Cohomology of group number 834 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 3 minimal generators and exponent 8.
  • 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t7  −  2·t6  +  t5  −  t4  +  t2  −  t  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-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 10:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. b_2_3, an element of degree 2
  5. a_3_3, a nilpotent element of degree 3
  6. a_3_4, a nilpotent element of degree 3
  7. a_4_3, a nilpotent element of degree 4
  8. b_4_4, an element of degree 4
  9. b_4_6, an element of degree 4
  10. b_5_7, an element of degree 5
  11. a_6_3, a nilpotent element of degree 6
  12. a_6_5, a nilpotent element of degree 6
  13. a_6_6, a nilpotent element of degree 6
  14. b_6_10, an element of degree 6
  15. a_7_11, a nilpotent element of degree 7
  16. b_7_12, an element of degree 7
  17. a_8_10, a nilpotent element of degree 8
  18. c_8_15, a Duflot regular element of degree 8
  19. a_9_16, a nilpotent element of degree 9
  20. a_10_14, a nilpotent element of degree 10

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

Ring relations

There are 150 minimal relations of maximal degree 20:

  1. a_1_1·a_1_2
  2. a_1_12 + a_1_0·a_1_2
  3. a_1_22 + a_1_12 + a_1_0·a_1_1 + a_1_02
  4. a_1_02·a_1_1
  5. b_2_3·a_1_0·a_1_1 + b_2_3·a_1_02
  6. a_1_1·a_3_3 + b_2_3·a_1_0·a_1_2 + b_2_3·a_1_0·a_1_1
  7. a_1_2·a_3_3 + b_2_3·a_1_0·a_1_1
  8. a_1_1·a_3_4
  9. a_1_0·a_3_4 + a_1_0·a_3_3 + b_2_3·a_1_0·a_1_1
  10. a_1_2·a_3_4 + a_1_0·a_3_3 + b_2_3·a_1_0·a_1_1
  11. a_4_3·a_1_1
  12. b_2_3·a_3_4 + b_2_32·a_1_2 + b_2_32·a_1_1 + a_4_3·a_1_0
  13. b_2_3·a_3_4 + b_2_32·a_1_2 + b_2_32·a_1_1 + a_4_3·a_1_2
  14. b_4_4·a_1_1 + b_2_32·a_1_2 + b_2_32·a_1_0
  15. b_4_4·a_1_0 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_2 + b_2_32·a_1_1
  16. b_4_4·a_1_2 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_1 + b_2_32·a_1_0
  17. b_4_6·a_1_1 + b_2_32·a_1_1
  18. a_3_3·a_3_4 + a_3_32
  19. a_3_42
  20. a_3_32 + b_4_6·a_1_02
  21. a_1_1·b_5_7
  22. a_4_3·a_3_4 + b_2_3·a_4_3·a_1_0
  23. b_4_4·a_3_3 + b_2_3·b_4_6·a_1_0 + b_2_32·a_3_3 + b_2_33·a_1_0 + a_4_3·a_3_3
       + b_2_3·a_4_3·a_1_0
  24. b_4_4·a_3_4 + b_4_4·a_3_3 + b_2_3·b_4_6·a_1_2 + b_2_33·a_1_2
  25. b_4_4·a_3_4 + b_2_32·a_3_3 + a_1_02·b_5_7 + a_4_3·a_3_3 + b_2_3·a_4_3·a_1_0
  26. b_4_4·a_3_4 + b_2_32·a_3_3 + a_6_3·a_1_1 + b_2_3·a_4_3·a_1_0
  27. b_4_4·a_3_4 + b_2_32·a_3_3 + a_6_3·a_1_0 + a_4_3·a_3_3 + b_2_3·a_4_3·a_1_0
  28. a_6_3·a_1_2 + a_4_3·a_3_3
  29. a_6_5·a_1_1 + b_2_3·a_4_3·a_1_0
  30. b_4_4·a_3_4 + b_2_32·a_3_3 + a_6_5·a_1_0 + b_2_3·a_4_3·a_1_0
  31. a_6_5·a_1_2
  32. a_6_6·a_1_1 + b_2_3·a_4_3·a_1_0
  33. b_4_4·a_3_4 + b_2_32·a_3_3 + a_6_6·a_1_0 + a_4_3·a_3_3
  34. a_6_6·a_1_2 + a_4_3·a_3_3 + b_2_3·a_4_3·a_1_0
  35. b_6_10·a_1_1 + b_4_4·a_3_4 + b_2_32·a_3_3 + b_2_33·a_1_2 + b_2_33·a_1_0
       + b_2_3·a_4_3·a_1_0
  36. b_6_10·a_1_0 + b_4_6·a_3_3
  37. b_6_10·a_1_2 + b_4_6·a_3_4 + b_4_6·a_3_3 + b_4_4·a_3_4 + b_4_4·a_3_3 + b_2_33·a_1_1
       + b_2_33·a_1_0 + a_4_3·a_3_3 + b_2_3·a_4_3·a_1_0
  38. a_4_32
  39. b_4_42 + b_2_32·b_4_6 + b_2_32·b_4_4 + b_2_34
  40. b_2_3·a_1_2·b_5_7 + b_2_3·a_1_0·b_5_7
  41. a_4_3·b_4_4 + b_2_3·a_6_6 + b_2_3·a_6_5 + b_2_32·a_4_3
  42. b_4_4·b_4_6 + b_2_3·b_6_10 + a_3_3·b_5_7 + b_2_3·a_1_0·b_5_7 + b_2_3·a_6_3 + b_2_32·a_4_3
       + b_4_6·a_1_0·a_3_3
  43. a_1_1·a_7_11
  44. a_3_4·b_5_7 + a_4_3·b_4_6 + b_2_32·a_4_3 + a_1_0·a_7_11 + b_4_6·a_1_0·a_3_3
  45. a_3_4·b_5_7 + a_4_3·b_4_6 + b_2_32·a_4_3 + a_1_2·a_7_11 + b_4_6·a_1_0·a_3_3
  46. a_1_1·b_7_12
  47. a_3_3·b_5_7 + a_1_0·b_7_12 + b_4_6·a_1_0·a_3_3
  48. a_3_4·b_5_7 + a_3_3·b_5_7 + a_1_2·b_7_12 + b_2_3·a_1_0·b_5_7 + b_4_6·a_1_0·a_3_3
  49. a_6_3·a_3_3 + a_4_3·b_4_6·a_1_0 + b_2_32·a_4_3·a_1_0
  50. a_6_3·a_3_4
  51. a_6_5·a_3_3 + b_2_3·a_4_3·a_3_3
  52. a_6_5·a_3_4 + b_2_32·a_4_3·a_1_0
  53. a_6_6·a_3_3 + a_4_3·b_4_6·a_1_0 + b_2_3·a_4_3·a_3_3 + b_2_32·a_4_3·a_1_0
  54. a_6_6·a_3_4 + b_2_3·a_4_3·a_3_3
  55. b_6_10·a_3_3 + b_4_62·a_1_0 + b_2_3·b_4_6·a_3_3 + b_2_32·b_4_6·a_1_0
       + a_4_3·b_4_6·a_1_0
  56. b_6_10·a_3_4 + b_4_62·a_1_2 + b_4_62·a_1_0 + b_2_3·b_4_6·a_3_3 + b_2_34·a_1_2
       + b_2_34·a_1_0 + b_2_3·a_4_3·a_3_3
  57. b_4_4·b_5_7 + b_2_3·b_7_12 + b_2_3·b_4_6·a_3_3 + b_2_34·a_1_2 + b_2_34·a_1_0
       + b_2_3·a_4_3·a_3_3
  58. a_8_10·a_1_1 + b_2_3·a_4_3·a_3_3
  59. a_4_3·b_5_7 + b_2_32·b_4_6·a_1_0 + b_2_34·a_1_0 + a_8_10·a_1_0 + b_2_3·a_4_3·a_3_3
       + b_2_32·a_4_3·a_1_0
  60. a_4_3·b_5_7 + b_2_32·b_4_6·a_1_0 + b_2_34·a_1_0 + a_8_10·a_1_2 + b_2_32·a_4_3·a_1_0
  61. a_4_3·a_6_3
  62. a_4_3·a_6_5
  63. b_4_6·a_6_5 + b_2_32·a_1_0·b_5_7 + b_2_32·a_6_5 + b_4_62·a_1_02
  64. b_4_4·a_6_5 + b_2_32·a_6_6 + b_2_32·a_6_5 + b_2_32·a_6_3 + b_2_33·a_4_3
  65. a_4_3·a_6_6
  66. b_4_4·a_6_6 + b_4_4·a_6_5 + b_4_4·a_6_3 + b_2_32·a_1_0·b_5_7 + b_2_32·a_6_3
  67. b_4_6·a_6_6 + b_4_4·a_6_3 + a_4_3·b_6_10 + b_2_32·a_6_5 + b_2_32·a_6_3 + b_2_33·a_4_3
       + b_4_62·a_1_02
  68. b_4_4·b_6_10 + b_2_3·b_4_62 + b_2_32·b_6_10 + b_2_33·b_4_6 + b_4_6·a_1_0·b_5_7
       + b_4_6·a_6_6 + b_4_6·a_6_3 + b_2_32·a_6_5 + b_2_32·a_6_3
  69. b_4_6·a_1_2·b_5_7 + b_4_6·a_1_0·b_5_7 + b_4_6·a_6_3 + b_2_32·a_6_3 + a_3_3·a_7_11
       + b_4_62·a_1_02
  70. b_4_4·a_6_3 + b_2_32·a_6_3 + b_2_3·a_1_0·a_7_11
  71. b_4_4·a_6_3 + b_2_32·a_6_3 + a_3_4·a_7_11
  72. a_3_3·b_7_12 + b_4_6·a_1_0·b_5_7 + b_4_6·a_6_6 + b_4_6·a_6_3 + b_4_4·a_6_5 + b_4_4·a_6_3
       + b_2_32·a_1_0·b_5_7 + b_2_32·a_6_5 + b_2_32·a_6_3 + b_2_33·a_4_3
  73. b_4_6·a_6_6 + b_4_6·a_6_3 + b_4_4·a_6_5 + b_4_4·a_6_3 + b_2_3·a_1_0·b_7_12 + b_2_32·a_6_5
       + b_2_32·a_6_3 + b_2_33·a_4_3 + b_4_62·a_1_02
  74. a_3_4·b_7_12 + b_4_6·a_1_2·b_5_7 + b_4_6·a_1_0·b_5_7 + b_4_6·a_6_6 + b_4_6·a_6_3
       + b_4_4·a_6_5 + b_4_4·a_6_3 + b_2_32·a_6_5 + b_2_32·a_6_3 + b_2_33·a_4_3
       + b_4_62·a_1_02
  75. b_5_72 + b_2_3·b_4_62 + b_2_35 + b_4_6·a_1_2·b_5_7 + b_2_32·a_1_0·b_5_7
       + b_4_62·a_1_02 + c_8_15·a_1_0·a_1_2 + c_8_15·a_1_0·a_1_1 + c_8_15·a_1_02
  76. a_1_1·a_9_16 + c_8_15·a_1_0·a_1_1
  77. b_5_72 + b_2_3·b_4_62 + b_2_35 + b_4_6·a_1_0·b_5_7 + b_4_6·a_6_3
       + b_2_32·a_1_0·b_5_7 + b_2_32·a_6_3 + a_1_0·a_9_16 + c_8_15·a_1_0·a_1_1
  78. b_4_6·a_1_2·b_5_7 + b_4_6·a_1_0·b_5_7 + b_4_6·a_6_3 + b_2_32·a_6_3 + a_1_2·a_9_16
       + b_4_62·a_1_02 + c_8_15·a_1_0·a_1_1 + c_8_15·a_1_02
  79. a_6_5·b_5_7 + b_2_33·b_4_6·a_1_0 + b_2_35·a_1_0 + a_4_3·b_4_6·a_3_3
       + b_2_33·a_4_3·a_1_0
  80. a_6_6·b_5_7 + a_6_3·b_5_7 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + a_4_3·a_7_11
       + a_4_3·b_4_6·a_3_3 + b_2_32·a_4_3·a_3_3 + b_2_33·a_4_3·a_1_0
  81. a_6_3·b_5_7 + a_4_3·b_7_12 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + b_2_32·a_4_3·a_3_3
  82. b_6_10·b_5_7 + b_4_6·b_7_12 + a_6_6·b_5_7 + b_4_62·a_3_4 + b_2_32·b_4_6·a_3_3
       + b_2_33·b_4_6·a_1_0 + b_2_35·a_1_1 + a_4_3·b_4_6·a_3_3 + b_2_32·a_4_3·a_3_3
  83. b_4_4·b_7_12 + b_2_3·b_4_6·b_5_7 + b_2_32·b_7_12 + b_2_33·b_5_7 + b_2_3·b_4_62·a_1_0
       + b_2_33·b_4_6·a_1_0 + a_4_3·b_4_6·a_3_3 + b_2_33·a_4_3·a_1_0
  84. a_6_3·b_5_7 + a_8_10·a_3_3
  85. a_6_6·b_5_7 + a_6_3·b_5_7 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + a_4_3·b_4_6·a_3_3
       + b_2_3·a_8_10·a_1_0 + b_2_32·a_4_3·a_3_3
  86. a_6_6·b_5_7 + a_6_3·b_5_7 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + a_8_10·a_3_4
       + a_4_3·b_4_6·a_3_3 + b_2_32·a_4_3·a_3_3
  87. a_6_3·b_5_7 + b_4_4·a_7_11 + b_2_3·a_9_16 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3
       + b_2_35·a_1_2 + a_4_3·b_4_6·a_3_3 + b_2_32·a_4_3·a_3_3 + b_2_3·c_8_15·a_1_2
       + b_2_3·c_8_15·a_1_0
  88. a_10_14·a_1_1 + b_2_33·a_4_3·a_1_0
  89. a_6_6·b_5_7 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + a_10_14·a_1_0 + a_4_3·b_4_6·a_3_3
  90. a_6_6·b_5_7 + b_2_32·b_4_6·a_3_3 + b_2_34·a_3_3 + a_10_14·a_1_2 + a_4_3·b_4_6·a_3_3
       + b_2_32·a_4_3·a_3_3
  91. a_6_32
  92. a_6_3·a_6_5
  93. a_6_52
  94. a_6_3·a_6_6
  95. a_6_62
  96. a_6_5·a_6_6
  97. a_6_6·b_6_10 + a_6_5·b_6_10 + a_4_3·b_4_62 + b_2_33·a_1_0·b_5_7 + b_2_34·a_4_3
       + b_2_32·a_1_0·a_7_11
  98. a_6_5·b_6_10 + b_2_32·a_1_0·b_7_12 + b_2_33·a_6_6 + b_2_33·a_6_5 + b_2_33·a_6_3
       + b_2_34·a_4_3 + b_4_62·a_1_0·a_3_3
  99. b_6_102 + b_4_63 + b_2_3·b_4_6·b_6_10 + b_2_32·b_4_62 + a_6_3·b_6_10
       + b_4_6·a_1_0·b_7_12 + a_4_3·b_4_62 + b_4_62·a_1_0·a_3_3
  100. b_5_7·b_7_12 + b_6_102 + b_4_63 + b_2_32·b_4_62 + b_2_34·b_4_4 + a_6_5·b_6_10
       + b_2_33·a_6_6 + b_2_33·a_6_5 + b_2_33·a_6_3 + b_2_34·a_4_3 + c_8_15·a_1_0·a_3_3
       + b_2_3·c_8_15·a_1_0·a_1_2
  101. a_6_6·b_6_10 + a_6_5·b_6_10 + a_4_3·b_4_62 + b_2_33·a_1_0·b_5_7 + b_2_34·a_4_3
       + a_4_3·a_8_10
  102. b_5_7·a_7_11 + b_4_6·a_8_10 + a_4_3·b_4_62 + b_2_32·a_8_10 + b_2_33·a_1_0·b_5_7
       + b_2_34·a_4_3 + b_4_6·a_1_0·a_7_11 + b_4_62·a_1_0·a_3_3 + c_8_15·a_1_0·a_3_3
       + b_2_3·c_8_15·a_1_02
  103. a_6_3·b_6_10 + a_4_3·b_4_62 + b_2_3·b_4_6·a_1_0·b_5_7 + b_2_33·a_1_0·b_5_7
       + b_2_33·a_6_3 + b_2_34·a_4_3 + a_3_3·a_9_16 + b_4_6·a_1_0·a_7_11 + c_8_15·a_1_0·a_3_3
       + b_2_3·c_8_15·a_1_02
  104. b_5_7·b_7_12 + b_6_102 + b_4_63 + b_2_32·b_4_62 + b_2_34·b_4_4 + a_6_6·b_6_10
       + a_6_3·b_6_10 + b_2_3·b_4_6·a_1_0·b_5_7 + b_2_33·a_6_6 + b_2_33·a_6_5 + b_2_34·a_4_3
       + b_2_3·a_1_0·a_9_16 + c_8_15·a_1_0·a_3_3 + b_2_3·c_8_15·a_1_02
  105. a_6_6·b_6_10 + a_6_5·b_6_10 + a_6_3·b_6_10 + b_2_3·b_4_6·a_1_0·b_5_7 + b_2_33·a_6_3
       + a_3_4·a_9_16
  106. b_5_7·b_7_12 + b_6_102 + b_4_63 + b_2_32·b_4_62 + b_2_34·b_4_4 + a_6_5·b_6_10
       + a_6_3·b_6_10 + b_4_4·a_8_10 + a_4_3·b_4_62 + b_2_3·a_10_14 + b_2_3·b_4_6·a_1_0·b_5_7
       + b_2_32·a_8_10 + b_2_33·a_6_6 + c_8_15·a_1_0·a_3_3 + b_2_3·c_8_15·a_1_02
  107. a_6_3·a_7_11 + b_2_33·a_4_3·a_3_3
  108. b_6_10·b_7_12 + b_4_62·b_5_7 + b_2_3·b_4_6·b_7_12 + b_2_32·b_4_6·b_5_7 + a_6_3·b_7_12
       + b_4_63·a_1_2 + b_4_63·a_1_0 + b_2_33·b_4_6·a_3_3 + b_2_34·b_4_6·a_1_0
       + b_2_35·a_3_3 + b_2_36·a_1_2 + a_6_6·a_7_11 + a_4_3·b_4_62·a_1_0
  109. a_6_6·b_7_12 + a_6_3·b_7_12 + b_2_32·b_4_62·a_1_0 + b_2_33·b_4_6·a_3_3
       + b_2_35·a_3_3 + b_2_36·a_1_0 + a_6_6·a_7_11 + a_4_3·b_4_62·a_1_0
       + b_2_33·a_4_3·a_3_3 + b_2_34·a_4_3·a_1_0
  110. a_6_5·b_7_12 + b_2_33·b_4_6·a_3_3 + b_2_35·a_3_3 + a_4_3·b_4_62·a_1_0
       + b_2_34·a_4_3·a_1_0
  111. a_6_6·a_7_11 + b_2_3·a_8_10·a_3_3
  112. a_6_5·a_7_11 + b_2_32·a_8_10·a_1_0 + b_2_33·a_4_3·a_3_3 + b_2_34·a_4_3·a_1_0
  113. a_6_3·b_7_12 + a_6_6·a_7_11 + a_6_5·a_7_11 + b_4_6·a_8_10·a_1_0 + b_2_34·a_4_3·a_1_0
  114. a_8_10·b_5_7 + a_6_3·b_7_12 + b_2_3·b_4_6·a_7_11 + b_2_32·b_4_62·a_1_0
       + b_2_33·a_7_11 + b_2_34·b_4_6·a_1_0 + a_6_6·a_7_11 + a_6_5·a_7_11
       + a_4_3·b_4_62·a_1_0 + b_2_33·a_4_3·a_3_3 + a_4_3·c_8_15·a_1_0
  115. a_6_6·a_7_11 + a_4_3·a_9_16 + b_2_33·a_4_3·a_3_3 + b_2_34·a_4_3·a_1_0
  116. b_6_10·a_7_11 + a_6_3·b_7_12 + b_4_6·a_9_16 + b_2_3·b_4_62·a_3_3 + b_2_33·b_4_6·a_3_3
       + b_2_34·b_4_6·a_1_0 + b_2_36·a_1_2 + b_2_36·a_1_0 + a_6_5·a_7_11
       + b_2_33·a_4_3·a_3_3 + b_4_6·c_8_15·a_1_2 + b_4_6·c_8_15·a_1_0
  117. a_6_3·b_7_12 + b_4_4·a_9_16 + b_2_3·b_4_6·a_7_11 + b_2_32·a_9_16
       + b_2_32·b_4_62·a_1_0 + b_2_33·a_7_11 + b_2_35·a_3_3 + a_6_6·a_7_11
       + a_4_3·b_4_62·a_1_0 + b_2_33·a_4_3·a_3_3
  118. a_6_3·b_7_12 + a_10_14·a_3_3 + a_6_6·a_7_11
  119. a_10_14·a_3_4 + a_6_6·a_7_11 + a_6_5·a_7_11 + b_2_33·a_4_3·a_3_3
  120. a_7_112
  121. a_6_3·a_8_10
  122. a_6_5·a_8_10 + b_2_33·a_1_0·a_7_11
  123. a_6_6·a_8_10 + b_2_32·a_1_0·a_9_16
  124. b_7_122 + b_2_3·b_4_63 + b_2_32·b_4_6·b_6_10 + b_2_33·b_4_62 + b_2_35·b_4_6
       + b_2_35·b_4_4 + b_2_37 + b_4_62·a_1_0·b_5_7 + a_4_3·b_4_6·b_6_10
       + b_2_3·b_4_6·a_1_0·b_7_12 + b_2_32·b_4_6·a_1_0·b_5_7 + b_2_34·a_6_6 + b_2_34·a_6_5
       + b_2_34·a_6_3 + b_4_6·a_1_0·a_9_16 + b_2_33·a_1_0·a_7_11
  125. a_7_11·b_7_12 + b_5_7·a_9_16 + b_2_3·b_4_6·a_1_0·b_7_12 + b_2_33·a_1_0·b_7_12
       + b_2_34·a_1_0·b_5_7 + a_6_6·a_8_10 + c_8_15·a_1_2·b_5_7 + c_8_15·a_1_0·b_5_7
  126. b_7_122 + b_2_3·b_4_63 + b_2_32·b_4_6·b_6_10 + b_2_33·b_4_62 + b_2_35·b_4_6
       + b_2_35·b_4_4 + b_2_37 + a_7_11·b_7_12 + b_6_10·a_8_10 + b_4_62·a_1_0·b_5_7
       + b_2_3·b_4_6·a_1_0·b_7_12 + b_2_32·a_10_14 + b_2_32·b_4_6·a_1_0·b_5_7
       + b_2_33·a_1_0·b_7_12 + b_2_33·a_8_10 + b_2_34·a_1_0·b_5_7 + b_2_34·a_6_5
       + b_2_34·a_6_3 + b_2_35·a_4_3 + b_4_63·a_1_02 + b_2_3·b_4_6·a_1_0·a_7_11
  127. a_6_6·a_8_10 + a_4_3·a_10_14 + b_2_33·a_1_0·a_7_11
  128. b_6_10·a_8_10 + b_4_6·a_10_14 + b_2_3·b_4_6·a_8_10 + b_2_32·b_4_6·a_1_0·b_5_7
       + b_2_34·a_1_0·b_5_7 + b_2_34·a_6_5 + a_6_6·a_8_10
  129. b_4_4·a_10_14 + b_2_3·b_4_6·a_8_10 + b_2_33·a_1_0·b_7_12 + b_2_33·a_8_10
       + b_2_34·a_6_6 + b_2_34·a_6_5 + b_2_34·a_6_3 + b_2_35·a_4_3
       + b_2_3·b_4_6·a_1_0·a_7_11 + b_2_33·a_1_0·a_7_11
  130. a_8_10·a_7_11 + b_2_3·b_4_6·a_8_10·a_1_0 + b_2_3·a_4_3·c_8_15·a_1_0
  131. a_8_10·b_7_12 + b_6_10·a_9_16 + b_4_62·a_7_11 + b_2_3·b_4_63·a_1_0
       + b_2_32·b_4_6·a_7_11 + b_2_33·a_9_16 + b_2_35·b_4_6·a_1_0 + b_2_36·a_3_3
       + b_2_37·a_1_0 + a_8_10·a_7_11 + b_4_6·a_8_10·a_3_3 + b_2_33·a_8_10·a_1_0
       + b_2_34·a_4_3·a_3_3 + b_4_6·c_8_15·a_3_4 + b_2_3·b_4_6·c_8_15·a_1_0
       + b_2_33·c_8_15·a_1_2 + b_2_33·c_8_15·a_1_1 + b_2_33·c_8_15·a_1_0
       + a_6_3·c_8_15·a_1_0
  132. a_6_3·a_9_16 + a_6_3·c_8_15·a_1_0 + a_4_3·c_8_15·a_3_3
  133. a_8_10·b_7_12 + b_2_3·b_4_6·a_9_16 + b_2_33·a_9_16 + b_2_34·b_4_6·a_3_3
       + b_2_35·b_4_6·a_1_0 + b_2_36·a_3_3 + b_2_37·a_1_0 + b_4_6·a_8_10·a_3_3
       + a_4_3·b_4_62·a_3_3 + b_2_32·a_8_10·a_3_3 + b_2_34·a_4_3·a_3_3 + a_6_3·c_8_15·a_1_0
       + a_4_3·c_8_15·a_3_3
  134. a_8_10·a_7_11 + a_6_6·a_9_16 + b_2_32·a_8_10·a_3_3 + b_2_33·a_8_10·a_1_0
       + b_2_34·a_4_3·a_3_3 + b_2_35·a_4_3·a_1_0 + a_6_3·c_8_15·a_1_0 + a_4_3·c_8_15·a_3_3
       + b_2_3·a_4_3·c_8_15·a_1_0
  135. a_6_5·a_9_16 + b_2_32·a_8_10·a_3_3 + a_6_3·c_8_15·a_1_0 + a_4_3·c_8_15·a_3_3
  136. a_10_14·b_5_7 + a_8_10·b_7_12 + b_2_32·b_4_6·a_7_11 + b_2_34·a_7_11
       + b_2_32·a_8_10·a_3_3 + b_2_33·a_8_10·a_1_0 + b_2_34·a_4_3·a_3_3
       + b_2_35·a_4_3·a_1_0 + b_2_3·a_4_3·c_8_15·a_1_0
  137. a_8_102
  138. a_7_11·a_9_16 + b_2_3·b_4_6·a_1_0·a_9_16 + b_2_33·a_1_0·a_9_16 + b_2_34·a_1_0·a_7_11
  139. b_6_10·a_10_14 + b_4_62·a_8_10 + b_2_32·b_4_6·a_1_0·b_7_12 + b_2_32·b_4_6·a_8_10
       + b_2_34·a_1_0·b_7_12 + b_2_35·a_6_6 + b_2_35·a_6_5 + b_2_35·a_6_3 + b_2_36·a_4_3
       + a_7_11·a_9_16 + b_2_33·a_1_0·a_9_16
  140. a_6_3·a_10_14
  141. b_7_12·a_9_16 + b_4_62·a_8_10 + a_4_3·b_4_63 + b_2_3·b_4_6·a_10_14
       + b_2_3·b_4_62·a_1_0·b_5_7 + b_2_32·b_4_6·a_8_10 + b_2_33·a_10_14
       + b_2_33·b_4_6·a_1_0·b_5_7 + b_2_36·a_4_3 + a_7_11·a_9_16 + b_4_62·a_1_0·a_7_11
       + b_4_63·a_1_0·a_3_3 + b_2_34·a_1_0·a_7_11 + a_4_3·b_4_6·c_8_15
       + b_2_3·c_8_15·a_1_0·b_5_7 + b_2_32·a_4_3·c_8_15 + c_8_15·a_1_0·a_7_11
  142. a_6_6·a_10_14 + b_2_32·b_4_6·a_1_0·a_7_11 + b_2_34·a_1_0·a_7_11
  143. a_6_5·a_10_14 + b_2_33·a_1_0·a_9_16 + b_2_34·a_1_0·a_7_11
  144. a_8_10·a_9_16 + b_2_33·a_8_10·a_3_3 + b_2_34·a_8_10·a_1_0 + b_2_35·a_4_3·a_3_3
  145. a_10_14·b_7_12 + b_2_3·b_4_62·a_7_11 + b_2_32·b_4_63·a_1_0 + b_2_33·b_4_62·a_3_3
       + b_2_35·a_7_11 + b_2_35·b_4_6·a_3_3 + b_2_36·b_4_6·a_1_0 + b_4_62·a_8_10·a_1_0
       + a_4_3·b_4_63·a_1_0 + b_2_34·a_8_10·a_1_0 + b_2_36·a_4_3·a_1_0
       + a_4_3·b_4_6·c_8_15·a_1_0 + b_2_32·a_4_3·c_8_15·a_1_0
  146. a_10_14·a_7_11 + b_2_3·b_4_6·a_8_10·a_3_3 + b_2_35·a_4_3·a_3_3 + b_2_36·a_4_3·a_1_0
       + b_2_3·a_4_3·c_8_15·a_3_3 + b_2_32·a_4_3·c_8_15·a_1_0
  147. a_9_162 + c_8_152·a_1_0·a_1_2 + c_8_152·a_1_0·a_1_1
  148. a_8_10·a_10_14 + b_2_33·b_4_6·a_1_0·a_7_11
  149. a_10_14·a_9_16 + b_2_32·b_4_6·a_8_10·a_3_3 + b_2_33·b_4_6·a_8_10·a_1_0
       + b_2_34·a_8_10·a_3_3 + b_2_32·a_4_3·c_8_15·a_3_3
  150. a_10_142

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 20.
  • 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_15, a Duflot regular element of degree 8
    2. b_4_6, an element of degree 4
    3. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 11].
  • 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_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. a_3_4 → 0, an element of degree 3
  7. a_4_3 → 0, an element of degree 4
  8. b_4_4 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. b_5_7 → 0, an element of degree 5
  11. a_6_3 → 0, an element of degree 6
  12. a_6_5 → 0, an element of degree 6
  13. a_6_6 → 0, an element of degree 6
  14. b_6_10 → 0, an element of degree 6
  15. a_7_11 → 0, an element of degree 7
  16. b_7_12 → 0, an element of degree 7
  17. a_8_10 → 0, an element of degree 8
  18. c_8_15 → c_1_08, an element of degree 8
  19. a_9_16 → 0, an element of degree 9
  20. a_10_14 → 0, an element of degree 10

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_1_2 → 0, an element of degree 1
  4. b_2_3 → c_1_22, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. a_3_4 → 0, an element of degree 3
  7. a_4_3 → 0, an element of degree 4
  8. b_4_4 → c_1_12·c_1_22, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  10. b_5_7 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  11. a_6_3 → 0, an element of degree 6
  12. a_6_5 → 0, an element of degree 6
  13. a_6_6 → 0, an element of degree 6
  14. b_6_10 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_16, an element of degree 6
  15. a_7_11 → 0, an element of degree 7
  16. b_7_12 → c_1_14·c_1_23 + c_1_16·c_1_2, an element of degree 7
  17. a_8_10 → 0, an element of degree 8
  18. c_8_15 → c_1_12·c_1_26 + c_1_16·c_1_22 + c_1_18 + 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
  19. a_9_16 → 0, an element of degree 9
  20. a_10_14 → 0, an element of degree 10

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