Cohomology of group number 124 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 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 3 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  −  t4  +  2·t2  −  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 19 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_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_3, an element of degree 2
  7. b_3_3, an element of degree 3
  8. b_3_4, an element of degree 3
  9. b_3_5, an element of degree 3
  10. b_3_6, an element of degree 3
  11. b_3_7, an element of degree 3
  12. b_4_9, an element of degree 4
  13. b_4_10, an element of degree 4
  14. b_5_12, an element of degree 5
  15. b_5_13, an element of degree 5
  16. a_6_7, a nilpotent element of degree 6
  17. b_6_17, an element of degree 6
  18. b_7_21, an element of degree 7
  19. c_8_26, a Duflot regular element of degree 8

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

Ring relations

There are 116 minimal relations of maximal degree 14:

  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_2·a_1_1 + b_2_1·a_1_1
  7. b_2_2·a_1_0 + b_2_1·a_1_1
  8. b_2_3·a_1_0 + b_2_1·a_1_1
  9. a_2_02
  10. b_2_22 + b_2_1·b_2_3
  11. a_1_1·b_3_3 + a_2_0·b_2_2
  12. a_1_0·b_3_3 + a_2_0·b_2_1
  13. a_1_1·b_3_4
  14. a_1_0·b_3_4 + a_2_0·b_2_2 + a_2_0·b_2_1
  15. a_1_1·b_3_5 + a_2_0·b_2_3 + a_2_0·b_2_2
  16. a_1_0·b_3_5 + a_2_0·b_2_2 + a_2_0·b_2_1
  17. b_2_2·b_2_3 + b_2_22 + a_1_1·b_3_6 + a_2_0·b_2_3 + a_2_0·b_2_2
  18. b_2_22 + b_2_1·b_2_2 + a_1_0·b_3_6
  19. b_2_2·b_2_3 + b_2_1·b_2_2 + a_1_0·b_3_7 + a_2_0·b_2_2
  20. b_2_12·a_1_0 + a_2_0·b_3_3
  21. b_2_2·b_3_3 + b_2_1·b_3_4 + b_2_1·b_3_3 + b_2_12·a_1_1
  22. b_2_3·b_3_3 + b_2_2·b_3_4 + b_2_2·b_3_3 + b_2_12·a_1_1
  23. b_2_12·a_1_1 + b_2_12·a_1_0 + a_2_0·b_3_4
  24. b_2_2·b_3_3 + b_2_1·b_3_5 + b_2_1·b_3_3 + b_2_12·a_1_1
  25. b_2_3·b_3_3 + b_2_2·b_3_5 + b_2_2·b_3_3 + b_2_12·a_1_1
  26. b_2_32·a_1_1 + b_2_12·a_1_0 + a_2_0·b_3_5
  27. b_2_3·b_3_6 + b_2_3·b_3_5 + b_2_3·b_3_4 + b_2_2·b_3_6
  28. b_2_3·b_3_3 + b_2_2·b_3_3 + b_2_32·a_1_1 + b_2_12·a_1_1 + a_2_0·b_3_6
  29. b_2_3·b_3_6 + b_2_3·b_3_5 + b_2_3·b_3_4 + b_2_2·b_3_3 + b_2_1·b_3_7 + b_2_1·b_3_6
       + b_2_12·a_1_1
  30. b_2_3·b_3_3 + b_2_2·b_3_7 + b_2_12·a_1_1
  31. b_2_3·b_3_4 + b_2_3·b_3_3 + b_2_2·b_3_3 + b_2_32·a_1_1 + b_2_12·a_1_1 + a_2_0·b_3_7
  32. b_2_3·b_3_4 + b_4_9·a_1_1 + b_2_12·a_1_1
  33. b_2_3·b_3_3 + b_2_2·b_3_3 + b_4_9·a_1_0 + b_2_12·a_1_1 + b_2_12·a_1_0
  34. b_2_3·b_3_3 + b_2_2·b_3_3 + b_4_10·a_1_0 + b_2_12·a_1_1 + b_2_12·a_1_0
  35. b_3_42 + b_3_3·b_3_4 + b_3_32 + b_2_13 + a_2_0·b_2_1·b_2_2
  36. b_3_3·b_3_5 + b_3_3·b_3_4
  37. b_3_52 + b_3_3·b_3_4 + b_2_33 + b_2_12·b_2_2 + a_2_0·b_2_32
  38. b_3_32 + b_2_13 + b_2_1·a_1_0·b_3_6
  39. b_3_5·b_3_6 + b_3_4·b_3_6 + b_3_32 + b_2_33 + b_2_12·b_2_2 + b_2_13 + a_2_0·b_2_32
       + a_2_0·b_2_1·b_2_2
  40. b_3_3·b_3_4 + b_2_12·b_2_2 + b_2_13 + b_2_1·a_1_0·b_3_7
  41. b_3_4·b_3_5 + b_3_3·b_3_4 + b_3_32 + b_2_13 + b_2_3·a_1_1·b_3_7 + a_2_0·b_2_32
       + a_2_0·b_2_1·b_2_2
  42. b_3_4·b_3_6 + b_3_4·b_3_5 + b_3_3·b_3_7 + b_3_3·b_3_4 + b_3_32 + b_2_12·b_2_2 + b_2_13
  43. b_3_3·b_3_6 + b_3_32 + b_2_1·b_4_9 + b_2_12·b_2_2 + a_2_0·b_2_1·b_2_2
  44. b_3_5·b_3_7 + b_3_4·b_3_5 + b_3_3·b_3_6 + b_3_3·b_3_4 + b_3_32 + b_2_3·b_4_9 + b_2_33
       + b_2_12·b_2_2 + b_2_13 + a_2_0·b_2_1·b_2_2
  45. b_3_4·b_3_6 + b_3_4·b_3_5 + b_3_3·b_3_6 + b_3_3·b_3_4 + b_2_2·b_4_9
  46. b_3_4·b_3_5 + b_3_3·b_3_4 + a_2_0·b_4_9 + a_2_0·b_2_12
  47. b_3_62 + b_3_3·b_3_6 + b_3_32 + b_2_33 + b_2_1·b_4_10 + a_2_0·b_2_32
       + a_2_0·b_2_1·b_2_2
  48. b_3_72 + b_3_62 + b_3_5·b_3_7 + b_3_4·b_3_7 + b_3_4·b_3_6 + b_3_4·b_3_5 + b_3_3·b_3_6
       + b_2_3·b_4_10 + b_2_33 + b_2_12·b_2_2 + b_2_13 + a_2_0·b_2_32 + a_2_0·b_2_1·b_2_2
  49. b_3_6·b_3_7 + b_3_62 + b_3_5·b_3_7 + b_3_4·b_3_7 + b_3_3·b_3_4 + b_3_32 + b_2_33
       + b_2_2·b_4_10 + a_2_0·b_2_32
  50. b_3_4·b_3_7 + b_3_4·b_3_6 + b_3_4·b_3_5 + b_2_12·b_2_2 + b_2_13 + a_2_0·b_4_10
       + a_2_0·b_2_12
  51. b_3_4·b_3_7 + b_3_4·b_3_6 + b_3_3·b_3_4 + b_2_12·b_2_2 + b_2_13 + a_1_1·b_5_12
       + a_2_0·b_2_1·b_2_2
  52. b_3_3·b_3_4 + b_2_12·b_2_2 + b_2_13 + a_1_0·b_5_12
  53. b_3_4·b_3_7 + b_3_4·b_3_6 + b_3_4·b_3_5 + b_3_3·b_3_4 + a_1_1·b_5_13 + a_2_0·b_2_32
  54. b_3_3·b_3_4 + b_3_32 + b_2_12·b_2_2 + a_1_0·b_5_13 + a_2_0·b_2_1·b_2_2 + a_2_0·b_2_12
  55. b_4_9·b_3_3 + b_2_12·b_3_6 + b_2_12·b_3_4 + a_2_0·b_2_1·b_3_6
  56. b_4_9·b_3_5 + b_2_32·b_3_7 + b_2_32·b_3_5 + b_2_12·b_3_7 + b_2_12·b_3_4
       + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_1·b_3_6
  57. b_4_9·b_3_4 + b_2_12·b_3_7 + b_2_12·b_3_3 + b_2_3·b_4_10·a_1_1 + a_2_0·b_2_3·b_3_7
       + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6 + a_2_0·b_2_1·b_3_4
       + a_2_0·b_2_1·b_3_3
  58. b_4_10·b_3_3 + b_4_9·b_3_6 + b_4_9·b_3_4 + b_2_32·b_3_7 + b_2_32·b_3_5 + b_2_12·b_3_6
       + b_2_12·b_3_4 + b_2_12·b_3_3 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_1·b_3_7
  59. b_4_10·b_3_5 + b_4_9·b_3_7 + b_2_12·b_3_7 + b_2_12·b_3_6 + b_2_12·b_3_3
       + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6
  60. b_2_2·b_5_12 + b_2_1·b_5_12 + b_2_12·b_3_7 + b_2_12·b_3_4 + b_2_12·b_3_3
       + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6 + a_2_0·b_2_1·b_3_3
  61. b_4_9·b_3_4 + b_2_12·b_3_7 + b_2_12·b_3_3 + a_2_0·b_5_12 + a_2_0·b_2_1·b_3_7
       + a_2_0·b_2_1·b_3_4 + a_2_0·b_2_1·b_3_3
  62. b_4_9·b_3_4 + b_2_3·b_5_13 + b_2_3·b_5_12 + b_2_32·b_3_7 + b_2_1·b_5_12 + b_2_12·b_3_7
       + b_2_12·b_3_6 + b_2_12·b_3_3 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_1·b_3_3
  63. b_2_2·b_5_13 + b_2_12·b_3_7 + b_2_12·b_3_6 + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6
       + a_2_0·b_2_1·b_3_4 + a_2_0·b_2_1·b_3_3
  64. b_4_9·b_3_4 + b_2_12·b_3_7 + b_2_12·b_3_3 + a_2_0·b_5_13 + a_2_0·b_2_3·b_3_7
       + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6 + a_2_0·b_2_1·b_3_4
  65. a_6_7·a_1_1
  66. a_6_7·a_1_0
  67. b_4_10·b_3_4 + b_4_9·b_3_7 + b_2_3·b_5_12 + b_2_32·b_3_7 + b_2_32·b_3_5 + b_2_1·b_5_12
       + b_2_12·b_3_6 + b_2_12·b_3_3 + b_6_17·a_1_1 + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_3·b_3_5
       + a_2_0·b_2_1·b_3_6 + a_2_0·b_2_1·b_3_4
  68. b_6_17·a_1_0 + a_2_0·b_2_1·b_3_7 + a_2_0·b_2_1·b_3_6 + a_2_0·b_2_1·b_3_4
       + a_2_0·b_2_1·b_3_3
  69. b_4_92 + b_2_32·b_4_10 + b_2_32·b_4_9 + b_2_1·b_2_2·b_4_10 + b_2_1·b_2_2·b_4_9
       + b_2_12·b_4_10 + b_2_12·b_4_9 + b_2_13·b_2_2 + b_2_14 + a_2_0·b_2_33
  70. b_3_4·b_5_12 + b_2_1·b_2_2·b_4_9 + b_2_12·b_4_9 + b_2_13·b_2_2 + b_2_14
       + b_4_10·a_1_1·b_3_7 + a_2_0·b_2_3·b_4_9 + a_2_0·b_2_13
  71. b_3_4·b_5_13 + b_3_3·b_5_13 + b_2_1·b_2_2·b_4_9 + b_2_13·b_2_2 + b_4_10·a_1_1·b_3_7
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_3·b_4_9 + a_2_0·b_2_2·b_4_9 + a_2_0·b_2_1·b_4_9
       + a_2_0·b_2_13
  72. b_3_5·b_5_13 + b_3_5·b_5_12 + b_3_3·b_5_13 + b_2_32·b_4_9 + b_2_34 + b_2_1·b_2_2·b_4_9
       + b_2_12·b_4_9 + b_2_13·b_2_2 + b_2_14 + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_33
       + a_2_0·b_2_1·b_4_9 + a_2_0·b_2_12·b_2_2
  73. b_3_7·b_5_13 + b_3_7·b_5_12 + b_3_6·b_5_13 + b_3_5·b_5_12 + b_3_3·b_5_12 + b_4_92
       + b_2_32·b_4_9 + b_2_1·b_2_2·b_4_9 + b_2_14 + a_2_0·b_2_3·b_4_9 + a_2_0·b_2_33
       + a_2_0·b_2_2·b_4_9 + a_2_0·b_2_1·b_4_9 + a_2_0·b_2_13
  74. b_3_3·b_5_13 + b_3_3·b_5_12 + b_2_12·b_4_10 + b_2_1·a_6_7 + a_2_0·b_2_1·b_4_9
       + a_2_0·b_2_13
  75. b_3_5·b_5_12 + b_3_3·b_5_12 + b_2_32·b_4_10 + b_2_32·b_4_9 + b_2_13·b_2_2 + b_2_3·a_6_7
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_33 + a_2_0·b_2_2·b_4_9 + a_2_0·b_2_1·b_4_9
       + a_2_0·b_2_13
  76. b_3_3·b_5_12 + b_4_92 + b_2_32·b_4_10 + b_2_32·b_4_9 + b_2_1·b_2_2·b_4_9
       + b_2_12·b_4_10 + b_2_13·b_2_2 + b_2_2·a_6_7 + a_2_0·b_2_33 + a_2_0·b_2_1·b_4_9
       + a_2_0·b_2_12·b_2_2
  77. a_2_0·a_6_7
  78. b_3_6·b_5_13 + b_3_6·b_5_12 + b_4_92 + b_2_32·b_4_10 + b_2_34 + b_2_1·b_6_17
       + b_2_1·b_2_2·b_4_9 + b_2_12·b_4_10 + b_2_13·b_2_2 + a_2_0·b_2_2·b_4_9
       + a_2_0·b_2_1·b_4_9
  79. b_3_6·b_5_12 + b_3_5·b_5_12 + b_4_92 + b_2_32·b_4_10 + b_2_32·b_4_9 + b_2_2·b_6_17
       + b_2_1·b_2_2·b_4_9 + b_2_12·b_4_9 + b_4_10·a_1_1·b_3_7 + a_2_0·b_2_3·b_4_9
       + a_2_0·b_2_33 + a_2_0·b_2_12·b_2_2 + a_2_0·b_2_13
  80. b_3_5·b_5_12 + b_4_92 + b_2_1·b_2_2·b_4_9 + b_2_12·b_4_10 + b_4_10·a_1_1·b_3_7
       + a_2_0·b_6_17 + a_2_0·b_2_3·b_4_9 + a_2_0·b_2_33 + a_2_0·b_2_2·b_4_9 + a_2_0·b_2_13
  81. b_3_5·b_5_12 + b_4_92 + b_2_1·b_2_2·b_4_9 + b_2_12·b_4_10 + a_1_1·b_7_21
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_3·b_4_9 + a_2_0·b_2_33 + a_2_0·b_2_2·b_4_9
       + a_2_0·b_2_13
  82. a_1_0·b_7_21 + a_2_0·b_2_1·b_4_9
  83. b_2_1·b_4_9·b_3_6 + b_2_12·b_5_13 + b_2_12·b_5_12 + b_2_13·b_3_7 + b_2_13·b_3_6
       + b_2_13·b_3_3 + a_6_7·b_3_3
  84. b_2_1·b_4_9·b_3_7 + b_2_12·b_5_13 + b_2_13·b_3_7 + b_2_13·b_3_4 + a_6_7·b_3_4
       + a_2_0·b_2_12·b_3_4
  85. b_4_9·b_5_13 + b_4_9·b_5_12 + b_2_32·b_5_12 + b_2_33·b_3_7 + b_2_33·b_3_5
       + b_2_1·b_4_10·b_3_6 + b_2_12·b_5_13 + b_2_12·b_5_12 + b_2_13·b_3_7 + b_2_13·b_3_6
       + b_2_13·b_3_4 + b_2_13·b_3_3 + a_6_7·b_3_6 + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_12·b_3_7
  86. b_4_9·b_5_13 + b_2_3·b_4_10·b_3_7 + b_2_32·b_5_12 + b_2_1·b_4_10·b_3_7
       + b_2_1·b_4_9·b_3_7 + b_2_12·b_5_13 + b_2_13·b_3_6 + a_6_7·b_3_7 + a_2_0·b_4_10·b_3_7
       + a_2_0·b_2_32·b_3_5 + a_2_0·b_2_12·b_3_4 + a_2_0·b_2_12·b_3_3
  87. b_2_1·b_4_9·b_3_7 + b_2_12·b_5_13 + b_2_13·b_3_7 + b_2_13·b_3_4 + a_6_7·b_3_5
       + b_2_3·b_6_17·a_1_1 + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_3·b_5_12 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_12·b_3_7 + a_2_0·b_2_12·b_3_4
  88. b_6_17·b_3_3 + b_4_9·b_5_13 + b_4_9·b_5_12 + b_2_32·b_5_12 + b_2_33·b_3_7
       + b_2_33·b_3_5 + b_2_1·b_4_9·b_3_6 + b_2_12·b_5_12 + b_2_13·b_3_7 + b_2_13·b_3_6
       + b_2_13·b_3_4 + a_6_7·b_3_5 + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_12·b_3_4 + a_2_0·b_2_12·b_3_3
  89. b_6_17·b_3_4 + b_4_9·b_5_13 + b_2_3·b_4_10·b_3_7 + b_2_32·b_5_12 + b_2_12·b_5_12
       + b_2_13·b_3_4 + a_6_7·b_3_5 + b_4_102·a_1_1 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_12·b_3_7 + a_2_0·b_2_12·b_3_3
  90. b_6_17·b_3_6 + b_6_17·b_3_5 + b_4_10·b_5_13 + b_4_10·b_5_12 + b_4_9·b_5_12 + b_2_33·b_3_7
       + b_2_33·b_3_5 + b_2_1·b_4_10·b_3_7 + b_2_1·b_4_10·b_3_6 + b_2_12·b_5_13
       + b_2_13·b_3_6 + a_2_0·b_2_12·b_3_4
  91. b_2_1·b_7_21 + b_2_1·b_4_9·b_3_6 + b_2_12·b_5_13 + b_2_12·b_5_12 + b_2_13·b_3_7
       + b_2_13·b_3_4 + b_2_13·b_3_3 + a_2_0·b_2_12·b_3_7 + a_2_0·b_2_12·b_3_4
       + a_2_0·b_2_12·b_3_3
  92. b_6_17·b_3_5 + b_4_9·b_5_13 + b_2_3·b_7_21 + b_2_33·b_3_7 + b_2_33·b_3_5
       + b_2_1·b_4_9·b_3_7 + b_2_12·b_5_13 + b_2_12·b_5_12 + b_2_13·b_3_7
       + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_32·b_3_7 + a_2_0·b_2_12·b_3_7 + a_2_0·b_2_12·b_3_6
       + a_2_0·b_2_12·b_3_4 + a_2_0·b_2_12·b_3_3
  93. b_2_2·b_7_21 + b_2_1·b_4_9·b_3_7 + b_2_1·b_4_9·b_3_6 + b_2_12·b_5_12 + b_2_13·b_3_7
       + a_2_0·b_2_12·b_3_6 + a_2_0·b_2_12·b_3_3
  94. b_2_1·b_4_9·b_3_7 + b_2_12·b_5_13 + b_2_13·b_3_7 + b_2_13·b_3_4 + a_6_7·b_3_5
       + a_2_0·b_7_21 + a_2_0·b_2_32·b_3_7 + a_2_0·b_2_12·b_3_6 + a_2_0·b_2_12·b_3_4
       + a_2_0·b_2_12·b_3_3
  95. b_5_122 + b_2_3·b_4_102 + b_2_33·b_4_10 + b_2_33·b_4_9 + b_2_12·b_2_2·b_4_10
       + b_2_12·b_2_2·b_4_9 + b_2_13·b_4_10 + b_2_13·b_4_9 + b_2_14·b_2_2 + a_2_0·b_4_102
       + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_32·b_4_9 + a_2_0·b_2_34 + a_2_0·b_2_1·b_2_2·b_4_9
       + a_2_0·b_2_13·b_2_2 + a_2_0·b_2_14
  96. b_5_132 + b_2_3·b_4_102 + b_2_35 + b_2_1·b_4_102 + b_2_13·b_4_10 + b_2_13·b_4_9
       + a_2_0·b_4_102 + a_2_0·b_2_34 + a_2_0·b_2_13·b_2_2 + a_2_0·b_2_14
  97. b_5_12·b_5_13 + b_2_3·b_4_102 + b_2_3·b_4_9·b_4_10 + b_2_33·b_4_10 + b_2_33·b_4_9
       + b_2_2·b_4_102 + b_2_2·b_4_9·b_4_10 + b_2_1·b_4_9·b_4_10 + b_2_12·b_2_2·b_4_10
       + b_2_12·b_2_2·b_4_9 + b_2_13·b_4_9 + b_2_14·b_2_2 + b_4_9·a_6_7 + b_2_12·a_6_7
       + a_2_0·b_2_3·b_6_17 + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_12·b_4_9
  98. b_5_12·b_5_13 + b_2_3·b_4_102 + b_2_3·b_4_9·b_4_10 + b_2_33·b_4_10 + b_2_33·b_4_9
       + b_2_2·b_4_102 + b_2_2·b_4_9·b_4_10 + b_2_12·b_6_17 + b_2_12·b_2_2·b_4_10
       + b_2_13·b_4_10 + b_2_15 + b_6_17·a_1_1·b_3_7 + b_2_1·b_2_2·a_6_7 + a_2_0·b_4_102
       + a_2_0·b_4_9·b_4_10 + a_2_0·b_2_12·b_4_9 + a_2_0·b_2_14
  99. b_3_3·b_7_21 + b_2_13·b_4_9 + b_2_12·a_6_7 + a_2_0·b_2_12·b_4_9 + a_2_0·b_2_13·b_2_2
       + a_2_0·b_2_14
  100. b_3_4·b_7_21 + b_2_1·b_4_9·b_4_10 + b_2_12·b_6_17 + b_2_13·b_4_10 + b_2_14·b_2_2
       + b_2_15 + b_4_9·a_6_7 + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_12·b_4_9
       + a_2_0·b_2_13·b_2_2
  101. b_5_12·b_5_13 + b_3_5·b_7_21 + b_2_3·b_4_102 + b_2_32·b_6_17 + b_2_33·b_4_10
       + b_2_33·b_4_9 + b_2_2·b_4_102 + b_2_1·b_2_2·b_6_17 + b_2_12·b_6_17
       + b_2_12·b_2_2·b_4_10 + b_2_12·b_2_2·b_4_9 + b_2_13·b_4_10 + b_2_13·b_4_9 + b_2_15
       + b_2_12·a_6_7 + a_2_0·b_4_9·b_4_10 + a_2_0·b_2_12·b_4_9 + a_2_0·b_2_13·b_2_2
  102. b_5_12·b_5_13 + b_3_6·b_7_21 + b_2_3·b_4_102 + b_2_32·b_6_17 + b_2_33·b_4_10
       + b_2_33·b_4_9 + b_2_2·b_4_102 + b_2_1·b_2_2·b_6_17 + b_2_12·b_6_17
       + b_2_12·b_2_2·b_4_10 + b_2_12·b_2_2·b_4_9 + b_2_14·b_2_2 + b_4_9·a_6_7
       + b_2_12·a_6_7 + a_2_0·b_4_9·b_4_10 + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_1·b_2_2·b_4_9
       + a_2_0·b_2_12·b_4_9
  103. b_3_7·b_7_21 + b_4_9·b_6_17 + b_2_3·b_4_102 + b_2_32·b_6_17 + b_2_2·b_4_102
       + b_2_1·b_4_102 + b_2_1·b_2_2·b_6_17 + b_2_12·b_2_2·b_4_10 + b_2_12·b_2_2·b_4_9
       + b_2_13·b_4_10 + b_4_10·a_6_7 + b_4_9·a_6_7 + b_2_1·b_2_2·a_6_7 + a_2_0·b_4_9·b_4_10
       + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_1·b_2_2·b_4_9
  104. b_4_9·b_4_10·b_3_7 + b_2_3·b_4_10·b_5_12 + b_2_32·b_4_10·b_3_7 + b_2_33·b_5_12
       + b_2_34·b_3_7 + b_2_34·b_3_5 + b_2_1·b_4_10·b_5_13 + b_2_1·b_4_10·b_5_12
       + b_2_13·b_5_13 + b_2_13·b_5_12 + b_2_14·b_3_7 + b_2_14·b_3_6 + b_2_14·b_3_4
       + b_2_14·b_3_3 + a_6_7·b_5_13 + b_2_1·a_6_7·b_3_6 + b_2_1·a_6_7·b_3_3
       + a_2_0·b_2_32·b_5_12
  105. b_6_17·b_5_13 + b_6_17·b_5_12 + b_4_102·b_3_6 + b_4_9·b_4_10·b_3_7 + b_4_9·b_4_10·b_3_6
       + b_2_3·b_6_17·b_3_7 + b_2_32·b_4_10·b_3_7 + b_2_33·b_5_12 + b_2_34·b_3_7
       + b_2_34·b_3_5 + b_2_12·b_4_10·b_3_6 + b_2_13·b_5_12 + b_2_14·b_3_7 + a_6_7·b_5_12
       + b_2_1·a_6_7·b_3_4 + b_2_1·a_6_7·b_3_3 + a_2_0·b_2_3·b_4_10·b_3_7
       + a_2_0·b_2_32·b_5_12 + a_2_0·b_2_33·b_3_7 + a_2_0·b_2_13·b_3_7
       + a_2_0·b_2_13·b_3_4 + a_2_0·b_2_13·b_3_3
  106. b_4_9·b_4_10·b_3_6 + b_2_32·b_4_10·b_3_7 + b_2_33·b_5_12 + b_2_34·b_3_7
       + b_2_34·b_3_5 + b_2_1·b_4_10·b_5_13 + b_2_1·b_4_10·b_5_12 + b_2_12·b_4_10·b_3_7
       + b_2_12·b_4_10·b_3_6 + b_2_13·b_5_13 + b_2_13·b_5_12 + b_2_14·b_3_7 + b_2_14·b_3_6
       + b_2_14·b_3_4 + b_2_14·b_3_3 + a_6_7·b_5_13 + a_6_7·b_5_12 + b_2_1·a_6_7·b_3_4
       + a_2_0·b_6_17·b_3_7 + a_2_0·b_2_3·b_7_21 + a_2_0·b_2_3·b_4_10·b_3_7
       + a_2_0·b_2_32·b_5_12 + a_2_0·b_2_13·b_3_7 + a_2_0·b_2_13·b_3_6
       + a_2_0·b_2_13·b_3_4 + a_2_0·b_2_13·b_3_3
  107. b_6_17·b_5_13 + b_6_17·b_5_12 + b_4_102·b_3_6 + b_4_9·b_7_21 + b_4_9·b_4_10·b_3_7
       + b_4_9·b_4_10·b_3_6 + b_2_3·b_4_10·b_5_12 + b_2_32·b_7_21 + b_2_1·b_4_10·b_5_12
       + b_2_13·b_5_13 + b_2_13·b_5_12 + b_2_14·b_3_7 + b_2_14·b_3_4 + b_2_14·b_3_3
       + a_6_7·b_5_12 + b_2_1·a_6_7·b_3_7
  108. b_6_17·b_5_13 + b_4_10·b_7_21 + b_4_102·b_3_7 + b_4_9·b_4_10·b_3_7 + b_4_9·b_4_10·b_3_6
       + b_2_32·b_7_21 + b_2_1·b_4_10·b_5_13 + b_2_1·b_4_10·b_5_12 + b_2_12·b_4_10·b_3_7
       + b_2_13·b_5_13 + b_2_14·b_3_3 + a_6_7·b_5_13 + a_6_7·b_5_12 + b_2_1·a_6_7·b_3_4
       + b_2_1·a_6_7·b_3_3 + a_2_0·b_4_10·b_5_12 + a_2_0·b_2_33·b_3_5 + a_2_0·b_2_13·b_3_4
       + a_2_0·b_2_13·b_3_3 + b_2_3·c_8_26·a_1_1 + b_2_1·c_8_26·a_1_1
  109. a_6_72
  110. b_5_12·b_7_21 + b_4_9·b_4_102 + b_2_3·b_4_10·b_6_17 + b_2_3·b_4_9·b_6_17
       + b_2_32·b_4_102 + b_2_32·b_4_9·b_4_10 + b_2_2·b_4_10·b_6_17 + b_2_1·b_4_10·b_6_17
       + b_2_12·b_4_102 + b_2_12·b_2_2·b_6_17 + b_2_13·b_6_17 + b_2_13·b_2_2·b_4_9
       + b_2_14·b_4_10 + b_2_14·b_4_9 + a_6_7·b_6_17 + b_2_2·b_4_10·a_6_7 + b_2_1·b_4_9·a_6_7
       + b_2_13·a_6_7 + a_2_0·b_4_9·b_6_17 + a_2_0·b_2_32·b_6_17 + a_2_0·b_2_33·b_4_10
       + a_2_0·b_2_15
  111. b_5_13·b_7_21 + b_4_9·b_4_102 + b_2_3·b_4_10·b_6_17 + b_2_32·b_4_9·b_4_10
       + b_2_33·b_6_17 + b_2_2·b_4_10·b_6_17 + b_2_1·b_4_10·b_6_17 + b_2_12·b_4_102
       + b_2_15·b_2_2 + b_2_16 + a_6_7·b_6_17 + b_2_2·b_4_9·a_6_7 + b_2_1·b_4_10·a_6_7
       + b_2_1·b_4_9·a_6_7 + b_2_12·b_2_2·a_6_7 + a_2_0·b_4_9·b_6_17
       + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_12·b_2_2·b_4_9 + a_2_0·b_2_13·b_4_9
       + a_2_0·b_2_14·b_2_2 + a_2_0·b_2_15
  112. b_4_10·b_3_7·b_5_12 + b_4_9·b_4_102 + b_2_1·b_4_10·b_6_17 + b_2_12·b_2_2·b_6_17
       + b_2_13·b_2_2·b_4_10 + b_2_14·b_4_10 + a_6_7·b_6_17 + b_2_1·b_4_9·a_6_7
       + b_2_12·b_2_2·a_6_7 + b_2_13·a_6_7 + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_33·b_4_10
       + a_2_0·b_2_33·b_4_9 + a_2_0·b_2_12·b_2_2·b_4_9 + a_2_0·b_2_15 + a_2_0·b_2_3·c_8_26
       + a_2_0·b_2_2·c_8_26
  113. b_6_172 + b_4_103 + b_4_9·b_4_102 + b_2_3·b_4_10·b_6_17 + b_2_3·b_4_9·b_6_17
       + b_2_32·b_4_102 + b_2_33·b_6_17 + b_2_34·b_4_9 + b_2_2·b_4_10·b_6_17
       + b_2_1·b_2_2·b_4_102 + b_2_12·b_2_2·b_6_17 + b_2_13·b_2_2·b_4_10 + b_2_14·b_4_10
       + b_2_14·b_4_9 + b_2_3·b_4_10·a_6_7 + b_2_2·b_4_9·a_6_7 + a_2_0·b_4_10·b_6_17
       + a_2_0·b_2_3·b_4_102 + a_2_0·b_2_33·b_4_10 + a_2_0·b_2_33·b_4_9 + a_2_0·b_2_35
       + a_2_0·b_2_14·b_2_2 + b_2_32·c_8_26 + b_2_1·b_2_2·c_8_26 + c_8_26·a_1_0·b_3_6
  114. a_6_7·b_7_21 + b_2_12·a_6_7·b_3_6 + b_2_12·a_6_7·b_3_4 + a_2_0·b_4_10·b_7_21
       + a_2_0·b_4_102·b_3_7 + a_2_0·b_2_3·b_4_10·b_5_12 + a_2_0·b_2_32·b_4_10·b_3_7
       + a_2_0·b_2_33·b_5_12 + a_2_0·b_2_14·b_3_7 + a_2_0·b_2_14·b_3_4 + a_2_0·c_8_26·b_3_5
       + a_2_0·c_8_26·b_3_4
  115. b_6_17·b_7_21 + b_4_10·b_6_17·b_3_7 + b_4_102·b_5_13 + b_2_3·b_4_102·b_3_7
       + b_2_32·b_6_17·b_3_7 + b_2_32·b_4_10·b_5_12 + b_2_33·b_4_10·b_3_7 + b_2_34·b_5_12
       + b_2_12·b_4_10·b_5_13 + b_2_12·b_4_10·b_5_12 + b_2_13·b_4_10·b_3_7 + b_2_14·b_5_13
       + b_2_14·b_5_12 + b_2_15·b_3_7 + b_2_15·b_3_6 + b_2_15·b_3_4 + b_2_1·a_6_7·b_5_13
       + b_2_12·a_6_7·b_3_7 + b_2_12·a_6_7·b_3_6 + a_2_0·b_4_10·b_7_21 + a_2_0·b_2_34·b_3_5
       + a_2_0·b_2_14·b_3_6 + a_2_0·b_2_14·b_3_4 + b_2_3·c_8_26·b_3_5 + a_2_0·c_8_26·b_3_7
       + a_2_0·c_8_26·b_3_6
  116. b_7_212 + b_2_32·b_4_10·b_6_17 + b_2_32·b_4_9·b_6_17 + b_2_33·b_4_102
       + b_2_34·b_6_17 + b_2_35·b_4_9 + b_2_1·b_2_2·b_4_10·b_6_17 + b_2_13·b_2_2·b_6_17
       + b_2_14·b_2_2·b_4_9 + b_2_15·b_4_10 + b_2_15·b_4_9 + b_2_16·b_2_2 + b_2_17
       + b_2_1·b_2_2·b_4_10·a_6_7 + b_2_1·b_2_2·b_4_9·a_6_7 + a_2_0·b_4_9·b_4_102
       + a_2_0·b_2_3·b_4_10·b_6_17 + a_2_0·b_2_3·b_4_9·b_6_17 + a_2_0·b_2_32·b_4_102
       + a_2_0·b_2_32·b_4_9·b_4_10 + a_2_0·b_2_33·b_6_17 + a_2_0·b_2_36
       + a_2_0·b_2_14·b_4_9 + b_2_33·c_8_26 + b_2_12·b_2_2·c_8_26
       + b_2_1·c_8_26·a_1_0·b_3_6 + a_2_0·b_2_32·c_8_26 + a_2_0·b_2_1·b_2_2·c_8_26

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 14.
  • 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_26, a Duflot regular element of degree 8
    2. b_4_10 + b_4_9 + b_2_32 + b_2_1·b_2_2 + b_2_12, an element of degree 4
    3. b_2_3 + b_2_2 + b_2_1, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 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_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_3 → 0, an element of degree 2
  7. b_3_3 → 0, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. b_3_5 → 0, an element of degree 3
  10. b_3_6 → 0, an element of degree 3
  11. b_3_7 → 0, an element of degree 3
  12. b_4_9 → 0, an element of degree 4
  13. b_4_10 → 0, an element of degree 4
  14. b_5_12 → 0, an element of degree 5
  15. b_5_13 → 0, an element of degree 5
  16. a_6_7 → 0, an element of degree 6
  17. b_6_17 → 0, an element of degree 6
  18. b_7_21 → 0, an element of degree 7
  19. c_8_26 → 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_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_3 → c_1_12, an element of degree 2
  7. b_3_3 → 0, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. b_3_5 → c_1_13, an element of degree 3
  10. b_3_6 → c_1_13, an element of degree 3
  11. b_3_7 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_13, an element of degree 3
  12. b_4_9 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  13. b_4_10 → c_1_24 + c_1_13·c_1_2, an element of degree 4
  14. b_5_12 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  15. b_5_13 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  16. a_6_7 → 0, an element of degree 6
  17. b_6_17 → c_1_26 + c_1_15·c_1_2 + c_1_16 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  18. b_7_21 → c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_15·c_1_22 + c_1_16·c_1_2 + c_1_17
       + c_1_02·c_1_15 + c_1_04·c_1_13, an element of degree 7
  19. c_8_26 → c_1_13·c_1_25 + 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_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + 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_0 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. b_2_3 → c_1_22, an element of degree 2
  7. b_3_3 → c_1_23, an element of degree 3
  8. b_3_4 → 0, an element of degree 3
  9. b_3_5 → 0, an element of degree 3
  10. b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  11. b_3_7 → c_1_23, an element of degree 3
  12. b_4_9 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  13. b_4_10 → c_1_1·c_1_23 + c_1_14, an element of degree 4
  14. b_5_12 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  15. b_5_13 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  16. a_6_7 → 0, an element of degree 6
  17. b_6_17 → c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_15·c_1_2
       + c_1_16, an element of degree 6
  18. b_7_21 → c_1_1·c_1_26 + c_1_12·c_1_25, an element of degree 7
  19. c_8_26 → 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

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. b_2_1 → c_1_22, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. b_3_3 → c_1_23, an element of degree 3
  8. b_3_4 → c_1_23, an element of degree 3
  9. b_3_5 → c_1_23, an element of degree 3
  10. b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  11. b_3_7 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  12. b_4_9 → c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  13. b_4_10 → c_1_24 + c_1_1·c_1_23 + c_1_14, an element of degree 4
  14. b_5_12 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  15. b_5_13 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  16. a_6_7 → 0, an element of degree 6
  17. b_6_17 → c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  18. b_7_21 → c_1_27 + c_1_1·c_1_26 + c_1_12·c_1_25, an element of degree 7
  19. c_8_26 → c_1_1·c_1_27 + c_1_12·c_1_26 + 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