
  9. a_2_02
  10. a_2_12
  11. a_1_1·a_3_5 + a_2_0·a_2_1
  12. a_1_0·a_3_5
  13. a_1_1·a_3_6
  14. a_1_0·a_3_6
  15. a_1_1·a_3_7
  16. a_1_0·a_3_7 + a_2_0·a_2_1
  17. a_1_1·b_3_8 + a_2_0·b_2_2
  18. a_1_0·b_3_8 + a_2_0·a_2_1
  19. a_2_0·a_3_5
  20. a_2_1·a_3_5
  21. a_2_0·a_3_6
  22. a_2_1·a_3_6
  23. a_2_0·a_3_7
  24. b_2_2·a_3_6 + b_2_2·a_3_5 + a_2_1·a_3_7
  25. b_2_22·a_1_1 + a_2_0·b_3_8
  26. b_2_2·a_3_5 + a_2_1·b_3_8
  27. a_4_5·a_1_1
  28. a_4_5·a_1_0
  29. b_2_2·a_3_6 + b_2_2·a_3_5 + a_4_13·a_1_1
  30. a_4_13·a_1_0
  31. a_3_52
  32. a_3_62
  33. a_3_5·a_3_6
  34. a_3_72
  35. a_3_6·a_3_7 + a_3_5·a_3_7
  36. a_3_6·b_3_8 + a_2_1·b_2_22 + a_3_6·a_3_7
  37. a_3_5·b_3_8 + a_2_1·b_2_22
  38. b_3_82 + b_2_23 + a_2_0·b_2_22
  39. a_3_7·b_3_8 + b_2_2·a_4_5
  40. a_2_0·a_4_5
  41. a_3_6·a_3_7 + a_2_1·a_4_5
  42. a_3_6·a_3_7 + a_2_0·a_4_13
  43. a_2_1·a_4_13
  44. a_3_6·a_3_7 + a_1_1·a_5_22 + a_2_0·a_2_1·c_2_4
  45. a_1_0·a_5_22
  46. a_4_5·a_3_7
  47. a_4_5·a_3_6 + a_2_1·b_2_2·a_3_7
  48. a_4_5·a_3_5 + a_2_1·b_2_2·a_3_7
  49. a_4_5·b_3_8 + b_2_22·a_3_7
  50. a_4_13·a_3_7 + a_2_1·b_2_2·a_3_7
  51. a_4_13·a_3_6
  52. a_4_13·a_3_5
  53. a_4_13·b_3_8 + b_2_2·a_5_22 + a_2_1·b_2_2·b_3_8 + a_2_1·c_2_4·b_3_8
  54. a_2_1·b_2_2·a_3_7 + a_2_0·a_5_22
  55. a_2_1·a_5_22
  56. a_4_52
  57. a_4_5·a_4_13 + a_2_0·b_2_2·a_4_13 + a_2_0·a_2_1·c_4_14
  58. a_4_132
  59. a_3_7·a_5_22 + a_2_0·c_2_4·a_4_13 + a_2_0·a_2_1·c_4_14
  60. a_3_6·a_5_22
  61. a_3_5·a_5_22
  62. b_3_8·a_5_22 + b_2_22·a_4_13 + a_2_1·b_2_23 + a_2_0·b_2_2·a_4_13
       + a_2_1·b_2_22·c_2_4
  63. a_4_13·a_5_22
  64. a_4_5·a_5_22 + a_2_0·c_2_4·a_5_22
  65. a_5_222

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 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_2_3, a Duflot regular element of degree 2
    2. c_2_4, a Duflot regular element of degree 2
    3. c_4_14, a Duflot regular element of degree 4
    4. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is 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_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. c_2_3 → c_1_02, an element of degree 2
  7. c_2_4 → c_1_22, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. a_3_6 → 0, an element of degree 3
  10. a_3_7 → 0, an element of degree 3
  11. b_3_8 → 0, an element of degree 3
  12. a_4_5 → 0, an element of degree 4
  13. a_4_13 → 0, an element of degree 4
  14. c_4_14 → c_1_14, an element of degree 4
  15. a_5_22 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_32, an element of degree 2
  6. c_2_3 → c_1_02, an element of degree 2
  7. c_2_4 → c_1_22, an element of degree 2
  8. a_3_5 → 0, an element of degree 3
  9. a_3_6 → 0, an element of degree 3
  10. a_3_7 → 0, an element of degree 3
  11. b_3_8 → c_1_33, an element of degree 3
  12. a_4_5 → 0, an element of degree 4
  13. a_4_13 → 0, an element of degree 4
  14. c_4_14 → c_1_12·c_1_32 + c_1_14, an element of degree 4
  15. a_5_22 → 0, an element of degree 5

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


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