Cohomology of group number 137 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 2.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    (t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)
    (t  −  1)2 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-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. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. a_2_2, a nilpotent element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_4, a nilpotent element of degree 3
  8. a_4_5, a nilpotent element of degree 4
  9. b_4_6, an element of degree 4
  10. a_5_7, a nilpotent element of degree 5
  11. b_6_8, an element of degree 6
  12. a_7_9, a nilpotent element of degree 7
  13. c_8_10, a Duflot regular element of degree 8

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Ring relations

There are 57 minimal relations of maximal degree 14:

  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. a_2_2·a_1_0
  6. a_2_1·a_2_2
  7. a_1_0·a_3_2
  8. b_1_1·a_3_2
  9. a_1_0·a_3_3
  10. a_1_0·a_3_4 + a_2_12
  11. b_1_1·a_3_4 + b_1_1·a_3_3 + a_2_22
  12. a_2_2·a_3_2
  13. a_2_1·a_3_2
  14. a_2_1·a_3_3
  15. a_2_2·a_3_4 + a_2_2·a_3_3 + a_2_1·a_3_4
  16. a_4_5·a_1_0
  17. b_1_12·a_3_3 + a_4_5·b_1_1 + a_2_2·a_3_3
  18. b_4_6·b_1_1 + b_4_6·a_1_0
  19. a_3_22
  20. a_3_2·a_3_3
  21. a_3_32 + a_2_2·b_1_1·a_3_3 + a_2_22·b_1_12
  22. a_3_2·a_3_4
  23. a_3_3·a_3_4 + a_2_2·a_4_5 + a_2_22·b_1_12
  24. a_2_1·a_4_5
  25. a_2_2·b_4_6 + a_2_1·b_4_6 + a_3_42 + a_3_32
  26. a_1_0·a_5_7
  27. b_1_1·a_5_7 + a_3_32
  28. a_4_5·a_3_2
  29. a_4_5·a_3_3 + a_2_2·a_4_5·b_1_1 + a_2_22·b_1_13
  30. b_4_6·a_3_3 + b_4_6·a_3_2 + a_4_5·a_3_4 + a_4_5·a_3_3 + a_2_22·a_3_3
  31. a_4_5·a_3_4 + a_4_5·a_3_3 + a_2_2·a_5_7
  32. a_4_5·a_3_4 + a_4_5·a_3_3 + a_2_1·a_5_7 + a_2_22·a_3_3
  33. b_6_8·a_1_0 + b_4_6·a_3_3
  34. b_6_8·b_1_1 + b_4_6·a_3_3 + a_4_5·b_1_13 + a_4_5·a_3_3 + a_2_22·a_3_3
  35. a_4_52 + a_2_2·a_4_5·b_1_12 + a_2_22·b_1_14 + a_2_22·a_4_5 + a_2_1·a_3_42
  36. a_3_3·a_5_7
  37. a_3_2·a_5_7 + a_2_1·a_3_42
  38. a_4_5·b_4_6 + a_2_2·b_6_8 + a_4_52 + a_2_22·b_1_14 + a_2_1·a_3_42
  39. a_4_5·b_4_6 + a_2_1·b_6_8 + a_3_4·a_5_7
  40. a_1_0·a_7_9 + a_2_1·a_3_42
  41. b_1_1·a_7_9 + a_4_5·b_1_14 + a_2_22·b_1_14 + a_2_1·a_3_42
  42. a_4_5·a_5_7 + a_2_1·b_4_6·a_3_4
  43. b_6_8·a_3_3 + b_4_62·a_1_0 + a_2_2·a_4_5·b_1_13 + a_2_22·b_1_15
  44. b_6_8·a_3_2 + b_4_62·a_1_0 + a_2_1·b_4_6·a_3_4
  45. a_2_2·a_7_9 + a_2_2·a_4_5·b_1_13
  46. a_2_1·a_7_9
  47. a_5_72 + b_4_6·a_3_42
  48. a_4_5·b_6_8 + a_2_1·b_4_62 + b_4_6·a_3_42 + a_2_2·a_4_5·b_1_14 + a_2_22·b_1_16
       + a_2_22·a_4_5·b_1_12 + a_2_1·a_3_4·a_5_7
  49. a_3_3·a_7_9 + a_2_2·a_4_5·b_1_14 + a_2_22·b_1_16 + a_2_22·a_4_5·b_1_12
       + a_2_1·a_3_4·a_5_7
  50. a_3_2·a_7_9 + a_2_1·a_3_4·a_5_7
  51. a_3_4·a_7_9 + a_2_2·a_4_5·b_1_14 + a_2_22·b_1_16 + a_2_1·a_3_4·a_5_7
  52. a_4_5·a_7_9 + a_2_2·a_4_5·b_1_15 + a_2_22·b_1_17
  53. b_6_8·a_5_7 + b_4_6·a_7_9 + b_4_62·a_3_4 + a_2_1·b_4_6·a_5_7
  54. a_5_7·a_7_9 + a_2_1·b_4_6·a_3_42
  55. b_6_82 + b_4_63 + b_4_6·a_3_4·a_5_7 + a_2_2·a_4_5·b_1_16 + a_2_22·b_1_18
       + a_2_22·a_4_5·b_1_14 + a_2_12·c_8_10
  56. b_6_8·a_7_9 + b_4_6·b_6_8·a_3_4 + b_4_62·a_5_7 + a_2_2·a_4_5·b_1_17
       + a_2_22·b_1_19 + a_2_1·b_4_62·a_3_4
  57. a_7_92 + a_2_2·a_4_5·b_1_18 + a_2_22·b_1_110 + a_2_22·a_4_5·b_1_16

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 14.
  • 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_1_14 + b_4_6, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_4_5 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. a_5_7 → 0, an element of degree 5
  11. b_6_8 → 0, an element of degree 6
  12. a_7_9 → 0, an element of degree 7
  13. c_8_10 → 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. b_1_1 → c_1_1, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_4_5 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. a_5_7 → 0, an element of degree 5
  11. b_6_8 → 0, an element of degree 6
  12. a_7_9 → 0, an element of degree 7
  13. c_8_10 → c_1_04·c_1_14 + 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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_4_5 → 0, an element of degree 4
  9. b_4_6 → c_1_14, an element of degree 4
  10. a_5_7 → 0, an element of degree 5
  11. b_6_8 → c_1_16, an element of degree 6
  12. a_7_9 → 0, an element of degree 7
  13. c_8_10 → c_1_18 + c_1_04·c_1_14 + 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