Cohomology of group number 34 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 2.
  • 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 coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t4  +  t3  +  t2  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • 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 18 minimal generators of maximal degree 6:

  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. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. a_3_1, a nilpotent element of degree 3
  7. a_3_2, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. b_3_3, an element of degree 3
  10. a_4_2, a nilpotent element of degree 4
  11. a_4_3, a nilpotent element of degree 4
  12. a_4_5, a nilpotent element of degree 4
  13. b_4_6, an element of degree 4
  14. c_4_7, a Duflot regular element of degree 4
  15. c_4_8, a Duflot regular element of degree 4
  16. a_5_10, a nilpotent element of degree 5
  17. b_5_11, an element of degree 5
  18. a_6_11, a nilpotent element of degree 6

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 119 minimal relations of maximal degree 12:

  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. a_2_1·a_1_1
  7. a_2_1·a_1_0
  8. b_2_2·a_1_0
  9. a_2_02
  10. a_2_0·a_2_1
  11. a_2_12
  12. a_1_1·a_3_1
  13. a_1_0·a_3_1
  14. a_1_1·a_3_2
  15. a_1_0·a_3_2
  16. a_2_1·b_2_2 + a_1_1·a_3_4
  17. a_1_0·a_3_4
  18. a_1_1·b_3_3 + a_2_0·b_2_2
  19. a_1_0·b_3_3
  20. a_2_0·a_3_1
  21. a_2_1·a_3_1 + a_2_0·a_3_2
  22. a_2_1·a_3_2
  23. b_2_2·a_3_1 + a_2_0·a_3_4
  24. a_2_1·a_3_4
  25. b_2_22·a_1_1 + a_2_0·b_3_3
  26. b_2_2·a_3_1 + a_2_1·b_3_3
  27. a_4_2·a_1_1 + a_2_1·a_3_1
  28. a_4_2·a_1_0
  29. a_4_3·a_1_1 + a_2_1·a_3_1
  30. a_4_3·a_1_0 + a_2_1·a_3_1
  31. b_2_2·a_3_1 + a_4_5·a_1_1 + a_2_1·a_3_1
  32. a_4_5·a_1_0
  33. b_4_6·a_1_1 + b_2_2·a_3_2 + b_2_2·a_3_1
  34. b_4_6·a_1_0 + a_2_1·a_3_1
  35. a_3_12
  36. a_3_22
  37. a_3_1·a_3_2
  38. a_3_1·a_3_4
  39. a_3_42
  40. a_3_1·b_3_3 + b_2_2·a_1_1·a_3_4
  41. b_3_32 + b_2_23
  42. a_3_2·b_3_3 + b_2_2·a_4_2
  43. a_2_0·a_4_2
  44. a_2_1·a_4_2
  45. a_3_2·b_3_3 + b_2_2·a_4_3 + a_2_0·b_2_22 + a_3_2·a_3_4
  46. a_2_0·a_4_3
  47. a_2_1·a_4_3
  48. a_3_4·b_3_3 + a_3_2·b_3_3 + b_2_2·a_4_5 + a_2_0·b_2_22
  49. b_2_2·a_1_1·a_3_4 + a_2_0·a_4_5
  50. a_2_1·a_4_5
  51. a_3_2·b_3_3 + a_2_0·b_4_6 + b_2_2·a_1_1·a_3_4
  52. a_2_1·b_4_6 + a_3_2·a_3_4
  53. a_3_2·a_3_4 + a_1_1·a_5_10
  54. a_1_0·a_5_10
  55. a_3_2·b_3_3 + a_1_1·b_5_11 + a_2_0·b_2_22 + a_3_2·a_3_4 + b_2_2·a_1_1·a_3_4
  56. a_1_0·b_5_11
  57. a_4_2·a_3_2
  58. a_4_2·a_3_1
  59. a_4_2·b_3_3 + b_2_22·a_3_2
  60. a_4_3·a_3_2
  61. a_4_3·a_3_1
  62. a_4_3·a_3_4 + a_4_2·a_3_4 + a_2_0·b_2_2·a_3_4
  63. a_4_3·b_3_3 + b_2_22·a_3_2 + a_2_0·b_2_2·b_3_3 + a_4_2·a_3_4
  64. a_4_5·a_3_2 + a_4_2·a_3_4
  65. a_4_5·a_3_1
  66. a_4_5·a_3_4 + a_4_2·a_3_4 + a_2_0·b_2_2·a_3_4
  67. a_4_5·b_3_3 + b_2_22·a_3_4 + b_2_22·a_3_2 + a_2_0·b_2_2·b_3_3
  68. b_4_6·a_3_2 + a_4_2·a_3_4 + a_2_0·b_2_2·a_3_4 + b_2_2·c_4_7·a_1_1
  69. b_4_6·a_3_1 + a_4_2·a_3_4
  70. b_4_6·a_3_4 + b_2_2·a_5_10 + a_2_0·b_2_2·b_3_3 + a_2_0·b_2_2·a_3_4 + b_2_2·c_4_8·a_1_1
  71. a_4_2·a_3_4 + a_2_0·a_5_10
  72. a_2_1·a_5_10
  73. b_4_6·b_3_3 + b_2_2·b_5_11 + b_2_22·b_3_3 + b_4_6·a_3_4 + b_2_22·a_3_2 + a_4_2·a_3_4
  74. b_2_22·a_3_2 + a_2_0·b_5_11 + a_2_0·b_2_2·b_3_3 + a_4_2·a_3_4 + a_2_0·b_2_2·a_3_4
  75. a_2_1·b_5_11 + a_4_2·a_3_4 + a_2_0·b_2_2·a_3_4
  76. a_6_11·a_1_1 + a_4_2·a_3_4
  77. a_6_11·a_1_0
  78. a_4_22
  79. a_4_32
  80. a_4_2·a_4_3
  81. a_4_52
  82. a_4_3·a_4_5 + a_4_2·a_4_5 + a_2_0·b_2_2·a_4_5
  83. a_4_3·b_4_6 + a_2_0·b_2_2·b_4_6 + a_4_3·a_4_5 + a_2_0·b_2_2·c_4_7 + c_4_7·a_1_1·a_3_4
  84. b_4_62 + b_2_22·a_4_5 + a_2_0·b_2_23 + a_4_3·a_4_5 + a_2_0·b_2_2·a_4_5
       + b_2_22·c_4_7
  85. a_4_2·b_4_6 + a_4_3·a_4_5 + a_2_0·b_2_2·c_4_7
  86. a_3_2·a_5_10 + c_4_7·a_1_1·a_3_4
  87. a_3_1·a_5_10
  88. a_4_3·a_4_5 + b_2_2·a_1_1·a_5_10 + a_2_0·b_2_2·a_4_5
  89. a_3_4·a_5_10 + a_2_0·b_2_2·a_4_5 + c_4_8·a_1_1·a_3_4
  90. b_3_3·a_5_10 + a_4_5·b_4_6 + a_4_3·b_4_6 + a_2_0·b_2_23 + a_2_0·b_2_2·a_4_5
       + a_2_0·b_2_2·c_4_8 + c_4_7·a_1_1·a_3_4
  91. a_3_2·b_5_11 + a_4_3·b_4_6 + a_2_0·b_2_2·a_4_5
  92. a_3_1·b_5_11 + a_4_3·a_4_5
  93. a_3_4·b_5_11 + a_4_5·b_4_6 + b_2_22·a_4_5 + a_2_0·b_2_23 + a_2_0·b_2_2·c_4_7
  94. b_3_3·b_5_11 + b_2_22·b_4_6 + b_2_24 + a_4_5·b_4_6 + a_2_0·b_2_2·c_4_7
  95. a_4_5·b_4_6 + b_2_2·a_6_11 + a_4_3·a_4_5 + a_2_0·b_2_2·c_4_8 + c_4_8·a_1_1·a_3_4
       + c_4_7·a_1_1·a_3_4
  96. a_4_3·a_4_5 + a_2_0·a_6_11 + a_2_0·b_2_2·a_4_5
  97. a_2_1·a_6_11
  98. a_4_3·a_5_10 + a_2_0·b_2_2·a_5_10 + a_2_0·c_4_7·a_3_4
  99. a_4_5·a_5_10 + a_4_3·a_5_10 + a_2_0·b_2_22·a_3_4 + a_2_0·c_4_8·a_3_4
       + a_2_0·c_4_7·a_3_2
  100. a_4_2·a_5_10 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_4
  101. a_4_3·b_5_11 + a_2_0·b_2_22·b_3_3 + a_4_3·a_5_10 + a_2_0·c_4_7·b_3_3
       + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_4
  102. b_4_6·a_5_10 + a_2_0·b_2_2·b_5_11 + a_2_0·b_2_22·b_3_3 + a_4_3·a_5_10
       + b_2_2·c_4_8·a_3_2 + b_2_2·c_4_7·a_3_4 + a_2_0·c_4_8·a_3_4 + a_2_0·c_4_7·a_3_4
       + a_2_0·c_4_7·a_3_2
  103. a_4_5·b_5_11 + b_2_22·a_5_10 + b_2_23·a_3_4 + a_4_3·a_5_10 + a_2_0·b_2_22·a_3_4
       + a_2_0·c_4_8·b_3_3 + a_2_0·c_4_7·b_3_3 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_2
  104. b_4_6·b_5_11 + b_2_22·b_5_11 + b_2_23·b_3_3 + b_2_22·a_5_10 + b_2_23·a_3_4
       + a_2_0·b_2_22·b_3_3 + b_2_2·c_4_7·b_3_3 + b_2_2·c_4_7·a_3_4 + a_2_0·c_4_8·b_3_3
       + a_2_0·c_4_7·b_3_3 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_4
  105. b_4_6·a_5_10 + a_4_2·b_5_11 + a_4_3·a_5_10 + b_2_2·c_4_8·a_3_2 + b_2_2·c_4_7·a_3_4
       + a_2_0·c_4_7·b_3_3 + a_2_0·c_4_8·a_3_4 + a_2_0·c_4_7·a_3_2
  106. a_6_11·a_3_2 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_4 + a_2_0·c_4_7·a_3_2
  107. a_6_11·a_3_1 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_2
  108. a_6_11·a_3_4 + a_4_3·a_5_10 + a_2_0·c_4_8·a_3_4 + a_2_0·c_4_8·a_3_2 + a_2_0·c_4_7·a_3_2
  109. a_6_11·b_3_3 + b_4_6·a_5_10 + b_2_22·a_5_10 + a_2_0·b_2_22·b_3_3 + a_2_0·b_2_22·a_3_4
       + b_2_2·c_4_8·a_3_2 + b_2_2·c_4_7·a_3_4 + a_2_0·c_4_7·b_3_3 + a_2_0·c_4_7·a_3_4
       + a_2_0·c_4_7·a_3_2
  110. a_5_102
  111. b_5_112 + b_2_25 + b_2_23·a_4_5 + a_2_0·b_2_24 + a_4_3·a_6_11 + b_2_23·c_4_7
       + a_2_0·a_4_5·c_4_7
  112. b_5_112 + b_2_25 + a_5_10·b_5_11 + b_2_22·a_6_11 + b_2_23·a_4_5
       + a_2_0·b_2_22·a_4_5 + b_2_23·c_4_7 + b_2_2·a_4_5·c_4_7 + a_2_0·b_4_6·c_4_8
       + a_2_0·b_4_6·c_4_7 + c_4_8·a_1_1·a_5_10 + a_2_0·a_4_5·c_4_8 + a_2_0·a_4_5·c_4_7
  113. b_5_112 + b_2_25 + b_2_23·a_4_5 + a_2_0·b_2_24 + a_2_0·b_2_2·a_6_11 + b_2_23·c_4_7
  114. a_4_5·a_6_11 + a_2_0·a_4_5·c_4_8
  115. b_5_112 + b_2_25 + b_4_6·a_6_11 + b_2_23·a_4_5 + a_2_0·b_2_24 + a_2_0·b_2_22·a_4_5
       + b_2_23·c_4_7 + b_2_2·a_4_5·c_4_7 + a_2_0·b_4_6·c_4_8 + c_4_8·a_1_1·a_5_10
       + c_4_7·a_1_1·a_5_10 + a_2_0·a_4_5·c_4_7
  116. a_4_2·a_6_11 + a_2_0·a_4_5·c_4_7
  117. a_6_11·b_5_11 + b_2_23·a_5_10 + a_2_0·b_2_22·b_5_11 + b_2_22·c_4_7·a_3_4
       + a_2_0·c_4_8·b_5_11 + a_2_0·c_4_7·b_5_11 + a_2_0·b_2_2·c_4_8·b_3_3
       + a_2_0·b_2_2·c_4_7·b_3_3 + a_2_0·c_4_8·a_5_10 + a_2_0·c_4_7·a_5_10
       + a_2_0·b_2_2·c_4_8·a_3_4 + a_2_0·b_2_2·c_4_7·a_3_4
  118. a_6_11·a_5_10 + a_2_0·b_2_22·a_5_10 + a_2_0·c_4_7·a_5_10 + a_2_0·b_2_2·c_4_7·a_3_4
  119. a_6_112

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 12.
  • 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_4_7, a Duflot regular element of degree 4
    2. c_4_8, a Duflot regular element of degree 4
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • 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 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. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. a_3_1 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_3 → 0, an element of degree 3
  10. a_4_2 → 0, an element of degree 4
  11. a_4_3 → 0, an element of degree 4
  12. a_4_5 → 0, an element of degree 4
  13. b_4_6 → 0, an element of degree 4
  14. c_4_7 → c_1_14 + c_1_04, an element of degree 4
  15. c_4_8 → c_1_04, an element of degree 4
  16. a_5_10 → 0, an element of degree 5
  17. b_5_11 → 0, an element of degree 5
  18. a_6_11 → 0, an element of degree 6

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_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. a_3_1 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_3 → c_1_23, an element of degree 3
  10. a_4_2 → 0, an element of degree 4
  11. a_4_3 → 0, an element of degree 4
  12. a_4_5 → 0, an element of degree 4
  13. b_4_6 → c_1_24 + c_1_12·c_1_22 + c_1_02·c_1_22, an element of degree 4
  14. c_4_7 → c_1_24 + c_1_14 + c_1_04, an element of degree 4
  15. c_4_8 → c_1_12·c_1_22 + c_1_04, an element of degree 4
  16. a_5_10 → 0, an element of degree 5
  17. b_5_11 → c_1_12·c_1_23 + c_1_02·c_1_23, an element of degree 5
  18. a_6_11 → 0, an element of degree 6

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