Cohomology of group number 122 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 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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
    t3  +  t2  +  1
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)
  • 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_0, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_4, an element of degree 2
  7. c_2_3, a Duflot regular element of degree 2
  8. b_3_5, an element of degree 3
  9. b_3_6, an element of degree 3
  10. b_3_7, an element of degree 3
  11. b_3_8, an element of degree 3
  12. b_4_7, an element of degree 4
  13. b_4_14, an element of degree 4
  14. c_4_15, a Duflot regular element of degree 4
  15. b_5_23, 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_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. b_2_1·a_1_1
  7. b_2_2·a_1_0
  8. b_2_4·a_1_0
  9. a_2_02
  10. b_2_1·b_2_2
  11. a_1_1·b_3_5
  12. a_1_0·b_3_5 + a_2_0·b_2_1
  13. a_1_1·b_3_6 + a_2_0·b_2_2
  14. a_1_0·b_3_6
  15. a_1_1·b_3_7
  16. a_1_0·b_3_7
  17. a_1_1·b_3_8 + a_2_0·b_2_4
  18. a_1_0·b_3_8
  19. b_2_2·b_3_5
  20. a_2_0·b_3_5 + b_2_1·c_2_3·a_1_0
  21. b_2_1·b_3_6
  22. b_2_22·a_1_1 + a_2_0·b_3_6
  23. b_2_2·b_3_7
  24. a_2_0·b_3_7
  25. b_2_1·b_3_8
  26. b_2_2·b_2_4·a_1_1 + a_2_0·b_3_8
  27. b_4_7·a_1_1
  28. b_4_7·a_1_0
  29. b_2_4·b_3_6 + b_2_2·b_3_8 + b_4_14·a_1_1 + b_2_42·a_1_1
  30. b_4_14·a_1_0
  31. b_3_52 + b_2_12·c_2_3
  32. b_3_5·b_3_6
  33. b_3_62 + b_2_23 + a_2_0·b_2_22
  34. b_3_6·b_3_7
  35. b_3_72 + b_2_1·b_2_42
  36. b_3_5·b_3_8
  37. b_3_7·b_3_8
  38. b_3_82 + b_2_2·b_2_42 + a_2_0·b_2_42
  39. b_3_5·b_3_7 + b_2_1·b_4_7
  40. b_3_6·b_3_8 + b_2_2·b_4_7 + b_2_22·b_2_4 + a_2_0·b_2_2·b_2_4
  41. a_2_0·b_4_7
  42. b_2_1·b_4_14 + b_2_1·b_2_42
  43. b_3_6·b_3_8 + b_2_22·b_2_4 + a_2_0·b_4_14 + a_2_0·b_2_42 + a_2_0·b_2_2·b_2_4
  44. b_3_6·b_3_8 + b_2_22·b_2_4 + a_1_1·b_5_23 + a_2_0·b_2_42 + a_2_0·b_2_2·b_2_4
  45. a_1_0·b_5_23
  46. b_4_7·b_3_5 + b_2_1·c_2_3·b_3_7
  47. b_4_7·b_3_7 + b_2_42·b_3_5
  48. b_4_7·b_3_6 + b_2_2·b_4_14·a_1_1 + a_2_0·b_2_4·b_3_8
  49. b_4_7·b_3_8 + b_2_4·b_4_14·a_1_1 + b_2_43·a_1_1
  50. b_4_14·b_3_5 + b_2_42·b_3_5
  51. b_4_14·b_3_7 + b_2_42·b_3_7
  52. b_2_1·b_5_23
  53. b_4_14·b_3_6 + b_4_7·b_3_8 + b_2_2·b_5_23 + a_2_0·b_2_4·b_3_8
  54. b_4_14·b_3_8 + b_4_7·b_3_8 + b_2_4·b_5_23 + b_2_2·c_4_15·a_1_1
  55. b_4_7·b_3_6 + a_2_0·b_5_23 + a_2_0·b_2_4·b_3_8
  56. b_4_72 + b_2_1·b_2_42·c_2_3
  57. b_4_142 + b_2_44 + b_2_2·b_2_4·b_4_14 + a_2_0·b_2_43 + b_2_22·c_4_15
  58. b_4_7·b_4_14 + b_2_42·b_4_7 + a_2_0·b_2_4·b_4_14 + a_2_0·b_2_2·c_4_15
  59. b_3_5·b_5_23
  60. b_3_6·b_5_23 + b_2_22·b_4_14 + a_2_0·b_2_4·b_4_14 + a_2_0·b_2_43 + a_2_0·b_2_2·b_4_14
       + a_2_0·b_2_2·b_2_42
  61. b_3_7·b_5_23
  62. b_3_8·b_5_23 + b_4_142 + b_2_44 + b_2_22·c_4_15 + a_2_0·b_2_2·c_4_15
  63. b_4_14·b_5_23 + b_2_43·b_3_8 + b_2_2·b_2_4·b_5_23 + b_2_42·b_4_14·a_1_1
       + b_2_44·a_1_1 + b_2_2·c_4_15·b_3_6 + a_2_0·c_4_15·b_3_8
  64. b_4_7·b_5_23 + b_2_42·b_4_14·a_1_1 + b_2_44·a_1_1 + a_2_0·b_2_4·b_5_23
       + a_2_0·c_4_15·b_3_6
  65. b_5_232 + b_2_2·b_2_44 + b_2_22·b_2_4·b_4_14 + a_2_0·b_2_44
       + a_2_0·b_2_2·b_2_4·b_4_14 + a_2_0·b_2_2·b_2_43 + b_2_23·c_4_15
       + a_2_0·b_2_22·c_4_15

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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_2_3, a Duflot regular element of degree 2
    2. c_4_15, a Duflot regular element of degree 4
    3. b_2_4 + b_2_2 + b_2_1, an element of degree 2
    4. b_3_8 + b_3_7, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. c_2_3 → c_1_02, an element of degree 2
  8. b_3_5 → 0, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_3_7 → 0, an element of degree 3
  11. b_3_8 → 0, an element of degree 3
  12. b_4_7 → 0, an element of degree 4
  13. b_4_14 → 0, an element of degree 4
  14. c_4_15 → c_1_14, an element of degree 4
  15. b_5_23 → 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_0 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  7. c_2_3 → c_1_02, an element of degree 2
  8. b_3_5 → 0, an element of degree 3
  9. b_3_6 → c_1_23, an element of degree 3
  10. b_3_7 → 0, an element of degree 3
  11. b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  12. b_4_7 → 0, an element of degree 4
  13. b_4_14 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_23
       + c_1_12·c_1_22, an element of degree 4
  14. c_4_15 → 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, an element of degree 4
  15. b_5_23 → 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_1·c_1_24
       + c_1_12·c_1_23, 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_0 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  7. c_2_3 → c_1_02, an element of degree 2
  8. b_3_5 → c_1_0·c_1_22, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_3_7 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  11. b_3_8 → 0, an element of degree 3
  12. b_4_7 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
  13. b_4_14 → c_1_34 + c_1_22·c_1_32, an element of degree 4
  14. c_4_15 → 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, an element of degree 4
  15. b_5_23 → 0, 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