Cohomology of group number 47 of order 128

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

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 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
    ( − 1) · (t4  +  t3  +  t2  +  1)
    (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 14 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, 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. b_3_3, an element of degree 3
  8. b_3_4, an element of degree 3
  9. a_4_5, a nilpotent element of degree 4
  10. b_4_6, an element of degree 4
  11. c_4_7, a Duflot regular element of degree 4
  12. c_4_8, a Duflot regular element of degree 4
  13. a_5_10, a nilpotent element of degree 5
  14. b_5_11, an element of degree 5

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

Ring relations

There are 65 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. a_2_1·b_1_1
  5. b_2_2·a_1_0
  6. b_2_2·b_1_1
  7. a_2_12
  8. a_1_0·a_3_1
  9. a_1_0·a_3_2
  10. b_1_1·a_3_2
  11. a_1_0·b_3_3
  12. a_1_0·b_3_4
  13. b_1_1·b_3_4
  14. a_2_1·a_3_1
  15. a_2_1·a_3_2
  16. b_2_2·a_3_2 + a_2_1·b_3_3
  17. b_2_2·b_3_4 + b_2_2·b_3_3
  18. b_2_2·a_3_1 + a_2_1·b_3_4
  19. a_4_5·a_1_0
  20. a_4_5·b_1_1 + b_2_2·a_3_2 + b_2_2·a_3_1
  21. b_4_6·a_1_0 + b_2_2·a_3_2 + b_2_2·a_3_1
  22. b_1_12·b_3_3 + b_4_6·b_1_1 + b_2_2·a_3_2 + b_2_2·a_3_1
  23. a_3_12
  24. a_3_22
  25. a_3_1·a_3_2
  26. a_3_2·b_3_3 + a_2_1·b_2_22
  27. a_3_2·b_3_4 + a_2_1·b_2_22
  28. a_3_1·b_3_4 + a_2_1·b_2_22
  29. b_3_42 + b_2_23
  30. b_3_3·b_3_4 + b_2_23
  31. b_3_32 + b_2_23 + c_4_7·b_1_12
  32. a_2_1·a_4_5
  33. b_2_2·a_4_5 + a_2_1·b_4_6
  34. a_1_0·a_5_10
  35. a_3_1·b_3_3 + b_1_1·a_5_10 + a_2_1·b_2_22
  36. a_1_0·b_5_11
  37. b_3_32 + b_1_1·b_5_11 + b_2_23
  38. a_4_5·a_3_2
  39. a_4_5·a_3_1
  40. a_4_5·b_3_4 + a_4_5·b_3_3
  41. b_4_6·a_3_2 + a_4_5·b_3_4
  42. b_4_6·b_3_4 + b_4_6·b_3_3 + c_4_7·b_1_13
  43. a_4_5·b_3_4 + b_2_2·a_5_10
  44. a_2_1·a_5_10
  45. b_1_12·a_5_10 + b_4_6·a_3_1 + a_4_5·b_3_4
  46. b_4_6·b_3_4 + b_2_2·b_5_11 + a_4_5·b_3_4 + a_2_1·b_2_2·b_3_3
  47. a_4_5·b_3_4 + a_2_1·b_5_11
  48. a_4_52
  49. a_4_5·b_4_6 + a_2_1·b_2_23 + a_2_1·b_2_2·c_4_7
  50. b_4_62 + b_2_24 + a_2_1·b_2_2·b_4_6 + c_4_7·b_1_14 + b_2_22·c_4_7
  51. a_3_2·a_5_10
  52. a_3_1·a_5_10
  53. b_3_4·a_5_10 + a_2_1·b_2_2·b_4_6
  54. b_3_3·a_5_10 + a_2_1·b_2_2·b_4_6 + c_4_7·b_1_1·a_3_1
  55. a_3_2·b_5_11 + a_2_1·b_2_2·b_4_6
  56. a_3_1·b_5_11 + a_2_1·b_2_2·b_4_6 + c_4_7·b_1_1·a_3_1
  57. b_3_4·b_5_11 + b_2_22·b_4_6 + a_2_1·b_2_2·b_4_6 + a_2_1·b_2_23
  58. b_3_3·b_5_11 + b_2_22·b_4_6 + a_2_1·b_2_2·b_4_6 + a_2_1·b_2_23 + c_4_7·b_1_1·b_3_3
  59. a_4_5·a_5_10
  60. b_4_6·a_5_10 + a_2_1·b_2_22·b_3_3 + c_4_7·b_1_12·a_3_1 + a_2_1·c_4_7·b_3_4
  61. a_4_5·b_5_11 + a_2_1·b_2_22·b_3_3 + a_2_1·c_4_7·b_3_3
  62. b_4_6·b_5_11 + b_2_23·b_3_3 + a_2_1·b_2_22·b_3_3 + b_4_6·c_4_7·b_1_1
       + b_2_2·c_4_7·b_3_3 + a_2_1·c_4_7·b_3_3
  63. a_5_102
  64. b_5_112 + b_2_25 + a_2_1·b_2_22·b_4_6 + b_2_23·c_4_7 + c_4_72·b_1_12
  65. a_5_10·b_5_11 + a_2_1·b_2_24 + c_4_7·b_1_1·a_5_10 + a_2_1·b_2_22·c_4_7

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_4_7, a Duflot regular element of degree 4
    2. c_4_8, a Duflot regular element of degree 4
    3. b_1_12 + b_2_2, 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].

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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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 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. b_3_3 → 0, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. a_4_5 → 0, an element of degree 4
  10. b_4_6 → 0, an element of degree 4
  11. c_4_7 → c_1_04, an element of degree 4
  12. c_4_8 → c_1_14, an element of degree 4
  13. a_5_10 → 0, an element of degree 5
  14. b_5_11 → 0, an element of degree 5

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_2, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 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. b_3_3 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. a_4_5 → 0, an element of degree 4
  10. b_4_6 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  11. c_4_7 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  12. c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  13. a_5_10 → 0, an element of degree 5
  14. b_5_11 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5

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 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 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. b_3_3 → c_1_23, an element of degree 3
  8. b_3_4 → c_1_23, an element of degree 3
  9. a_4_5 → 0, an element of degree 4
  10. b_4_6 → c_1_24 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  11. c_4_7 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  12. c_4_8 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  13. a_5_10 → 0, an element of degree 5
  14. b_5_11 → c_1_25 + 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 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