Cohomology of group number 98 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 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t7  +  t5  +  t4  +  t2  +  1
    (t  −  1)2 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. a_3_1, a nilpotent element of degree 3
  5. a_4_1, a nilpotent element of degree 4
  6. b_4_2, an element of degree 4
  7. a_5_2, a nilpotent element of degree 5
  8. a_5_3, a nilpotent element of degree 5
  9. a_5_4, a nilpotent element of degree 5
  10. a_6_4, a nilpotent element of degree 6
  11. a_7_3, a nilpotent element of degree 7
  12. a_8_3, a nilpotent element of degree 8
  13. c_8_5, a Duflot regular element of degree 8

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Ring relations

There are 65 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. a_2_1·a_1_0
  5. a_2_12
  6. a_2_1·a_1_12
  7. a_1_0·a_3_1
  8. a_1_12·a_3_1
  9. a_4_1·a_1_1 + a_2_1·a_3_1
  10. a_4_1·a_1_0
  11. b_4_2·a_1_0
  12. a_2_1·a_4_1
  13. a_1_1·a_5_2 + b_4_2·a_1_12 + a_2_1·a_1_1·a_3_1
  14. a_1_0·a_5_2 + a_2_1·a_1_1·a_3_1
  15. a_2_1·b_4_2 + a_1_1·a_5_3
  16. a_1_0·a_5_3 + a_2_1·a_1_1·a_3_1
  17. a_2_1·b_4_2 + a_3_12 + a_1_1·a_5_4
  18. a_1_0·a_5_4
  19. a_2_1·a_5_3 + a_2_1·a_5_2
  20. a_2_1·a_5_2 + a_1_12·a_5_3
  21. a_4_1·a_3_1 + a_2_1·a_5_4 + a_2_1·a_5_2
  22. a_2_1·a_5_2 + a_1_12·a_5_4
  23. a_6_4·a_1_1 + a_4_1·a_3_1
  24. a_6_4·a_1_0
  25. a_4_12
  26. a_3_1·a_5_2 + b_4_2·a_1_1·a_3_1
  27. a_4_1·b_4_2 + a_3_1·a_5_3 + b_4_2·a_1_1·a_3_1
  28. a_2_1·a_1_1·a_5_4
  29. a_2_1·a_6_4
  30. a_4_1·b_4_2 + a_3_1·a_5_4 + a_1_1·a_7_3
  31. a_1_0·a_7_3
  32. a_4_1·a_5_3
  33. a_4_1·a_5_2 + a_1_1·a_3_1·a_5_3
  34. b_4_2·a_5_2 + b_4_22·a_1_1 + a_6_4·a_3_1 + a_4_1·a_5_4
  35. b_4_2·a_5_2 + b_4_22·a_1_1 + a_4_1·a_5_4 + a_4_1·a_5_2 + a_2_1·a_7_3
  36. b_4_2·a_5_2 + b_4_22·a_1_1 + a_1_12·a_7_3
  37. a_8_3·a_1_1 + a_4_1·a_5_4 + a_4_1·a_5_2
  38. a_8_3·a_1_0
  39. a_5_22 + b_4_22·a_1_12
  40. a_5_32 + b_4_2·a_1_1·a_5_3
  41. a_5_32 + a_5_2·a_5_4 + a_5_2·a_5_3 + b_4_2·a_1_1·a_5_4
  42. a_4_1·a_6_4
  43. b_4_2·a_6_4 + a_5_3·a_5_4 + a_5_32
  44. a_5_42 + a_5_2·a_5_4 + a_3_1·a_7_3
  45. a_5_32 + a_5_2·a_5_3 + a_2_1·a_1_1·a_7_3
  46. a_5_42 + a_5_32 + a_5_2·a_5_4 + a_5_2·a_5_3 + c_8_5·a_1_12
  47. a_2_1·a_8_3
  48. a_6_4·a_5_3 + a_1_1·a_5_3·a_5_4 + b_4_2·a_1_12·a_5_3
  49. a_6_4·a_5_2 + a_1_1·a_5_3·a_5_4 + b_4_2·a_1_12·a_5_3
  50. a_6_4·a_5_4 + a_4_1·a_7_3
  51. a_6_4·a_5_4 + a_2_1·c_8_5·a_1_1
  52. a_8_3·a_3_1 + a_6_4·a_5_4
  53. a_6_42
  54. a_5_2·a_7_3 + b_4_2·a_1_1·a_7_3
  55. a_5_4·a_7_3 + a_5_3·a_7_3 + c_8_5·a_1_1·a_3_1
  56. a_4_1·a_8_3
  57. b_4_2·a_8_3 + a_5_3·a_7_3 + a_5_2·a_7_3
  58. a_6_4·a_7_3 + a_2_1·c_8_5·a_3_1
  59. a_8_3·a_5_3
  60. a_8_3·a_5_2 + a_1_1·a_5_3·a_7_3 + b_4_2·a_1_12·a_7_3
  61. a_8_3·a_5_4 + a_6_4·a_7_3
  62. a_7_32 + a_1_12·a_5_3·a_7_3 + c_8_5·a_1_1·a_5_4 + c_8_5·a_1_1·a_5_3
       + b_4_2·c_8_5·a_1_12
  63. a_6_4·a_8_3
  64. a_8_3·a_7_3 + a_2_1·c_8_5·a_5_4
  65. a_8_32

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 16.
  • 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_8_5, a Duflot regular element of degree 8
    2. b_4_2, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, 6, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_3_1 → 0, an element of degree 3
  5. a_4_1 → 0, an element of degree 4
  6. b_4_2 → 0, an element of degree 4
  7. a_5_2 → 0, an element of degree 5
  8. a_5_3 → 0, an element of degree 5
  9. a_5_4 → 0, an element of degree 5
  10. a_6_4 → 0, an element of degree 6
  11. a_7_3 → 0, an element of degree 7
  12. a_8_3 → 0, an element of degree 8
  13. c_8_5 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. 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_1 → 0, an element of degree 2
  4. a_3_1 → 0, an element of degree 3
  5. a_4_1 → 0, an element of degree 4
  6. b_4_2 → c_1_14, an element of degree 4
  7. a_5_2 → 0, an element of degree 5
  8. a_5_3 → 0, an element of degree 5
  9. a_5_4 → 0, an element of degree 5
  10. a_6_4 → 0, an element of degree 6
  11. a_7_3 → 0, an element of degree 7
  12. a_8_3 → 0, an element of degree 8
  13. c_8_5 → c_1_18 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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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