Cohomology of group number 146 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 3.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t9  −  t8  −  t2  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-4,-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 8:

  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. a_3_3, a nilpotent element of degree 3
  6. b_3_2, an element of degree 3
  7. b_4_2, an element of degree 4
  8. b_4_4, an element of degree 4
  9. a_5_5, a nilpotent element of degree 5
  10. b_5_4, an element of degree 5
  11. b_6_6, an element of degree 6
  12. b_7_7, an element of degree 7
  13. b_8_8, an element of degree 8
  14. c_8_10, a Duflot regular element of degree 8

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Ring relations

There are 63 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_2·a_1_1 + a_1_13
  4. a_2_2·a_1_0 + a_1_13
  5. b_2_1·a_1_1 + a_1_13
  6. a_2_2·b_2_1 + a_2_22
  7. a_1_0·a_3_3 + a_2_22
  8. a_1_1·b_3_2 + a_2_22
  9. a_1_0·b_3_2 + a_2_22
  10. b_2_1·a_3_3 + a_2_2·a_3_3
  11. a_2_2·a_3_3 + a_1_12·a_3_3
  12. a_2_2·b_3_2 + a_2_2·a_3_3
  13. b_4_2·a_1_1 + a_2_2·a_3_3
  14. b_4_2·a_1_0
  15. b_4_4·a_1_0
  16. a_2_2·b_4_2
  17. a_3_3·b_3_2 + a_2_2·b_4_4
  18. b_3_22 + b_2_1·b_4_4
  19. a_3_3·b_3_2 + a_3_32 + a_1_1·a_5_5
  20. a_1_0·a_5_5
  21. a_1_1·b_5_4
  22. a_1_0·b_5_4
  23. b_4_2·a_3_3 + a_2_2·a_5_5
  24. b_4_2·a_3_3 + b_2_1·a_5_5
  25. b_4_2·a_3_3 + a_1_12·a_5_5
  26. b_4_2·a_3_3 + a_2_2·b_5_4
  27. b_6_6·a_1_1 + b_4_2·a_3_3
  28. b_4_2·b_3_2 + b_2_1·b_5_4 + b_6_6·a_1_0 + b_4_2·a_3_3
  29. b_4_22 + b_2_12·b_4_4
  30. b_3_2·a_5_5 + a_3_3·b_5_4
  31. b_3_2·b_5_4 + b_4_2·b_4_4 + b_3_2·a_5_5 + b_4_4·a_1_1·a_3_3
  32. a_2_2·b_6_6
  33. b_3_2·a_5_5 + a_1_1·b_7_7 + a_3_3·a_5_5
  34. a_1_0·b_7_7
  35. b_4_2·a_5_5 + a_1_1·a_3_3·a_5_5
  36. b_4_2·b_5_4 + b_2_1·b_4_4·b_3_2 + b_4_4·a_1_12·a_3_3
  37. b_6_6·a_3_3 + b_4_2·a_5_5 + b_4_4·a_1_12·a_3_3
  38. a_2_2·b_7_7
  39. b_6_6·b_3_2 + b_2_1·b_7_7 + b_2_1·b_4_4·b_3_2 + b_2_12·b_5_4 + b_2_13·b_3_2
       + b_2_1·b_6_6·a_1_0 + b_4_4·a_1_12·a_3_3
  40. b_8_8·a_1_1 + b_4_2·a_5_5
  41. b_8_8·a_1_0 + b_2_1·b_6_6·a_1_0
  42. a_5_5·b_5_4 + b_4_4·a_3_32 + b_4_4·a_1_1·a_5_5
  43. b_5_42 + b_2_1·b_4_42
  44. a_3_3·b_7_7 + a_5_52 + b_4_4·a_3_32 + b_4_4·a_1_1·a_5_5
  45. b_3_2·b_7_7 + b_4_4·b_6_6 + b_2_1·b_4_42 + b_2_1·b_4_2·b_4_4 + b_2_13·b_4_4
       + b_4_4·a_3_32 + b_4_42·a_1_12
  46. a_5_52 + b_4_4·a_3_32 + b_4_42·a_1_12 + c_8_10·a_1_12
  47. a_2_2·b_8_8
  48. b_4_2·b_6_6 + b_2_1·b_8_8 + b_2_1·b_4_2·b_4_4 + b_2_12·b_6_6 + b_2_13·b_4_4
       + b_2_13·b_4_2
  49. b_6_6·a_5_5 + c_8_10·a_1_13
  50. b_6_6·b_5_4 + b_4_2·b_7_7 + b_2_1·b_4_4·b_5_4 + b_2_12·b_4_4·b_3_2 + b_2_13·b_5_4
       + b_2_1·c_8_10·a_1_0
  51. b_8_8·a_3_3 + b_6_6·a_5_5 + b_4_4·a_1_12·a_5_5
  52. b_8_8·b_3_2 + b_4_2·b_7_7 + b_2_12·b_7_7 + b_2_12·b_4_4·b_3_2 + b_2_13·b_5_4
       + b_2_14·b_3_2 + b_6_6·a_5_5 + b_2_12·b_6_6·a_1_0 + b_4_4·a_1_12·a_5_5
  53. b_6_62 + b_2_12·b_4_42 + b_2_12·b_4_2·b_4_4 + b_2_13·b_6_6 + b_2_14·b_4_2
       + b_2_12·c_8_10 + a_2_22·c_8_10
  54. b_6_62 + b_2_12·b_4_42 + b_2_12·b_4_2·b_4_4 + b_2_13·b_6_6 + b_2_14·b_4_2
       + a_5_5·b_7_7 + b_4_4·a_1_1·b_7_7 + b_4_42·a_1_1·a_3_3 + b_2_12·c_8_10
       + c_8_10·a_1_1·a_3_3
  55. b_4_2·b_8_8 + b_2_1·b_4_4·b_6_6 + b_2_12·b_8_8 + b_2_12·b_4_42 + b_2_13·b_6_6
       + b_2_14·b_4_2
  56. b_5_4·b_7_7 + b_4_4·b_8_8 + b_2_1·b_4_4·b_6_6 + b_4_4·a_1_1·b_7_7
  57. b_6_6·b_7_7 + b_2_1·b_4_4·b_7_7 + b_2_1·b_4_2·b_7_7 + b_2_12·b_4_4·b_5_4
       + b_4_42·a_1_12·a_3_3 + b_2_1·c_8_10·b_3_2
  58. b_8_8·a_5_5 + b_4_4·a_1_1·a_3_3·a_5_5 + b_4_42·a_1_12·a_3_3 + c_8_10·a_1_12·a_3_3
  59. b_8_8·b_5_4 + b_2_1·b_4_4·b_7_7 + b_2_1·b_4_2·b_7_7 + b_2_12·b_4_4·b_5_4
       + b_2_13·b_4_4·b_3_2 + b_2_14·b_5_4 + b_2_12·c_8_10·a_1_0
  60. b_7_72 + b_2_1·b_4_2·b_4_42 + b_2_12·b_4_4·b_6_6 + b_2_13·b_4_42
       + b_2_13·b_4_2·b_4_4 + b_2_15·b_4_4 + b_4_43·a_1_12 + b_2_1·b_4_4·c_8_10
       + c_8_10·a_3_32 + b_4_4·c_8_10·a_1_12
  61. b_6_6·b_8_8 + b_2_1·b_4_4·b_8_8 + b_2_13·b_4_42 + b_2_14·b_6_6 + b_2_15·b_4_4
       + b_2_15·b_4_2 + b_2_1·b_4_2·c_8_10 + b_2_13·c_8_10
  62. b_8_8·b_7_7 + b_2_12·b_4_4·b_7_7 + b_2_12·b_4_42·b_3_2 + b_2_13·b_4_4·b_5_4
       + b_2_1·c_8_10·b_5_4 + b_2_12·c_8_10·b_3_2 + b_6_6·c_8_10·a_1_0
  63. b_8_82 + b_2_12·b_4_2·b_4_42 + b_2_13·b_4_4·b_6_6 + b_2_15·b_6_6 + b_2_16·b_4_4
       + b_2_16·b_4_2 + b_2_12·b_4_4·c_8_10 + b_2_14·c_8_10

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 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_10, a Duflot regular element of degree 8
    2. b_4_4 + b_2_12, an element of degree 4
    3. b_3_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 12].
  • 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 1

  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. a_3_3 → 0, an element of degree 3
  6. b_3_2 → 0, an element of degree 3
  7. b_4_2 → 0, an element of degree 4
  8. b_4_4 → 0, an element of degree 4
  9. a_5_5 → 0, an element of degree 5
  10. b_5_4 → 0, an element of degree 5
  11. b_6_6 → 0, an element of degree 6
  12. b_7_7 → 0, an element of degree 7
  13. b_8_8 → 0, an element of degree 8
  14. c_8_10 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. 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_2 → 0, an element of degree 2
  4. b_2_1 → c_1_12, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. b_3_2 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_4_2 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  8. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  9. a_5_5 → 0, an element of degree 5
  10. b_5_4 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  11. b_6_6 → c_1_12·c_1_24 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_12, an element of degree 6
  12. b_7_7 → c_1_15·c_1_22 + c_1_16·c_1_2 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2, an element of degree 7
  13. b_8_8 → c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14, an element of degree 8
  14. c_8_10 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8

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