Cohomology of group number 81 of order 128

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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_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. b_3_2, an 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. b_5_10, an element of degree 5
  14. b_5_11, an 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_0·b_1_1
  3. a_2_2·a_1_0
  4. a_2_2·b_1_1
  5. b_2_1·a_1_0
  6. b_2_1·b_1_1
  7. a_2_22
  8. a_1_0·a_3_1
  9. b_1_1·a_3_1
  10. a_1_0·b_3_2
  11. a_1_0·b_3_3
  12. b_1_1·b_3_3 + b_1_1·b_3_2
  13. a_1_0·b_3_4
  14. a_2_2·b_3_2
  15. b_2_1·b_3_2 + b_2_1·a_3_1
  16. b_2_1·a_3_1 + a_2_2·b_3_3 + a_2_2·a_3_1
  17. a_2_2·b_3_4
  18. b_2_1·b_3_4 + a_2_2·a_3_1
  19. a_4_5·a_1_0 + a_2_2·a_3_1
  20. a_4_5·b_1_1
  21. b_4_6·a_1_0 + a_2_2·a_3_1
  22. b_4_6·b_1_1 + a_2_2·a_3_1
  23. a_3_12
  24. a_3_1·b_3_2
  25. a_3_1·b_3_3 + a_2_2·b_2_12
  26. b_3_2·b_3_3 + b_3_22 + a_2_2·b_2_12
  27. b_3_32 + b_3_22 + b_2_13
  28. a_3_1·b_3_4
  29. b_3_3·b_3_4 + b_3_2·b_3_4
  30. b_3_42 + c_4_7·b_1_12
  31. b_3_42 + b_3_22 + c_4_8·b_1_12
  32. a_2_2·a_4_5
  33. b_2_1·a_4_5 + a_2_2·b_4_6
  34. a_1_0·b_5_10
  35. b_1_1·b_5_10
  36. a_1_0·b_5_11
  37. b_3_42 + b_3_2·b_3_4 + b_3_22 + b_1_1·b_5_11
  38. a_4_5·a_3_1
  39. a_4_5·b_3_2
  40. a_4_5·b_3_4
  41. b_4_6·a_3_1 + a_4_5·b_3_3
  42. b_4_6·b_3_2 + a_4_5·b_3_3
  43. b_4_6·b_3_4
  44. a_4_5·b_3_3 + a_2_2·b_5_10 + a_2_2·b_2_1·b_3_3
  45. b_4_6·b_3_3 + b_2_1·b_5_10 + b_2_12·b_3_3 + a_4_5·b_3_3
  46. a_2_2·b_5_11 + a_2_2·b_2_1·b_3_3
  47. b_2_1·b_5_11 + b_2_12·b_3_3 + a_4_5·b_3_3
  48. a_4_52
  49. a_4_5·b_4_6 + a_2_2·b_2_1·c_4_7
  50. b_4_62 + b_2_12·c_4_7
  51. a_3_1·b_5_10 + a_2_2·b_2_1·b_4_6 + a_2_2·b_2_13
  52. b_3_2·b_5_10 + a_2_2·b_2_1·b_4_6 + a_2_2·b_2_13
  53. b_3_3·b_5_10 + b_2_12·b_4_6 + b_2_14 + a_2_2·b_2_1·b_4_6
  54. b_3_4·b_5_10
  55. a_3_1·b_5_11 + a_2_2·b_2_13
  56. b_3_2·b_5_11 + a_2_2·b_2_13 + c_4_8·b_1_1·b_3_4 + c_4_8·b_1_1·b_3_2
       + c_4_7·b_1_1·b_3_4
  57. b_3_3·b_5_11 + b_2_14 + a_2_2·b_2_1·b_4_6 + c_4_8·b_1_1·b_3_4 + c_4_8·b_1_1·b_3_2
       + c_4_7·b_1_1·b_3_4
  58. b_3_4·b_5_11 + c_4_8·b_1_1·b_3_4 + c_4_7·b_1_1·b_3_2
  59. a_4_5·b_5_10 + a_2_2·b_2_1·b_5_10 + a_2_2·b_2_12·b_3_3 + a_2_2·c_4_7·b_3_3
       + a_2_2·c_4_8·a_3_1
  60. b_4_6·b_5_10 + b_2_12·b_5_10 + b_2_13·b_3_3 + a_2_2·b_2_1·b_5_10 + a_2_2·b_2_12·b_3_3
       + b_2_1·c_4_7·b_3_3 + a_2_2·c_4_7·b_3_3 + a_2_2·c_4_7·a_3_1
  61. a_4_5·b_5_11 + a_2_2·b_2_1·b_5_10 + a_2_2·b_2_12·b_3_3 + a_2_2·c_4_8·a_3_1
       + a_2_2·c_4_7·a_3_1
  62. b_4_6·b_5_11 + b_2_12·b_5_10 + b_2_13·b_3_3 + a_2_2·b_2_1·b_5_10 + a_2_2·b_2_12·b_3_3
       + a_2_2·c_4_7·b_3_3
  63. b_5_102 + b_2_15 + b_2_13·c_4_7
  64. b_5_10·b_5_11 + b_2_13·b_4_6 + b_2_15 + a_2_2·b_2_12·c_4_7
  65. b_5_112 + b_2_15 + c_4_82·b_1_12 + c_4_7·c_4_8·b_1_12 + c_4_72·b_1_12

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_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].

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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_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. b_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_14, an element of degree 4
  12. c_4_8 → c_1_14 + c_1_04, an element of degree 4
  13. b_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_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. b_3_2 → c_1_0·c_1_22 + c_1_02·c_1_2, 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 → c_1_1·c_1_22 + c_1_12·c_1_2, 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_12·c_1_22 + c_1_14, an element of degree 4
  12. c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. b_5_10 → 0, an element of degree 5
  14. b_5_11 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22
       + c_1_02·c_1_23 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + 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_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. b_3_2 → 0, an element of degree 3
  7. b_3_3 → c_1_23, 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_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  11. c_4_7 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  12. c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. b_5_10 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  14. b_5_11 → c_1_25, 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