Cohomology of group number 71 of order 128

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

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 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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
    ( − 1) · (t5  +  t3  +  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.

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

Ring generators

The cohomology ring has 20 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_2, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. b_3_2, an element of degree 3
  6. b_3_3, an element of degree 3
  7. b_3_4, an element of degree 3
  8. a_4_1, a nilpotent element of degree 4
  9. b_4_6, an element of degree 4
  10. a_5_1, a nilpotent element of degree 5
  11. b_5_5, an element of degree 5
  12. b_5_8, an element of degree 5
  13. b_5_9, an element of degree 5
  14. a_6_5, a nilpotent element of degree 6
  15. b_6_12, an element of degree 6
  16. b_7_13, an element of degree 7
  17. b_7_15, an element of degree 7
  18. a_8_4, a nilpotent element of degree 8
  19. c_8_19, a Duflot regular element of degree 8
  20. b_9_23, an element of degree 9

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

Ring relations

There are 152 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_2·a_1_0
  4. a_2_2·b_1_1
  5. b_2_1·b_1_1
  6. a_2_22
  7. a_1_0·b_3_2
  8. a_1_0·b_3_3 + a_2_2·b_2_1
  9. b_1_1·b_3_3 + b_1_1·b_3_2
  10. a_1_0·b_3_4 + a_2_2·b_2_1
  11. a_2_2·b_3_2
  12. b_2_1·b_3_2
  13. a_2_2·b_3_3
  14. a_2_2·b_3_4
  15. b_2_1·b_3_4 + b_2_1·b_3_3
  16. a_4_1·a_1_0
  17. a_4_1·b_1_1
  18. b_4_6·a_1_0
  19. b_3_2·b_3_3 + b_3_22
  20. a_2_2·a_4_1
  21. b_3_3·b_3_4 + b_3_32 + b_3_2·b_3_4 + b_3_22 + a_2_2·b_4_6
  22. b_3_32 + b_3_22 + b_2_1·b_4_6 + b_2_1·a_4_1
  23. b_3_22 + b_4_6·b_1_12
  24. b_2_1·a_4_1 + a_1_0·a_5_1
  25. b_1_1·a_5_1
  26. a_1_0·b_5_5
  27. b_3_22 + b_1_1·b_5_5 + b_1_13·b_3_2
  28. a_1_0·b_5_8 + b_2_1·a_4_1
  29. b_3_42 + b_3_32 + b_3_22 + b_1_1·b_5_8 + b_1_13·b_3_4 + b_1_13·b_3_2
  30. a_1_0·b_5_9 + a_2_2·b_2_12
  31. b_3_42 + b_3_32 + b_3_2·b_3_4 + b_1_1·b_5_9 + b_1_13·b_3_4
  32. a_4_1·b_3_2
  33. a_4_1·b_3_4 + a_4_1·b_3_3
  34. a_4_1·b_3_3 + a_2_2·a_5_1
  35. a_2_2·b_5_5
  36. a_4_1·b_3_3 + a_2_2·b_5_8
  37. b_2_1·b_5_8 + b_2_1·b_5_5 + b_2_1·a_5_1
  38. a_2_2·b_5_9
  39. b_2_1·b_5_9 + b_2_12·b_3_3 + a_4_1·b_3_3
  40. a_4_1·b_3_3 + a_6_5·a_1_0
  41. a_6_5·b_1_1
  42. b_6_12·a_1_0
  43. b_1_14·b_3_4 + b_6_12·b_1_1 + b_4_6·b_3_2 + b_4_6·b_1_13
  44. a_4_12
  45. b_3_2·a_5_1
  46. b_3_4·a_5_1 + b_3_3·a_5_1
  47. b_3_2·b_5_5 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14
  48. b_3_3·b_5_8 + b_3_3·b_5_5 + b_3_2·b_5_8 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14 + b_3_3·a_5_1
       + a_4_1·b_4_6
  49. b_3_4·b_5_5 + b_3_3·b_5_5 + b_1_13·b_5_9 + b_1_13·b_5_8 + b_1_15·b_3_2
       + b_4_6·b_1_1·b_3_4 + b_4_6·b_1_1·b_3_2 + a_4_1·b_4_6
  50. b_3_2·b_5_9 + b_3_2·b_5_8 + b_4_6·b_1_1·b_3_4 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14
  51. b_3_3·b_5_9 + b_3_2·b_5_8 + b_4_6·b_1_1·b_3_4 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14
       + b_2_12·b_4_6 + b_2_1·a_1_0·a_5_1
  52. b_3_4·b_5_9 + b_3_4·b_5_8 + b_3_3·b_5_5 + b_3_2·b_5_8 + b_4_6·b_1_1·b_3_4
       + b_4_6·b_1_1·b_3_2 + b_2_12·b_4_6 + b_3_3·a_5_1 + b_2_1·a_1_0·a_5_1
  53. a_2_2·a_6_5
  54. b_3_3·a_5_1 + b_2_1·a_6_5 + a_2_2·b_2_13 + b_2_1·a_1_0·a_5_1
  55. a_4_1·b_4_6 + a_2_2·b_6_12
  56. b_3_3·b_5_5 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14 + b_2_1·b_6_12 + b_2_12·b_4_6
  57. a_1_0·b_7_13 + a_2_2·b_2_13 + b_2_1·a_1_0·a_5_1
  58. b_3_4·b_5_5 + b_3_3·b_5_5 + b_3_2·b_5_8 + b_1_1·b_7_13 + b_6_12·b_1_12
       + b_4_6·b_1_1·b_3_4 + b_4_6·b_1_1·b_3_2 + b_4_6·b_1_14 + a_4_1·b_4_6
  59. a_1_0·b_7_15 + b_2_1·a_1_0·a_5_1
  60. b_3_4·b_5_8 + b_3_4·b_5_5 + b_1_1·b_7_15 + b_1_13·b_5_8 + b_4_6·b_1_1·b_3_4
       + b_4_6·b_1_1·b_3_2 + b_3_3·a_5_1 + a_4_1·b_4_6
  61. a_4_1·a_5_1
  62. a_4_1·b_5_5
  63. a_4_1·b_5_8
  64. a_4_1·b_5_9 + a_2_2·b_2_1·a_5_1
  65. a_6_5·b_3_2
  66. a_6_5·b_3_3 + b_4_6·a_5_1 + a_2_2·b_2_1·a_5_1
  67. a_6_5·b_3_4 + b_4_6·a_5_1 + a_2_2·b_2_1·a_5_1
  68. b_1_14·b_5_9 + b_1_14·b_5_8 + b_1_16·b_3_2 + b_6_12·b_3_2 + b_4_6·b_1_12·b_3_2
       + b_4_6·b_1_15 + b_4_62·b_1_1
  69. b_1_14·b_5_9 + b_1_14·b_5_8 + b_1_16·b_3_2 + b_6_12·b_3_3 + b_4_6·b_5_5
       + b_4_6·b_1_15 + b_2_1·b_4_6·b_3_3
  70. b_1_14·b_5_8 + b_1_16·b_3_2 + b_6_12·b_3_4 + b_6_12·b_1_13 + b_4_6·b_5_9 + b_4_6·b_5_8
       + b_4_6·b_1_12·b_3_4 + b_4_6·b_1_15 + b_4_62·b_1_1 + b_4_6·a_5_1
  71. a_2_2·b_7_13 + a_2_2·b_2_1·a_5_1
  72. b_2_1·b_7_13 + b_2_12·b_5_5 + b_2_13·b_3_3 + b_2_12·a_5_1 + a_2_2·b_2_1·a_5_1
  73. a_2_2·b_7_15 + a_2_2·b_2_1·a_5_1
  74. b_2_1·b_7_15 + b_2_1·b_4_6·b_3_3 + b_2_12·b_5_5 + b_4_6·a_5_1 + b_2_12·a_5_1
  75. a_8_4·a_1_0 + a_2_2·b_2_1·a_5_1
  76. a_8_4·b_1_1
  77. a_2_2·b_2_14 + a_5_12 + b_2_12·a_1_0·a_5_1
  78. b_5_52 + b_4_6·b_1_16 + b_4_62·b_1_12 + b_2_1·b_4_62
  79. a_5_1·b_5_8 + a_5_1·b_5_5 + a_5_12
  80. b_5_92 + b_5_82 + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_4 + b_4_6·b_1_13·b_3_2
       + b_4_6·b_1_16 + b_4_62·b_1_12 + b_2_1·b_4_62 + b_2_13·b_4_6 + a_2_2·b_2_14
  81. a_5_1·b_5_9 + b_2_12·a_6_5 + a_5_12
  82. a_4_1·a_6_5
  83. b_5_5·b_5_9 + b_5_5·b_5_8 + b_4_6·b_1_1·b_5_9 + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_4
       + b_4_6·b_1_13·b_3_2 + b_4_6·b_1_16 + b_2_1·b_4_62 + b_2_12·b_6_12 + b_2_13·b_4_6
       + a_5_1·b_5_5 + a_2_2·b_4_62
  84. a_4_1·b_6_12 + a_2_2·b_4_62
  85. b_5_5·b_5_8 + b_1_13·b_7_13 + b_6_12·b_1_1·b_3_2 + b_6_12·b_1_14 + b_4_6·b_1_1·b_5_8
       + b_4_6·b_1_13·b_3_2 + b_4_62·b_1_12 + b_2_1·b_4_62 + a_5_1·b_5_5 + a_2_2·b_4_62
  86. b_3_2·b_7_13 + b_6_12·b_1_1·b_3_2 + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_4
  87. b_5_5·b_5_9 + b_5_5·b_5_8 + b_3_3·b_7_13 + b_6_12·b_1_1·b_3_2 + b_4_6·b_1_1·b_5_9
       + b_4_6·b_1_13·b_3_2 + b_4_6·b_1_16 + b_2_1·b_4_62 + b_2_13·b_4_6 + a_5_1·b_5_9
       + a_5_1·b_5_5 + a_2_2·b_2_14 + a_5_12
  88. b_5_8·b_5_9 + b_5_82 + b_3_4·b_7_13 + b_6_12·b_1_1·b_3_4 + b_6_12·b_1_1·b_3_2
       + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_2 + b_4_6·b_1_16 + b_4_62·b_1_12
       + b_2_1·b_4_62 + b_2_13·b_4_6 + a_2_2·b_4_62 + a_2_2·b_2_14
  89. b_5_8·b_5_9 + b_5_82 + b_5_5·b_5_9 + b_5_5·b_5_8 + b_3_2·b_7_15 + b_4_6·b_1_1·b_5_9
       + b_4_6·b_1_13·b_3_2 + b_4_6·b_1_16 + b_4_62·b_1_12 + a_5_1·b_5_9 + a_5_1·b_5_5
       + a_2_2·b_4_62 + a_5_12
  90. b_5_8·b_5_9 + b_5_82 + b_3_3·b_7_15 + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_4
       + b_4_62·b_1_12 + b_4_6·a_6_5 + a_5_12
  91. b_5_82 + b_5_5·b_5_9 + b_5_5·b_5_8 + b_3_4·b_7_15 + b_6_12·b_1_1·b_3_2
       + b_4_6·b_1_13·b_3_4 + b_4_6·b_1_13·b_3_2 + b_2_1·b_4_62 + a_5_1·b_5_9 + a_5_1·b_5_5
       + b_4_6·a_6_5 + a_5_12
  92. b_5_82 + b_5_5·b_5_8 + b_1_13·b_7_15 + b_6_12·b_1_1·b_3_4 + b_6_12·b_1_1·b_3_2
       + b_6_12·b_1_14 + b_4_6·b_1_1·b_5_9 + b_4_6·b_1_1·b_5_8 + b_4_6·b_1_13·b_3_4
       + b_4_62·b_1_12 + a_5_1·b_5_5 + a_2_2·b_4_62 + a_5_12 + c_8_19·b_1_12
  93. a_2_2·a_8_4
  94. a_5_1·b_5_9 + a_5_1·b_5_5 + b_2_1·a_8_4 + a_2_2·b_2_14
  95. a_1_0·b_9_23 + a_5_12
  96. b_5_8·b_5_9 + b_5_82 + b_5_5·b_5_9 + b_1_1·b_9_23 + b_6_12·b_1_1·b_3_2
       + b_4_6·b_1_1·b_5_9 + b_4_6·b_1_13·b_3_4 + b_2_1·b_4_62 + a_5_1·b_5_9 + a_5_12
  97. a_6_5·a_5_1
  98. a_6_5·b_5_9 + b_2_1·b_4_6·a_5_1 + a_2_2·b_2_12·a_5_1
  99. a_6_5·b_5_8 + a_6_5·b_5_5
  100. b_6_12·a_5_1 + a_6_5·b_5_8 + b_2_1·b_4_6·a_5_1
  101. b_6_12·b_5_5 + b_6_12·b_1_12·b_3_2 + b_4_6·b_6_12·b_1_1 + b_4_62·b_3_3 + b_4_62·b_3_2
       + b_2_1·b_4_6·b_5_5
  102. b_1_14·b_7_13 + b_6_12·b_5_9 + b_6_12·b_5_8 + b_6_12·b_1_12·b_3_2 + b_6_12·b_1_15
       + b_4_6·b_1_12·b_5_9 + b_4_6·b_1_12·b_5_8 + b_4_6·b_1_14·b_3_2 + b_4_6·b_1_17
       + b_4_6·b_6_12·b_1_1 + b_4_62·b_3_4 + b_4_62·b_1_13 + b_2_12·b_4_6·b_3_3
       + a_6_5·b_5_8 + b_2_1·b_4_6·a_5_1
  103. a_4_1·b_7_13 + a_2_2·b_2_12·a_5_1
  104. b_1_14·b_7_15 + b_1_18·b_3_2 + b_6_12·b_5_8 + b_6_12·b_1_12·b_3_4
       + b_6_12·b_1_12·b_3_2 + b_6_12·b_1_15 + b_4_6·b_7_13 + b_4_6·b_1_12·b_5_8
       + b_4_6·b_1_14·b_3_2 + b_4_6·b_1_17 + b_4_62·b_3_3 + b_2_12·b_4_6·b_3_3
       + a_6_5·b_5_8
  105. a_4_1·b_7_15
  106. a_8_4·b_3_2
  107. a_8_4·b_3_3 + a_6_5·b_5_8 + b_2_1·b_4_6·a_5_1
  108. a_8_4·b_3_4 + a_6_5·b_5_8 + b_2_1·b_4_6·a_5_1
  109. a_2_2·b_9_23 + a_2_2·b_2_12·a_5_1
  110. b_2_1·b_9_23 + b_2_1·b_4_6·b_5_5 + b_2_14·b_3_3 + a_6_5·b_5_8 + b_2_1·b_4_6·a_5_1
       + b_2_13·a_5_1 + b_2_1·c_8_19·a_1_0
  111. a_6_52
  112. b_6_12·b_1_13·b_3_4 + b_6_122 + b_4_6·b_1_13·b_5_9 + b_4_6·b_1_13·b_5_8
       + b_4_6·b_1_15·b_3_2 + b_4_6·b_6_12·b_1_12 + b_4_62·b_1_1·b_3_2 + b_4_62·b_1_14
       + b_4_63 + b_2_12·b_4_62
  113. b_5_5·b_7_13 + b_6_12·b_1_13·b_3_4 + b_6_12·b_1_13·b_3_2 + b_6_122
       + b_4_6·b_1_1·b_7_13 + b_4_6·b_1_13·b_5_9 + b_4_6·b_1_15·b_3_2 + b_4_63
       + b_2_13·b_6_12 + b_2_14·b_4_6 + a_5_1·b_7_13 + b_2_13·a_6_5 + a_2_2·b_4_6·b_6_12
  114. a_5_1·b_7_15 + a_5_1·b_7_13 + b_2_1·b_4_6·a_6_5 + b_2_13·a_6_5 + b_2_1·a_5_12
  115. b_5_9·b_7_13 + b_5_8·b_7_13 + b_6_12·b_1_1·b_5_9 + b_6_12·b_1_1·b_5_8
       + b_6_12·b_1_13·b_3_4 + b_6_122 + b_4_6·b_1_1·b_7_15 + b_4_6·b_1_1·b_7_13
       + b_4_6·b_1_13·b_5_8 + b_4_6·b_1_15·b_3_2 + b_4_63 + b_2_14·b_4_6
       + a_2_2·b_4_6·b_6_12 + b_2_1·a_5_12 + b_2_13·a_1_0·a_5_1
  116. b_5_9·b_7_15 + b_5_9·b_7_13 + b_5_8·b_7_15 + b_6_12·b_1_1·b_5_8 + b_4_6·b_1_1·b_7_13
       + b_4_6·b_1_13·b_5_8 + b_4_6·b_1_15·b_3_2 + b_4_6·b_1_18 + b_4_62·b_1_1·b_3_2
       + b_2_1·b_4_6·b_6_12 + b_2_12·b_4_62 + b_2_14·b_4_6 + a_6_5·b_6_12
       + b_2_1·b_4_6·a_6_5 + a_2_2·b_4_6·b_6_12 + b_2_1·a_5_12 + b_2_13·a_1_0·a_5_1
  117. b_5_8·b_7_13 + b_6_12·b_1_1·b_5_8 + b_6_12·b_1_13·b_3_4 + b_6_122
       + b_4_6·b_1_1·b_7_13 + b_4_6·b_1_13·b_5_8 + b_4_6·b_1_15·b_3_2 + b_4_6·b_1_18
       + b_4_63 + b_2_13·b_6_12 + b_2_14·b_4_6 + b_2_13·a_6_5 + a_2_2·b_4_6·b_6_12
       + c_8_19·b_1_1·b_3_2
  118. b_5_9·b_7_13 + b_5_8·b_7_15 + b_5_8·b_7_13 + b_6_12·b_1_1·b_5_8 + b_6_12·b_1_13·b_3_4
       + b_6_12·b_1_13·b_3_2 + b_6_122 + b_4_6·b_1_1·b_7_13 + b_4_6·b_1_13·b_5_8
       + b_4_6·b_1_15·b_3_2 + b_4_6·b_1_18 + b_4_62·b_1_14 + b_4_63 + b_2_1·b_4_6·b_6_12
       + b_2_14·b_4_6 + a_6_5·b_6_12 + b_2_13·a_1_0·a_5_1 + c_8_19·b_1_1·b_3_4
  119. a_5_1·b_7_13 + b_2_12·a_8_4 + b_2_13·a_1_0·a_5_1
  120. a_4_1·a_8_4
  121. a_6_5·b_6_12 + b_4_6·a_8_4 + a_2_2·b_4_6·b_6_12
  122. b_5_9·b_7_13 + b_5_8·b_7_13 + b_5_5·b_7_15 + b_1_13·b_9_23 + b_6_12·b_1_13·b_3_4
       + b_6_12·b_1_13·b_3_2 + b_6_122 + b_4_6·b_1_1·b_7_13 + b_4_6·b_1_13·b_5_9
       + b_4_6·b_1_13·b_5_8 + b_4_62·b_1_1·b_3_4 + b_4_62·b_1_1·b_3_2 + b_4_63
       + b_2_1·b_4_6·b_6_12 + b_2_14·b_4_6 + a_5_1·b_7_13 + a_6_5·b_6_12 + b_2_1·b_4_6·a_6_5
       + b_2_13·a_6_5 + a_2_2·b_4_6·b_6_12 + b_2_1·a_5_12 + b_2_13·a_1_0·a_5_1
  123. b_5_9·b_7_13 + b_5_8·b_7_13 + b_3_2·b_9_23 + b_6_12·b_1_1·b_5_9 + b_6_12·b_1_1·b_5_8
       + b_6_12·b_1_13·b_3_4 + b_6_122 + b_4_62·b_1_1·b_3_2 + b_4_62·b_1_14 + b_4_63
       + b_2_14·b_4_6 + a_2_2·b_4_6·b_6_12 + b_2_1·a_5_12 + b_2_13·a_1_0·a_5_1
  124. b_5_9·b_7_13 + b_5_8·b_7_13 + b_3_3·b_9_23 + b_6_12·b_1_1·b_5_9 + b_6_12·b_1_1·b_5_8
       + b_6_12·b_1_13·b_3_4 + b_6_122 + b_4_62·b_1_1·b_3_2 + b_4_62·b_1_14 + b_4_63
       + b_2_1·b_4_6·b_6_12 + b_2_12·b_4_62 + a_6_5·b_6_12 + b_2_13·a_6_5
       + a_2_2·b_2_1·c_8_19
  125. b_5_9·b_7_13 + b_3_4·b_9_23 + b_6_12·b_1_1·b_5_8 + b_6_12·b_1_13·b_3_4
       + b_6_12·b_1_13·b_3_2 + b_6_122 + b_4_6·b_1_1·b_7_13 + b_4_6·b_1_13·b_5_8
       + b_4_6·b_1_15·b_3_2 + b_4_62·b_1_14 + b_4_63 + b_2_1·b_4_6·b_6_12 + b_2_13·b_6_12
       + b_2_14·b_4_6 + a_6_5·b_6_12 + a_2_2·b_2_1·c_8_19
  126. a_6_5·b_7_13 + b_2_1·a_6_5·b_5_5 + b_2_12·b_4_6·a_5_1 + a_2_2·b_2_13·a_5_1
  127. a_6_5·b_7_15 + b_4_62·a_5_1 + b_2_1·a_6_5·b_5_5
  128. a_8_4·a_5_1
  129. a_8_4·b_5_8 + b_4_62·a_5_1 + b_2_1·a_6_5·b_5_5
  130. a_8_4·b_5_9 + b_2_1·a_6_5·b_5_5 + b_2_12·b_4_6·a_5_1
  131. a_8_4·b_5_5 + b_4_62·a_5_1 + b_2_1·a_6_5·b_5_5
  132. b_1_14·b_9_23 + b_6_12·b_7_13 + b_6_12·b_1_12·b_5_9 + b_6_12·b_1_12·b_5_8
       + b_6_122·b_1_1 + b_4_6·b_1_12·b_7_13 + b_4_6·b_1_19 + b_4_6·b_6_12·b_3_4
       + b_4_6·b_6_12·b_3_2 + b_4_6·b_6_12·b_1_13 + b_4_62·b_5_9 + b_4_62·b_5_5
       + b_4_62·b_1_12·b_3_4 + b_4_62·b_1_15 + b_4_63·b_1_1 + b_2_1·b_4_62·b_3_3
       + b_2_13·b_4_6·b_3_3 + b_2_1·a_6_5·b_5_5 + b_2_12·b_4_6·a_5_1
  133. a_4_1·b_9_23 + a_2_2·b_2_13·a_5_1
  134. b_1_110·b_3_2 + b_6_12·b_7_15 + b_6_12·b_1_12·b_5_9 + b_6_12·b_1_14·b_3_2
       + b_4_6·b_9_23 + b_4_6·b_1_12·b_7_15 + b_4_6·b_1_19 + b_4_6·b_6_12·b_3_2
       + b_4_6·b_6_12·b_1_13 + b_4_62·b_5_8 + b_4_62·b_5_5 + b_2_12·b_4_6·b_5_5
       + b_2_13·b_4_6·b_3_3 + b_4_62·a_5_1 + b_2_1·a_6_5·b_5_5 + c_8_19·b_1_15
  135. b_7_132 + b_6_122·b_1_12 + b_4_6·b_1_13·b_7_15 + b_4_6·b_1_13·b_7_13
       + b_4_62·b_1_1·b_5_8 + b_4_63·b_1_12 + b_2_13·b_4_62 + b_2_15·b_4_6
       + b_2_12·a_5_12 + b_2_14·a_1_0·a_5_1 + b_4_6·c_8_19·b_1_12
  136. b_6_12·a_8_4 + b_4_62·a_6_5 + b_2_12·b_4_6·a_6_5 + a_2_2·b_4_63
  137. b_7_152 + b_7_13·b_7_15 + b_6_12·b_1_13·b_5_9 + b_6_12·b_1_13·b_5_8
       + b_6_12·b_1_15·b_3_2 + b_4_6·b_1_13·b_7_13 + b_4_6·b_1_17·b_3_2 + b_4_6·b_1_110
       + b_4_6·b_6_12·b_1_1·b_3_4 + b_4_6·b_6_12·b_1_1·b_3_2 + b_4_62·b_1_1·b_5_8
       + b_4_62·b_1_13·b_3_2 + b_4_62·b_1_16 + b_2_1·b_4_63 + b_2_12·b_4_6·b_6_12
       + b_2_14·b_6_12 + b_2_15·b_4_6 + b_2_1·b_4_6·a_8_4 + b_2_12·b_4_6·a_6_5
       + b_2_14·a_6_5 + a_2_2·b_4_63 + b_2_12·a_5_12 + c_8_19·b_1_1·b_5_9
  138. a_6_5·a_8_4
  139. b_7_152 + b_7_13·b_7_15 + b_6_12·b_1_13·b_5_9 + b_6_12·b_1_13·b_5_8
       + b_6_12·b_1_15·b_3_2 + b_4_6·b_1_13·b_7_13 + b_4_6·b_1_17·b_3_2 + b_4_6·b_1_110
       + b_4_6·b_6_12·b_1_1·b_3_4 + b_4_6·b_6_12·b_1_1·b_3_2 + b_4_62·b_1_1·b_5_8
       + b_4_62·b_1_13·b_3_2 + b_4_62·b_1_16 + b_2_1·b_4_63 + b_2_12·b_4_6·b_6_12
       + b_2_14·b_6_12 + b_2_15·b_4_6 + a_5_1·b_9_23 + a_2_2·b_4_63 + b_2_12·a_5_12
       + c_8_19·b_1_1·b_5_9 + c_8_19·a_1_0·a_5_1
  140. b_7_152 + b_7_132 + b_5_8·b_9_23 + b_6_12·b_1_1·b_7_15 + b_6_12·b_1_1·b_7_13
       + b_4_6·b_1_13·b_7_15 + b_4_6·b_1_110 + b_4_6·b_6_12·b_1_1·b_3_2
       + b_4_6·b_6_12·b_1_14 + b_4_62·b_1_1·b_5_9 + b_4_62·b_1_1·b_5_8 + b_4_63·b_1_12
       + b_2_14·b_6_12 + b_4_62·a_6_5 + b_2_13·a_8_4 + c_8_19·b_1_1·b_5_9
       + c_8_19·a_1_0·a_5_1
  141. b_7_152 + b_7_132 + b_6_12·b_1_1·b_7_15 + b_6_12·b_1_13·b_5_9
       + b_6_12·b_1_13·b_5_8 + b_6_12·b_1_15·b_3_2 + b_6_122·b_1_12 + b_4_6·b_1_1·b_9_23
       + b_4_6·b_1_13·b_7_15 + b_4_6·b_1_13·b_7_13 + b_4_6·b_1_17·b_3_2
       + b_4_6·b_6_12·b_1_1·b_3_4 + b_4_6·b_6_12·b_1_14 + b_4_62·b_1_1·b_5_9
       + b_4_62·b_1_1·b_5_8 + b_4_62·b_1_13·b_3_4 + b_2_1·b_4_63 + b_2_15·b_4_6
       + b_2_14·a_1_0·a_5_1 + c_8_19·b_1_1·b_5_8
  142. b_7_13·b_7_15 + b_5_9·b_9_23 + b_6_12·b_1_1·b_7_15 + b_6_12·b_1_1·b_7_13
       + b_6_12·b_1_13·b_5_9 + b_6_12·b_1_13·b_5_8 + b_6_12·b_1_15·b_3_2
       + b_6_122·b_1_12 + b_4_6·b_1_13·b_7_15 + b_4_6·b_1_13·b_7_13
       + b_4_6·b_6_12·b_1_1·b_3_4 + b_4_62·b_1_16 + b_4_63·b_1_12 + b_2_14·b_6_12
       + b_2_12·b_4_6·a_6_5 + a_2_2·b_4_63 + b_2_12·a_5_12 + b_2_14·a_1_0·a_5_1
       + a_2_2·b_2_12·c_8_19
  143. b_7_13·b_7_15 + b_7_132 + b_5_5·b_9_23 + b_6_12·b_1_1·b_7_15 + b_6_122·b_1_12
       + b_4_6·b_1_13·b_7_13 + b_4_6·b_1_110 + b_4_6·b_6_12·b_1_1·b_3_4
       + b_4_6·b_6_12·b_1_1·b_3_2 + b_4_62·b_1_1·b_5_8 + b_4_62·b_1_16 + b_4_63·b_1_12
       + b_2_1·b_4_63 + b_2_12·b_4_6·b_6_12 + b_2_15·b_4_6 + b_4_62·a_6_5 + b_2_13·a_8_4
       + b_2_12·a_5_12 + c_8_19·b_1_1·b_5_9 + c_8_19·b_1_1·b_5_8
  144. a_8_4·b_7_15 + b_4_6·a_6_5·b_5_5 + b_2_12·a_6_5·b_5_5
  145. a_8_4·b_7_13 + b_2_1·b_4_62·a_5_1 + b_2_13·b_4_6·a_5_1
  146. b_6_12·b_9_23 + b_6_122·b_3_2 + b_4_6·b_1_12·b_9_23 + b_4_6·b_1_18·b_3_2
       + b_4_6·b_1_111 + b_4_6·b_6_12·b_5_9 + b_4_6·b_6_12·b_1_15 + b_4_62·b_7_15
       + b_4_62·b_1_14·b_3_2 + b_4_62·b_6_12·b_1_1 + b_4_63·b_3_4 + b_4_63·b_3_3
       + b_4_63·b_1_13 + b_2_1·b_4_62·b_5_5 + b_2_12·b_4_62·b_3_3 + b_2_13·b_4_6·b_5_5
       + b_2_14·b_4_6·b_3_3 + b_2_12·a_6_5·b_5_5 + b_2_13·b_4_6·a_5_1
       + c_8_19·b_1_14·b_3_2
  147. a_6_5·b_9_23 + b_4_6·a_6_5·b_5_5 + b_2_13·b_4_6·a_5_1 + a_5_13 + a_2_2·c_8_19·a_5_1
  148. a_8_42
  149. b_7_15·b_9_23 + b_6_12·b_1_13·b_7_13 + b_6_122·b_1_1·b_3_2 + b_6_122·b_1_14
       + b_4_6·b_1_19·b_3_2 + b_4_6·b_6_12·b_1_1·b_5_9 + b_4_6·b_6_12·b_1_16
       + b_4_6·b_6_122 + b_4_62·b_1_1·b_7_15 + b_4_62·b_1_15·b_3_2 + b_4_63·b_1_1·b_3_2
       + b_4_64 + b_2_1·b_4_62·b_6_12 + b_2_12·b_4_63 + b_2_14·b_4_62 + b_2_15·b_6_12
       + b_2_16·b_4_6 + b_2_14·a_8_4 + a_2_2·b_4_62·b_6_12 + b_2_15·a_1_0·a_5_1
       + c_8_19·b_1_1·b_7_13 + b_6_12·c_8_19·b_1_12 + b_4_6·c_8_19·b_1_1·b_3_4
       + b_4_6·c_8_19·b_1_1·b_3_2 + b_4_6·c_8_19·b_1_14 + b_2_1·c_8_19·a_1_0·a_5_1
  150. b_7_13·b_9_23 + b_6_122·b_1_1·b_3_2 + b_4_6·b_1_19·b_3_2 + b_4_6·b_1_112
       + b_4_6·b_6_12·b_1_13·b_3_2 + b_4_6·b_6_12·b_1_16 + b_4_6·b_6_122
       + b_4_62·b_1_1·b_7_13 + b_4_62·b_1_18 + b_4_63·b_1_1·b_3_4 + b_4_64
       + b_2_13·b_4_6·b_6_12 + b_2_14·b_4_62 + b_2_15·b_6_12 + b_2_1·b_4_62·a_6_5
       + b_2_12·b_4_6·a_8_4 + b_2_14·a_8_4 + b_2_15·a_6_5 + a_2_2·b_4_62·b_6_12
       + b_2_13·a_5_12 + c_8_19·b_1_15·b_3_2 + b_4_6·c_8_19·b_1_1·b_3_4
       + b_4_6·c_8_19·b_1_1·b_3_2 + a_2_2·b_2_13·c_8_19 + b_2_1·c_8_19·a_1_0·a_5_1
  151. a_8_4·b_9_23 + b_4_63·a_5_1 + b_2_1·b_4_6·a_6_5·b_5_5 + b_2_13·a_6_5·b_5_5
       + b_2_14·b_4_6·a_5_1 + a_2_2·b_2_1·c_8_19·a_5_1
  152. b_9_232 + b_4_6·b_6_12·b_1_13·b_5_9 + b_4_6·b_6_12·b_1_13·b_5_8
       + b_4_6·b_6_122·b_1_12 + b_4_62·b_6_12·b_1_1·b_3_4 + b_4_62·b_6_12·b_1_1·b_3_2
       + b_4_63·b_1_1·b_5_9 + b_4_63·b_1_13·b_3_2 + b_4_63·b_1_16 + b_2_1·b_4_64
       + b_2_17·b_4_6 + b_2_14·a_5_12 + b_2_16·a_1_0·a_5_1 + b_4_6·c_8_19·b_1_1·b_5_8

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 18.
  • 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_19, a Duflot regular element of degree 8
    2. b_1_14 + b_4_6 + 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. b_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. b_3_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. a_4_1 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. a_5_1 → 0, an element of degree 5
  11. b_5_5 → 0, an element of degree 5
  12. b_5_8 → 0, an element of degree 5
  13. b_5_9 → 0, an element of degree 5
  14. a_6_5 → 0, an element of degree 6
  15. b_6_12 → 0, an element of degree 6
  16. b_7_13 → 0, an element of degree 7
  17. b_7_15 → 0, an element of degree 7
  18. a_8_4 → 0, an element of degree 8
  19. c_8_19 → c_1_08, an element of degree 8
  20. b_9_23 → 0, an element of degree 9

Restriction map to a maximal el. ab. subgp. of rank 3

  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_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_3_2 → c_1_1·c_1_22 + c_1_12·c_1_2, 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. b_3_4 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  8. a_4_1 → 0, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  10. a_5_1 → 0, an element of degree 5
  11. b_5_5 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  12. b_5_8 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_04·c_1_1, an element of degree 5
  13. b_5_9 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
       + c_1_0·c_1_14 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
  14. a_6_5 → 0, an element of degree 6
  15. b_6_12 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_0·c_1_15
       + c_1_02·c_1_14, an element of degree 6
  16. b_7_13 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23 + c_1_0·c_1_16
       + c_1_02·c_1_13·c_1_22 + c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15
       + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  17. b_7_15 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23
       + c_1_15·c_1_22 + c_1_16·c_1_2 + c_1_0·c_1_16 + c_1_02·c_1_15 + c_1_03·c_1_14
       + c_1_04·c_1_13 + c_1_05·c_1_12 + c_1_06·c_1_1, an element of degree 7
  18. a_8_4 → 0, an element of degree 8
  19. c_8_19 → c_1_28 + 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_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_03·c_1_15
       + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_05·c_1_13 + c_1_06·c_1_12
       + c_1_08, an element of degree 8
  20. b_9_23 → c_1_1·c_1_28 + c_1_13·c_1_26 + c_1_14·c_1_25 + c_1_15·c_1_24
       + c_1_16·c_1_23 + c_1_17·c_1_22 + c_1_0·c_1_16·c_1_22 + c_1_0·c_1_17·c_1_2
       + c_1_02·c_1_13·c_1_24 + c_1_02·c_1_15·c_1_22 + c_1_03·c_1_14·c_1_22
       + c_1_03·c_1_15·c_1_2 + c_1_04·c_1_1·c_1_24 + c_1_04·c_1_13·c_1_22
       + c_1_05·c_1_12·c_1_22 + c_1_05·c_1_13·c_1_2 + c_1_06·c_1_1·c_1_22
       + c_1_06·c_1_12·c_1_2, an element of degree 9

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. b_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. b_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. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. a_4_1 → 0, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  10. a_5_1 → 0, an element of degree 5
  11. b_5_5 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  12. b_5_8 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  13. b_5_9 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  14. a_6_5 → 0, an element of degree 6
  15. b_6_12 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_14·c_1_22, an element of degree 6
  16. b_7_13 → c_1_13·c_1_24 + c_1_16·c_1_2, an element of degree 7
  17. b_7_15 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_15·c_1_22, an element of degree 7
  18. a_8_4 → 0, an element of degree 8
  19. c_8_19 → c_1_28 + 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_16·c_1_22 + 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
  20. b_9_23 → c_1_1·c_1_28 + c_1_15·c_1_24 + c_1_17·c_1_22 + c_1_18·c_1_2, an element of degree 9

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