Cohomology of group number 72 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 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 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    t7  −  t5  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 16 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_2, a nilpotent element of degree 3
  6. b_3_3, an element of degree 3
  7. a_4_2, a nilpotent element of degree 4
  8. b_4_4, an element of degree 4
  9. a_5_4, a nilpotent element of degree 5
  10. a_5_5, a nilpotent element of degree 5
  11. b_5_6, an element of degree 5
  12. a_6_7, a nilpotent element of degree 6
  13. b_6_8, an element of degree 6
  14. a_7_7, a nilpotent element of degree 7
  15. a_8_5, a nilpotent element of degree 8
  16. c_8_11, a Duflot regular element of degree 8

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Ring relations

There are 92 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_2·a_1_1
  4. a_2_2·a_1_0
  5. b_2_1·a_1_1
  6. a_1_14
  7. a_2_22
  8. a_1_0·a_3_2
  9. a_1_1·b_3_3 + a_1_1·a_3_2
  10. a_1_0·b_3_3 + a_2_2·b_2_1
  11. a_2_2·a_3_2
  12. b_2_1·a_3_2
  13. a_2_2·b_3_3 + a_1_12·a_3_2
  14. a_4_2·a_1_1 + a_1_12·a_3_2
  15. a_4_2·a_1_0
  16. b_4_4·a_1_0
  17. a_3_2·b_3_3 + a_3_22
  18. a_2_2·a_4_2
  19. b_3_32 + b_2_1·b_4_4 + b_2_1·a_4_2 + a_2_2·b_2_12 + a_3_22
  20. a_3_22 + b_4_4·a_1_12
  21. a_1_0·a_5_4
  22. a_1_1·a_5_5 + a_1_1·a_5_4
  23. b_2_1·a_4_2 + a_1_0·a_5_5
  24. a_1_1·b_5_6 + a_3_22
  25. a_1_0·b_5_6 + b_2_1·a_4_2 + a_2_2·b_2_12
  26. a_4_2·a_3_2 + b_4_4·a_1_13
  27. a_2_2·a_5_4
  28. b_2_1·a_5_4 + a_4_2·a_3_2
  29. a_4_2·b_3_3 + a_2_2·a_5_5
  30. a_4_2·b_3_3 + a_2_2·b_5_6 + a_4_2·a_3_2
  31. a_6_7·a_1_1 + a_1_12·a_5_4
  32. a_4_2·b_3_3 + a_6_7·a_1_0
  33. b_6_8·a_1_1 + b_4_4·a_3_2 + a_4_2·a_3_2
  34. b_6_8·a_1_0
  35. a_4_22
  36. a_1_13·a_5_4
  37. b_3_3·a_5_4 + a_4_2·b_4_4 + a_3_2·a_5_4 + b_4_4·a_1_1·a_3_2
  38. a_3_2·a_5_5 + a_3_2·a_5_4
  39. a_3_2·b_5_6 + b_4_4·a_1_1·a_3_2
  40. a_2_2·a_6_7
  41. b_3_3·a_5_5 + a_4_2·b_4_4 + b_2_1·a_6_7 + a_2_2·b_2_13 + a_3_2·a_5_4 + b_4_4·a_1_1·a_3_2
  42. a_4_2·b_4_4 + a_2_2·b_6_8 + b_4_4·a_1_1·a_3_2
  43. b_3_3·b_5_6 + b_2_1·b_6_8 + b_2_12·b_4_4 + b_3_3·a_5_5 + a_4_2·b_4_4 + a_3_2·a_5_4
       + b_2_1·a_1_0·a_5_5
  44. a_3_2·a_5_4 + a_1_1·a_7_7 + b_4_4·a_1_1·a_3_2
  45. a_1_0·a_7_7 + b_2_1·a_1_0·a_5_5
  46. a_4_2·a_5_5 + a_4_2·a_5_4
  47. a_4_2·b_5_6 + a_2_2·b_2_1·a_5_5
  48. a_6_7·a_3_2 + a_4_2·a_5_4
  49. a_6_7·b_3_3 + b_4_4·a_5_5 + b_4_4·a_5_4 + a_2_2·b_2_1·a_5_5 + b_4_4·a_1_12·a_3_2
  50. b_6_8·a_3_2 + b_4_42·a_1_1 + b_4_4·a_1_12·a_3_2
  51. b_6_8·b_3_3 + b_4_4·b_5_6 + b_2_1·b_4_4·b_3_3 + b_4_4·a_5_5 + b_4_4·a_5_4 + a_4_2·a_5_4
       + b_4_4·a_1_12·a_3_2
  52. a_2_2·a_7_7 + a_2_2·b_2_1·a_5_5
  53. b_2_1·a_7_7 + b_2_12·a_5_5 + b_4_4·a_1_12·a_3_2
  54. a_4_2·a_5_4 + a_1_12·a_7_7 + b_4_4·a_1_12·a_3_2
  55. a_8_5·a_1_1 + a_4_2·a_5_4 + b_4_4·a_1_12·a_3_2
  56. a_8_5·a_1_0
  57. a_5_4·a_5_5 + a_5_42
  58. a_2_2·b_2_14 + a_5_52 + a_5_42 + b_2_12·a_1_0·a_5_5
  59. a_5_4·b_5_6 + a_2_2·b_4_42 + b_4_4·a_1_1·a_5_4
  60. b_5_62 + b_2_1·b_4_42 + b_2_13·b_4_4 + b_4_42·a_1_12
  61. a_4_2·a_6_7
  62. a_4_2·b_6_8 + a_2_2·b_4_42 + b_4_42·a_1_12
  63. a_3_2·a_7_7 + b_4_4·a_1_1·a_5_4 + b_4_42·a_1_12
  64. b_3_3·a_7_7 + b_2_12·a_6_7 + a_2_2·b_4_42 + a_2_2·b_2_14 + b_4_4·a_1_1·a_5_4
       + b_4_42·a_1_12
  65. a_5_42 + b_4_4·a_1_1·a_5_4 + c_8_11·a_1_12
  66. a_2_2·a_8_5
  67. a_5_5·b_5_6 + b_2_1·a_8_5 + b_2_12·a_6_7 + a_2_2·b_4_42 + a_2_2·b_2_14
       + b_4_4·a_1_1·a_5_4
  68. a_6_7·a_5_5 + a_6_7·a_5_4
  69. b_6_8·a_5_5 + b_6_8·a_5_4 + a_6_7·b_5_6 + b_2_1·b_4_4·a_5_5
  70. b_6_8·b_5_6 + b_4_42·b_3_3 + b_2_1·b_4_4·b_5_6 + b_2_12·b_4_4·b_3_3 + a_6_7·b_5_6
       + b_4_4·a_1_12·a_5_4
  71. a_4_2·a_7_7 + b_4_4·a_1_12·a_5_4 + b_4_42·a_1_13
  72. b_6_8·a_5_5 + a_6_7·b_5_6 + b_4_4·a_7_7 + b_4_42·a_3_2 + b_4_42·a_1_13
  73. a_6_7·a_5_5 + b_4_4·a_1_12·a_5_4 + c_8_11·a_1_13
  74. a_8_5·a_3_2 + b_4_4·a_1_12·a_5_4 + b_4_42·a_1_13
  75. a_8_5·b_3_3 + a_6_7·b_5_6 + b_2_1·b_4_4·a_5_5 + b_4_42·a_1_13
  76. a_6_72
  77. b_6_82 + b_4_43
  78. a_5_5·a_7_7 + a_5_4·a_7_7 + b_2_1·a_5_52
  79. a_5_5·a_7_7 + b_2_1·a_5_52 + c_8_11·a_1_1·a_3_2
  80. b_5_6·a_7_7 + b_2_12·a_8_5 + b_2_13·a_6_7 + a_2_2·b_4_4·b_6_8 + b_4_4·a_1_1·a_7_7
       + b_2_1·a_5_52 + b_2_13·a_1_0·a_5_5
  81. a_4_2·a_8_5
  82. a_6_7·b_6_8 + b_4_4·a_8_5 + a_2_2·b_4_4·b_6_8 + b_4_42·a_1_1·a_3_2
  83. b_6_8·a_7_7 + b_4_42·a_5_4 + b_4_43·a_1_1 + b_2_1·a_6_7·b_5_6 + b_2_12·b_4_4·a_5_5
       + b_4_42·a_1_12·a_3_2
  84. a_6_7·a_7_7 + c_8_11·a_1_12·a_3_2
  85. a_8_5·a_5_5 + a_6_7·a_7_7 + a_2_2·b_2_13·a_5_5
  86. a_8_5·a_5_4 + a_6_7·a_7_7
  87. a_8_5·b_5_6 + b_4_42·a_5_5 + b_4_42·a_5_4 + b_2_1·a_6_7·b_5_6 + b_2_12·b_4_4·a_5_5
       + a_2_2·b_2_13·a_5_5 + b_4_42·a_1_12·a_3_2
  88. a_7_72 + b_4_42·a_1_1·a_5_4 + b_4_43·a_1_12 + b_2_12·a_5_52
       + b_4_4·c_8_11·a_1_12
  89. b_6_8·a_8_5 + b_4_42·a_6_7 + a_2_2·b_4_43 + b_4_43·a_1_12
  90. a_6_7·a_8_5
  91. a_8_5·a_7_7 + a_5_53 + b_4_43·a_1_13 + c_8_11·a_1_12·a_5_4
  92. a_8_52

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_11, a Duflot regular element of degree 8
    2. b_4_4 + b_2_12, an element of degree 4
    3. b_3_3, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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 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_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. a_4_2 → 0, an element of degree 4
  8. b_4_4 → 0, an element of degree 4
  9. a_5_4 → 0, an element of degree 5
  10. a_5_5 → 0, an element of degree 5
  11. b_5_6 → 0, an element of degree 5
  12. a_6_7 → 0, an element of degree 6
  13. b_6_8 → 0, an element of degree 6
  14. a_7_7 → 0, an element of degree 7
  15. a_8_5 → 0, an element of degree 8
  16. c_8_11 → 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_2 → 0, an element of degree 3
  6. b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. a_4_2 → 0, 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_4 → 0, an element of degree 5
  10. a_5_5 → 0, an element of degree 5
  11. b_5_6 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  12. a_6_7 → 0, an element of degree 6
  13. b_6_8 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23, an element of degree 6
  14. a_7_7 → 0, an element of degree 7
  15. a_8_5 → 0, an element of degree 8
  16. c_8_11 → 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_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + 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