Cohomology of group number 83 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 2.
  • 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 exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t5  −  t4  +  t3  −  2·t2  +  t  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • 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_2, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. b_2_3, an element of degree 2
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_5, a nilpotent element of degree 3
  8. b_3_4, an element of degree 3
  9. b_3_6, an element of degree 3
  10. b_4_6, an element of degree 4
  11. b_4_9, an element of degree 4
  12. c_4_10, a Duflot regular element of degree 4
  13. c_4_11, a Duflot regular element of degree 4
  14. b_5_14, an element of degree 5
  15. b_5_16, 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·a_1_1
  3. a_1_13
  4. a_2_2·a_1_0
  5. b_2_1·a_1_1
  6. b_2_3·a_1_0
  7. a_1_1·a_3_3
  8. a_1_0·a_3_3
  9. a_1_1·a_3_5 + b_2_3·a_1_12 + a_2_22 + a_2_2·a_1_12
  10. a_1_0·a_3_5
  11. a_1_1·b_3_4 + b_2_3·a_1_12
  12. a_1_0·b_3_4
  13. a_1_1·b_3_6
  14. a_1_0·b_3_6 + a_2_2·b_2_1 + a_2_2·a_1_12
  15. a_2_2·a_3_3
  16. b_2_3·a_3_3 + a_2_22·a_1_1
  17. b_2_1·a_3_5 + a_2_22·a_1_1
  18. a_2_2·b_3_4 + a_2_2·b_2_3·a_1_1
  19. b_2_1·a_3_3 + a_2_2·b_3_6
  20. b_4_6·a_1_1 + a_2_22·a_1_1
  21. b_4_6·a_1_0 + b_2_1·a_3_3
  22. b_4_9·a_1_1 + b_2_32·a_1_1 + a_2_2·a_3_5 + a_2_2·b_2_3·a_1_1 + a_2_22·a_1_1
  23. b_4_9·a_1_0
  24. a_3_32
  25. a_3_3·a_3_5
  26. a_3_5·b_3_4 + b_2_32·a_1_12 + a_2_22·b_2_3 + a_2_2·b_2_3·a_1_12
  27. a_3_3·b_3_4
  28. b_3_42 + b_2_1·b_2_32
  29. a_3_5·b_3_6
  30. a_3_52 + a_2_22·b_2_3 + a_2_23 + c_4_11·a_1_12 + c_4_10·a_1_12
  31. a_3_3·b_3_6 + a_2_2·b_4_6 + a_2_2·b_2_3·a_1_12 + a_2_23
  32. b_3_62 + b_2_1·b_4_6
  33. a_2_2·b_4_9 + a_2_2·b_2_32 + a_3_52 + b_2_32·a_1_12
  34. b_3_4·b_3_6 + b_2_1·b_4_9 + b_2_1·b_2_32 + a_2_23
  35. a_1_1·b_5_14 + a_3_52 + a_2_2·b_2_3·a_1_12 + a_2_23 + c_4_10·a_1_12
  36. a_1_0·b_5_14
  37. a_1_1·b_5_16 + a_2_2·b_2_3·a_1_12 + c_4_10·a_1_12
  38. a_3_3·b_3_6 + a_1_0·b_5_16 + a_2_2·b_2_12
  39. b_4_6·a_3_5 + a_2_22·b_2_3·a_1_1
  40. b_4_6·a_3_3 + b_2_1·c_4_10·a_1_0
  41. b_4_9·a_3_5 + b_2_32·a_3_5 + a_2_2·b_2_32·a_1_1 + a_2_2·c_4_11·a_1_1
       + a_2_2·c_4_10·a_1_1
  42. b_4_9·a_3_3 + a_2_22·b_2_3·a_1_1
  43. b_4_9·b_3_4 + b_2_32·b_3_6 + b_2_32·b_3_4 + a_2_2·b_2_3·a_3_5 + a_2_2·b_2_32·a_1_1
       + a_2_22·b_2_3·a_1_1
  44. b_4_9·b_3_6 + b_4_6·b_3_4 + b_2_32·b_3_6 + a_2_22·b_2_3·a_1_1
  45. a_2_2·b_5_14 + a_2_2·b_2_3·a_3_5 + a_2_2·b_2_32·a_1_1 + a_2_22·b_2_3·a_1_1
       + a_2_2·c_4_11·a_1_1
  46. b_4_6·b_3_4 + b_2_1·b_5_14 + b_2_12·b_3_4 + a_2_2·b_2_1·b_3_6
  47. a_2_2·b_5_16 + a_2_2·b_2_1·b_3_6 + b_2_1·c_4_10·a_1_0 + a_2_2·c_4_10·a_1_1
  48. b_4_6·b_3_6 + b_4_6·b_3_4 + b_2_1·b_5_16 + b_2_12·b_3_6 + a_2_22·b_2_3·a_1_1
       + b_2_1·c_4_11·a_1_0
  49. b_4_62 + b_2_12·c_4_10
  50. b_4_92 + b_2_32·b_4_6 + b_2_34 + a_2_22·b_2_32 + b_2_3·c_4_11·a_1_12
       + b_2_3·c_4_10·a_1_12 + a_2_22·c_4_11 + a_2_22·c_4_10 + a_2_2·c_4_11·a_1_12
       + a_2_2·c_4_10·a_1_12
  51. a_3_5·b_5_14 + b_2_33·a_1_12 + a_2_23·b_2_3 + b_2_3·c_4_10·a_1_12 + a_2_22·c_4_11
       + a_2_2·c_4_11·a_1_12
  52. a_3_3·b_5_14
  53. b_3_4·b_5_14 + b_2_32·b_4_6 + b_2_12·b_2_32 + b_2_33·a_1_12
       + b_2_3·c_4_11·a_1_12
  54. b_3_6·b_5_14 + b_4_6·b_4_9 + b_2_32·b_4_6 + b_2_12·b_4_9 + b_2_12·b_2_32
       + a_2_2·b_2_1·b_4_6 + a_2_2·b_2_32·a_1_12 + a_2_2·c_4_11·a_1_12
       + a_2_2·c_4_10·a_1_12
  55. a_3_5·b_5_16 + a_2_23·b_2_3 + b_2_3·c_4_10·a_1_12 + a_2_22·c_4_10
       + a_2_2·c_4_10·a_1_12
  56. a_3_3·b_5_16 + a_2_2·b_2_1·b_4_6 + a_2_2·b_2_1·c_4_10 + a_2_2·c_4_10·a_1_12
  57. b_3_4·b_5_16 + b_4_6·b_4_9 + b_2_12·b_4_9 + b_2_12·b_2_32 + b_2_33·a_1_12
       + a_2_22·b_2_32 + a_2_2·b_2_32·a_1_12 + b_2_3·c_4_10·a_1_12
       + a_2_2·c_4_11·a_1_12 + a_2_2·c_4_10·a_1_12
  58. b_3_6·b_5_16 + b_4_6·b_4_9 + b_2_32·b_4_6 + b_2_12·b_4_6 + a_2_2·b_2_32·a_1_12
       + b_2_12·c_4_10 + a_2_2·b_2_1·c_4_11 + a_2_2·c_4_10·a_1_12
  59. b_4_6·b_5_14 + b_2_12·b_5_14 + b_2_13·b_3_4 + a_2_2·b_2_12·b_3_6 + b_2_1·c_4_10·b_3_4
       + b_2_12·c_4_10·a_1_0 + a_2_22·c_4_11·a_1_1
  60. b_4_9·b_5_14 + b_2_32·b_5_16 + b_2_1·b_2_32·b_3_4 + b_2_33·a_3_5 + b_2_34·a_1_1
       + a_2_2·b_2_32·a_3_5 + a_2_2·b_2_33·a_1_1 + a_2_22·b_2_32·a_1_1
       + b_2_32·c_4_11·a_1_1 + b_2_32·c_4_10·a_1_1 + a_2_2·c_4_11·a_3_5
       + a_2_2·b_2_3·c_4_10·a_1_1 + a_2_22·c_4_11·a_1_1
  61. b_4_6·b_5_16 + b_2_12·b_5_16 + b_2_12·b_5_14 + b_2_13·b_3_6 + b_2_13·b_3_4
       + a_2_2·b_2_12·b_3_6 + b_2_1·c_4_10·b_3_6 + b_2_1·c_4_10·b_3_4 + b_2_12·c_4_11·a_1_0
       + a_2_2·c_4_11·b_3_6 + a_2_22·c_4_10·a_1_1
  62. b_4_9·b_5_16 + b_2_32·b_5_14 + b_2_1·b_2_32·b_3_6 + b_2_1·b_2_32·b_3_4
       + b_2_12·b_5_14 + b_2_13·b_3_4 + b_2_33·a_3_5 + b_2_34·a_1_1 + a_2_2·b_2_12·b_3_6
       + a_2_2·b_2_33·a_1_1 + a_2_22·b_2_32·a_1_1 + b_2_1·c_4_10·b_3_4
       + b_2_32·c_4_11·a_1_1 + b_2_32·c_4_10·a_1_1 + a_2_2·c_4_10·a_3_5
       + a_2_2·b_2_3·c_4_10·a_1_1 + a_2_22·c_4_10·a_1_1
  63. b_5_142 + b_2_13·b_2_32 + b_2_34·a_1_12 + a_2_22·b_2_33 + a_2_23·b_2_32
       + b_2_1·b_2_32·c_4_10 + b_2_32·c_4_11·a_1_12 + c_4_112·a_1_12
  64. b_5_14·b_5_16 + b_2_1·b_2_32·b_4_6 + b_2_13·b_4_9 + b_2_13·b_2_32
       + a_2_2·b_2_12·b_4_6 + a_2_2·b_2_33·a_1_12 + a_2_23·b_2_32 + b_2_1·b_4_9·c_4_10
       + a_2_2·b_2_12·c_4_10 + b_2_32·c_4_10·a_1_12 + a_2_22·b_2_3·c_4_10
       + a_2_2·b_2_3·c_4_11·a_1_12 + a_2_2·b_2_3·c_4_10·a_1_12 + a_2_23·c_4_10
       + c_4_10·c_4_11·a_1_12
  65. b_5_162 + b_2_13·b_4_6 + b_2_1·b_4_6·c_4_10 + b_2_1·b_2_32·c_4_10
       + b_2_32·c_4_10·a_1_12 + c_4_102·a_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_10, a Duflot regular element of degree 4
    2. c_4_11, a Duflot regular element of degree 4
    3. b_2_3 + b_2_1, an element of degree 2
    4. b_3_4, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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. a_2_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_5 → 0, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_4_6 → 0, an element of degree 4
  11. b_4_9 → 0, an element of degree 4
  12. c_4_10 → c_1_04, an element of degree 4
  13. c_4_11 → c_1_14, an element of degree 4
  14. b_5_14 → 0, an element of degree 5
  15. b_5_16 → 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_2 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. b_2_3 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_5 → 0, an element of degree 3
  8. b_3_4 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  9. b_3_6 → c_1_22·c_1_3 + c_1_0·c_1_22, an element of degree 3
  10. b_4_6 → c_1_22·c_1_32 + c_1_02·c_1_22, an element of degree 4
  11. b_4_9 → c_1_34 + c_1_2·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
  12. c_4_10 → c_1_34 + c_1_04, an element of degree 4
  13. c_4_11 → c_1_34 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  14. b_5_14 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_24·c_1_3
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3, an element of degree 5
  15. b_5_16 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_24
       + c_1_02·c_1_2·c_1_32 + c_1_03·c_1_22, 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