Cohomology of group number 553 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 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t2  −  t  +  1
    (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-4,-4,-4. 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 18 minimal generators of maximal degree 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. a_2_3, a nilpotent element of degree 2
  5. a_2_4, a nilpotent element of degree 2
  6. b_2_5, an element of degree 2
  7. a_3_5, a nilpotent element of degree 3
  8. a_3_9, a nilpotent element of degree 3
  9. b_3_8, an element of degree 3
  10. b_3_10, an element of degree 3
  11. a_4_11, a nilpotent element of degree 4
  12. a_4_14, a nilpotent element of degree 4
  13. b_4_16, an element of degree 4
  14. c_4_17, a Duflot regular element of degree 4
  15. c_4_18, a Duflot regular element of degree 4
  16. a_5_16, a nilpotent element of degree 5
  17. b_5_29, an element of degree 5
  18. a_6_26, 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 103 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·a_1_2
  3. a_1_0·b_1_1 + a_1_22
  4. a_1_22·b_1_1
  5. a_2_3·a_1_0
  6. a_2_4·a_1_0 + a_2_3·a_1_2
  7. a_2_3·b_1_1 + a_2_4·a_1_2
  8. b_2_5·a_1_0
  9. a_2_32
  10. a_2_3·a_2_4 + a_2_4·a_1_22
  11. a_2_4·a_1_2·b_1_1 + a_2_42 + a_2_4·a_1_22
  12. a_1_0·a_3_5
  13. a_1_2·a_3_5 + a_2_4·a_1_2·b_1_1
  14. a_1_0·a_3_9
  15. a_2_3·b_2_5 + a_1_2·a_3_9
  16. a_1_0·b_3_8
  17. b_1_1·a_3_5 + a_1_2·b_3_8 + a_2_4·b_1_12 + a_2_4·a_1_22
  18. a_1_0·b_3_10
  19. b_1_1·a_3_9 + a_1_2·b_3_10 + a_2_4·b_2_5 + a_2_3·b_2_5
  20. a_2_3·a_3_5
  21. a_2_3·a_3_9
  22. a_2_3·b_3_8 + a_2_4·a_3_5 + a_2_42·b_1_1
  23. a_1_2·b_1_1·b_3_10 + b_2_5·a_3_5 + a_2_4·b_2_5·b_1_1 + a_2_4·b_2_5·a_1_2
  24. a_2_3·b_3_10 + a_2_4·a_3_9 + a_2_4·b_2_5·a_1_2
  25. a_4_11·b_1_1 + b_2_5·a_3_5 + b_2_5·a_1_2·b_1_12 + a_2_4·b_3_8
  26. a_4_11·a_1_0
  27. a_4_11·a_1_2 + a_2_4·a_3_5 + a_2_4·b_2_5·a_1_2 + a_2_42·b_1_1
  28. a_4_14·b_1_1 + b_2_52·a_1_2 + a_2_4·b_3_10 + a_2_4·a_3_9
  29. a_4_14·a_1_0
  30. a_4_14·a_1_2 + a_2_4·a_3_9 + a_2_4·b_2_5·a_1_2
  31. b_4_16·a_1_0
  32. b_1_12·b_3_10 + b_2_5·b_3_8 + b_4_16·a_1_2 + b_2_5·a_3_5 + b_2_5·a_1_2·b_1_12
       + a_2_4·b_2_5·a_1_2
  33. a_3_52 + a_2_42·b_1_12
  34. a_3_92 + b_2_5·a_1_2·a_3_9
  35. a_3_5·a_3_9 + a_2_42·b_2_5
  36. a_3_5·b_3_8 + b_2_5·a_1_2·b_1_13 + a_2_4·b_1_1·b_3_8
  37. a_3_5·b_3_10 + b_2_52·a_1_2·b_1_1 + a_2_4·b_1_1·b_3_10 + a_2_4·a_1_2·b_3_10
  38. b_3_82 + b_2_5·b_1_14 + a_1_2·b_1_12·b_3_8 + c_4_17·a_1_22
  39. b_3_102 + b_2_53 + b_2_5·a_1_2·b_3_10 + c_4_18·a_1_22
  40. a_3_9·b_3_8 + b_2_5·a_1_2·b_3_10 + b_2_5·a_4_11 + a_2_4·b_2_52 + a_3_92
  41. a_2_3·a_4_11
  42. a_3_5·b_3_10 + b_2_52·a_1_2·b_1_1 + a_2_4·b_1_1·b_3_10 + a_2_4·a_1_2·b_3_8
       + a_2_4·a_4_11
  43. a_3_9·b_3_10 + b_2_5·a_4_14 + a_3_92
  44. a_2_3·a_4_14
  45. a_3_5·b_3_10 + b_2_52·a_1_2·b_1_1 + a_2_4·b_1_1·b_3_10 + a_3_92 + a_2_4·a_4_14
  46. a_3_9·b_3_8 + a_3_5·b_3_10 + a_2_3·b_4_16
  47. b_3_8·b_3_10 + b_2_52·b_1_12 + a_3_9·b_3_8 + a_3_5·b_3_10 + b_1_1·a_5_16
       + b_2_5·a_1_2·b_3_10 + b_2_5·a_1_2·b_3_8 + b_2_52·a_1_2·b_1_1 + a_2_4·b_1_1·b_3_10
       + a_2_4·b_4_16 + a_2_4·b_2_5·b_1_12 + a_2_4·b_2_52 + a_3_92 + a_2_42·b_2_5
  48. a_1_0·a_5_16
  49. a_3_9·b_3_8 + a_3_5·b_3_10 + a_3_92 + a_1_2·a_5_16 + a_2_42·b_2_5
  50. a_1_0·b_5_29
  51. b_3_8·b_3_10 + b_2_52·b_1_12 + a_3_9·b_3_8 + a_1_2·b_5_29 + a_1_2·b_1_12·b_3_8
       + b_4_16·a_1_2·b_1_1 + b_2_5·a_1_2·b_3_8 + b_2_52·a_1_2·b_1_1 + a_2_4·a_1_2·b_3_8
  52. a_4_11·a_3_9 + a_2_4·b_2_52·a_1_2 + a_2_42·b_3_10
  53. a_4_11·a_3_5 + a_2_42·b_3_8 + a_2_42·b_2_5·b_1_1
  54. a_4_11·b_3_8 + b_2_5·a_1_2·b_1_1·b_3_8 + b_2_52·a_1_2·b_1_12 + a_2_4·b_2_5·b_3_8
       + a_2_4·b_2_5·b_1_13 + a_2_42·b_3_8 + a_2_3·c_4_17·a_1_2
  55. a_4_14·a_3_9 + a_2_4·b_2_5·a_3_9 + a_2_4·b_2_52·a_1_2
  56. a_4_14·b_3_10 + b_2_52·a_3_9 + a_2_3·c_4_18·a_1_2
  57. a_4_14·a_3_5 + a_2_42·b_3_10
  58. a_4_14·b_3_8 + a_4_11·b_3_10 + b_2_53·a_1_2 + a_2_4·b_2_5·b_3_10 + a_2_4·b_2_5·a_3_9
       + a_2_4·b_2_52·a_1_2 + a_2_42·b_3_10
  59. b_4_16·a_3_9 + b_2_5·a_5_16 + b_2_52·a_3_9 + b_2_52·a_3_5 + a_2_4·b_2_52·a_1_2
  60. a_2_3·a_5_16
  61. a_4_11·b_3_10 + b_2_52·a_3_5 + b_2_53·a_1_2 + a_2_4·b_2_5·b_3_10 + a_2_4·a_5_16
       + a_2_4·b_4_16·a_1_2 + a_2_4·b_2_52·a_1_2 + a_2_42·b_2_5·b_1_1
  62. b_1_12·b_5_29 + b_1_14·b_3_8 + b_4_16·b_3_8 + b_4_16·b_1_13 + b_2_5·b_1_12·b_3_8
       + b_2_52·b_1_13 + b_4_16·a_1_2·b_1_12 + b_2_5·a_1_2·b_1_14
       + b_2_52·a_1_2·b_1_12 + a_2_4·b_1_12·b_3_8 + a_2_4·b_2_5·b_3_8 + a_2_42·b_1_13
       + a_2_42·b_2_5·b_1_1 + b_2_5·c_4_17·a_1_2
  63. a_1_2·b_1_1·b_5_29 + a_1_2·b_1_13·b_3_8 + b_4_16·a_3_5 + b_4_16·a_1_2·b_1_12
       + b_2_5·a_1_2·b_1_1·b_3_8 + b_2_52·a_1_2·b_1_12 + a_2_4·b_4_16·b_1_1
       + a_2_42·b_3_10 + a_2_42·b_3_8
  64. b_4_16·b_3_10 + b_2_5·b_5_29 + b_2_5·b_1_12·b_3_8 + b_2_5·b_4_16·b_1_1 + b_2_52·b_3_8
       + b_2_53·b_1_1 + b_4_16·a_3_9 + b_2_5·b_4_16·a_1_2 + b_2_52·a_1_2·b_1_12
       + b_2_53·a_1_2 + a_2_4·b_2_5·b_3_10 + a_2_4·b_2_5·b_3_8 + a_2_4·b_2_5·a_3_9
       + a_2_4·b_2_52·a_1_2 + a_2_42·b_3_10 + a_2_42·b_2_5·b_1_1 + c_4_18·a_1_2·b_1_12
  65. a_4_11·b_3_10 + b_2_52·a_3_5 + b_2_53·a_1_2 + a_2_4·b_2_5·b_3_10 + a_2_3·b_5_29
       + a_2_4·b_4_16·a_1_2 + a_2_4·b_2_5·a_3_9 + a_2_42·b_3_8
  66. a_6_26·b_1_1 + b_4_16·a_1_2·b_1_12 + b_2_5·b_4_16·a_1_2 + b_2_52·a_3_5
       + b_2_52·a_1_2·b_1_12 + a_2_4·b_5_29 + a_2_4·b_1_12·b_3_8 + a_2_4·b_1_15
       + a_2_4·b_2_5·b_1_13 + a_2_4·b_2_52·b_1_1 + a_2_4·b_4_16·a_1_2 + a_2_42·b_3_8
       + c_4_17·a_1_2·b_1_12 + a_2_4·c_4_17·b_1_1
  67. a_6_26·a_1_0 + a_2_3·c_4_17·a_1_2
  68. a_4_11·b_3_10 + b_2_52·a_3_5 + b_2_53·a_1_2 + a_2_4·b_2_5·b_3_10 + a_6_26·a_1_2
       + a_2_4·b_4_16·a_1_2 + a_2_4·b_2_5·a_3_9 + a_2_42·b_1_13 + a_2_42·b_2_5·b_1_1
       + a_2_4·c_4_17·a_1_2
  69. a_4_112 + a_2_42·b_2_5·b_1_12 + a_2_42·b_2_52
  70. a_4_142 + a_2_4·b_2_5·a_4_14 + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_10
  71. a_4_11·a_4_14 + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_10 + a_2_42·b_2_52
  72. b_4_162 + b_2_5·b_4_16·b_1_12 + b_2_52·a_1_2·b_3_8 + b_2_52·a_1_2·b_1_13
       + b_2_53·a_1_2·b_1_1 + a_2_4·b_2_5·b_1_1·b_3_8 + a_2_4·b_2_52·b_1_12
       + a_2_42·b_4_16 + a_2_42·b_2_5·b_1_12 + a_2_42·b_2_52 + c_4_18·b_1_14
       + b_2_52·c_4_17
  73. b_3_10·a_5_16 + a_4_14·b_4_16 + b_2_52·a_4_14 + b_2_53·a_1_2·b_1_1
       + a_2_4·b_2_5·b_1_1·b_3_10 + a_3_9·a_5_16 + a_2_42·b_2_52
  74. a_3_9·a_5_16 + b_2_5·a_1_2·a_5_16
  75. b_4_162 + b_2_5·b_4_16·b_1_12 + b_2_52·a_1_2·b_3_8 + b_2_52·a_1_2·b_1_13
       + b_2_53·a_1_2·b_1_1 + a_2_4·b_2_5·b_1_1·b_3_8 + a_2_4·b_2_52·b_1_12 + a_3_5·a_5_16
       + c_4_18·b_1_14 + b_2_52·c_4_17
  76. b_3_8·a_5_16 + a_3_9·b_5_29 + a_4_14·b_4_16 + a_4_11·b_4_16 + b_2_52·a_4_11
       + b_2_53·a_1_2·b_1_1 + a_2_4·b_2_5·b_1_1·b_3_10 + a_2_42·c_4_18
       + a_2_4·c_4_18·a_1_22
  77. b_3_8·a_5_16 + a_4_11·b_4_16 + b_2_5·a_1_2·b_5_29 + b_2_5·a_1_2·b_1_12·b_3_8
       + b_2_5·b_4_16·a_1_2·b_1_1 + b_2_52·a_1_2·b_3_8 + b_2_52·a_1_2·b_1_13
       + a_2_4·b_2_5·b_1_1·b_3_10 + a_2_4·b_2_5·b_1_1·b_3_8 + a_2_4·b_2_5·b_4_16
       + a_3_9·a_5_16 + a_4_142 + a_2_42·b_2_52
  78. b_4_162 + b_2_5·b_4_16·b_1_12 + b_3_8·a_5_16 + a_3_5·b_5_29 + b_4_16·a_1_2·b_3_8
       + b_2_5·a_1_2·b_1_15 + b_2_52·a_1_2·b_1_13 + a_2_4·b_1_13·b_3_8
       + a_2_4·b_4_16·b_1_12 + a_2_4·b_2_5·b_1_1·b_3_10 + a_2_4·b_2_5·b_1_1·b_3_8
       + a_3_9·a_5_16 + a_4_142 + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_10
       + a_2_42·b_1_1·b_3_8 + a_2_42·b_2_5·b_1_12 + c_4_18·b_1_14 + b_2_52·c_4_17
       + c_4_17·a_1_2·a_3_9
  79. b_4_162 + b_2_5·b_4_16·b_1_12 + b_3_8·a_5_16 + b_2_5·b_4_16·a_1_2·b_1_1
       + b_2_52·a_1_2·b_3_8 + a_2_4·b_1_1·b_5_29 + a_2_4·b_1_13·b_3_8
       + a_2_4·b_4_16·b_1_12 + a_2_4·b_2_5·b_1_1·b_3_10 + a_2_4·b_2_5·b_1_1·b_3_8
       + a_3_9·a_5_16 + a_4_142 + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_10
       + a_2_42·b_1_1·b_3_8 + a_2_42·b_2_52 + c_4_18·b_1_14 + b_2_52·c_4_17
       + c_4_17·a_1_2·a_3_9
  80. b_3_10·b_5_29 + b_4_162 + b_2_5·b_1_1·b_5_29 + b_2_5·b_1_13·b_3_8
       + b_2_52·b_1_1·b_3_10 + b_2_52·b_1_1·b_3_8 + b_2_52·b_1_14 + b_2_52·b_4_16
       + b_4_16·a_1_2·b_3_8 + a_4_14·b_4_16 + b_2_5·b_4_16·a_1_2·b_1_1
       + b_2_52·a_1_2·b_1_13 + b_2_52·a_4_11 + a_2_4·b_2_5·b_1_1·b_3_10
       + a_2_4·b_2_5·b_1_1·b_3_8 + a_2_4·b_2_5·b_4_16 + a_3_9·a_5_16 + a_4_142
       + a_2_4·a_1_2·b_5_29 + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_8 + c_4_18·b_1_14
       + b_2_52·c_4_17 + c_4_18·a_1_2·b_3_8 + c_4_18·a_1_2·b_1_13 + a_2_42·c_4_18
       + a_2_4·c_4_18·a_1_22
  81. b_3_8·b_5_29 + b_4_16·b_1_1·b_3_8 + b_2_5·b_1_16 + b_2_5·b_4_16·b_1_12
       + b_2_52·b_1_1·b_3_8 + b_2_52·b_1_14 + a_1_2·b_1_14·b_3_8 + b_2_52·a_1_2·b_3_8
       + a_2_4·b_2_5·b_1_14 + a_2_4·b_2_52·b_1_12 + c_4_17·a_1_2·b_3_10
       + c_4_17·a_1_2·a_3_9 + a_2_4·c_4_17·a_1_22
  82. a_4_14·b_4_16 + b_2_5·a_6_26 + b_2_5·b_4_16·a_1_2·b_1_1 + b_2_52·a_4_11
       + a_2_4·b_2_5·b_1_14 + a_2_4·b_2_5·b_4_16 + a_2_4·b_2_52·b_1_12 + a_4_142
       + a_2_4·b_2_5·a_4_11 + a_2_42·b_1_1·b_3_10 + a_2_42·b_2_52
       + b_2_5·c_4_17·a_1_2·b_1_1 + a_2_4·b_2_5·c_4_17 + a_2_42·c_4_18
       + a_2_4·c_4_18·a_1_22
  83. a_2_3·a_6_26
  84. b_3_10·b_5_29 + b_2_5·b_1_1·b_5_29 + b_2_5·b_1_13·b_3_8 + b_2_5·b_4_16·b_1_12
       + b_2_52·b_1_1·b_3_10 + b_2_52·b_1_1·b_3_8 + b_2_52·b_1_14 + b_2_52·b_4_16
       + b_4_16·a_1_2·b_3_8 + a_4_14·b_4_16 + b_2_5·b_4_16·a_1_2·b_1_1 + b_2_52·a_1_2·b_3_8
       + b_2_52·a_4_11 + b_2_53·a_1_2·b_1_1 + a_2_4·b_2_5·b_1_1·b_3_10 + a_2_4·b_2_5·b_4_16
       + a_2_4·b_2_52·b_1_12 + a_2_4·a_6_26 + a_2_42·b_1_1·b_3_10 + a_2_42·b_1_14
       + a_2_42·b_2_52 + c_4_18·a_1_2·b_3_8 + c_4_18·a_1_2·b_1_13 + a_2_42·c_4_18
       + a_2_4·c_4_17·a_1_22
  85. a_4_14·a_5_16 + a_2_4·b_2_5·a_5_16 + a_2_4·b_2_5·b_4_16·a_1_2 + a_2_4·b_2_53·a_1_2
       + a_2_42·b_2_52·b_1_1
  86. b_4_16·a_5_16 + b_2_52·a_5_16 + b_2_53·a_3_9 + b_2_53·a_3_5
       + a_2_4·b_2_5·b_4_16·a_1_2 + a_2_42·b_2_5·b_3_8 + a_2_42·b_2_52·b_1_1
       + c_4_18·a_1_2·b_1_1·b_3_8 + b_2_5·c_4_17·a_3_9 + a_2_4·c_4_18·b_1_13
  87. a_4_14·b_5_29 + b_2_52·a_5_16 + b_2_52·a_1_2·b_1_1·b_3_8 + b_2_52·b_4_16·a_1_2
       + b_2_53·a_3_9 + b_2_54·a_1_2 + a_2_4·b_2_5·b_5_29 + a_2_4·b_2_5·b_1_12·b_3_8
       + a_2_4·b_2_5·b_4_16·b_1_1 + a_2_4·b_2_52·b_3_10 + a_2_4·b_2_52·b_3_8
       + a_2_4·b_2_52·b_1_13 + a_2_4·b_2_53·b_1_1 + a_4_14·a_5_16 + a_4_11·a_5_16
       + a_2_4·b_2_53·a_1_2 + a_2_42·b_2_5·b_3_10 + a_2_42·b_2_52·b_1_1
       + a_2_4·c_4_18·a_3_5
  88. a_4_14·b_5_29 + a_4_11·b_5_29 + b_2_5·a_1_2·b_1_13·b_3_8
       + b_2_5·b_4_16·a_1_2·b_1_12 + b_2_52·a_5_16 + b_2_52·a_1_2·b_1_14 + b_2_53·a_3_9
       + b_2_53·a_3_5 + b_2_54·a_1_2 + a_2_4·b_4_16·b_3_8 + a_2_4·b_2_5·b_1_12·b_3_8
       + a_2_4·b_2_5·b_1_15 + a_2_4·b_2_52·b_3_10 + a_4_14·a_5_16 + a_4_11·a_5_16
       + a_2_42·b_1_12·b_3_8 + a_2_42·b_2_5·b_3_10 + a_2_42·b_2_5·b_3_8
       + a_2_42·b_2_5·b_1_13 + a_2_42·b_2_52·b_1_1 + a_2_4·c_4_18·a_3_5
       + a_2_4·c_4_17·a_3_9 + a_2_4·b_2_5·c_4_17·a_1_2
  89. b_4_16·b_5_29 + b_4_16·b_1_12·b_3_8 + b_2_5·b_4_16·b_1_13 + b_2_52·b_4_16·b_1_1
       + a_4_14·b_5_29 + b_2_52·a_5_16 + b_2_52·a_1_2·b_1_1·b_3_8 + b_2_52·a_1_2·b_1_14
       + b_2_53·a_3_9 + b_2_53·a_3_5 + b_2_54·a_1_2 + a_2_4·b_4_16·b_3_8
       + a_2_4·b_2_5·b_1_12·b_3_8 + a_2_4·b_2_52·b_3_10 + a_2_4·b_2_52·b_3_8
       + a_2_4·b_2_52·b_1_13 + a_2_4·b_2_53·b_1_1 + a_4_14·a_5_16
       + a_2_4·b_2_5·b_4_16·a_1_2 + a_2_4·b_2_53·a_1_2 + a_2_42·b_2_5·b_3_8
       + a_2_42·b_2_5·b_1_13 + a_2_42·b_2_52·b_1_1 + c_4_18·b_1_12·b_3_8
       + c_4_18·b_1_15 + b_2_5·c_4_17·b_3_10 + b_2_52·c_4_17·b_1_1 + c_4_18·a_1_2·b_1_14
       + b_2_5·c_4_17·a_3_9 + a_2_4·c_4_18·a_3_5
  90. a_4_11·a_5_16 + a_2_4·b_2_5·b_4_16·a_1_2 + a_2_4·b_2_53·a_1_2 + a_2_42·b_5_29
       + a_2_42·b_1_12·b_3_8 + a_2_42·b_4_16·b_1_1 + a_2_42·b_2_5·b_3_10
  91. a_6_26·a_3_9 + a_2_4·b_2_53·a_1_2 + a_2_42·b_2_5·b_3_8 + a_2_42·b_2_5·b_1_13
       + a_2_42·b_2_52·b_1_1 + a_2_4·c_4_17·a_3_9 + a_2_4·b_2_5·c_4_17·a_1_2
  92. a_6_26·b_3_10 + a_4_14·b_5_29 + b_2_5·b_4_16·a_3_5 + b_2_52·a_1_2·b_1_1·b_3_8
       + b_2_52·b_4_16·a_1_2 + a_2_4·b_2_5·b_1_12·b_3_8 + a_2_4·b_2_5·b_4_16·b_1_1
       + a_2_4·b_2_52·b_3_8 + a_2_4·b_2_52·b_1_13 + a_4_14·a_5_16 + a_4_11·a_5_16
       + a_2_42·b_4_16·b_1_1 + a_2_42·b_2_5·b_3_10 + b_2_5·c_4_17·a_3_5
       + a_2_4·c_4_17·b_3_10 + a_2_4·b_2_5·c_4_17·b_1_1 + a_2_4·b_2_5·c_4_17·a_1_2
  93. a_6_26·a_3_5 + a_2_42·b_1_12·b_3_8 + a_2_42·b_1_15 + a_2_42·b_2_5·b_1_13
       + a_2_42·b_2_52·b_1_1 + a_2_4·c_4_17·a_3_5 + a_2_42·c_4_17·b_1_1
  94. a_6_26·b_3_8 + b_4_16·a_1_2·b_1_1·b_3_8 + b_2_5·b_4_16·a_3_5
       + b_2_52·a_1_2·b_1_1·b_3_8 + b_2_53·a_1_2·b_1_12 + a_2_4·b_1_14·b_3_8
       + a_2_4·b_4_16·b_3_8 + a_2_4·b_2_5·b_1_12·b_3_8 + a_2_4·b_2_52·b_3_8
       + a_2_4·b_2_52·b_1_13 + a_4_11·a_5_16 + a_2_4·b_2_5·b_4_16·a_1_2
       + a_2_4·b_2_53·a_1_2 + a_2_42·b_2_5·b_3_8 + c_4_17·a_1_2·b_1_1·b_3_8
       + a_2_4·c_4_17·b_3_8 + a_2_4·c_4_17·a_3_9 + a_2_4·b_2_5·c_4_17·a_1_2
  95. a_5_162 + a_2_4·b_2_52·a_4_14 + a_2_4·b_2_52·a_4_11 + a_2_42·b_2_5·b_1_1·b_3_10
       + a_2_42·b_2_5·b_4_16 + a_2_42·b_2_52·b_1_12 + a_2_4·c_4_17·a_1_2·b_3_8
       + a_2_4·a_4_14·c_4_17 + a_2_4·a_4_11·c_4_17 + a_2_42·c_4_18·b_1_12
  96. b_5_292 + b_2_5·b_1_18 + b_2_5·b_4_16·b_1_14 + b_2_52·b_4_16·b_1_12
       + b_2_53·b_1_14 + b_2_54·b_1_12 + a_1_2·b_1_16·b_3_8 + b_2_5·b_4_16·a_1_2·b_3_8
       + b_2_52·a_1_2·b_1_15 + b_2_53·a_1_2·b_3_8 + b_2_54·a_1_2·b_1_1
       + a_2_4·b_2_5·b_1_13·b_3_8 + a_2_4·b_2_52·b_1_1·b_3_8 + a_2_4·b_2_52·b_1_14
       + a_2_4·b_2_53·b_1_12 + a_2_42·b_4_16·b_1_12 + a_2_42·b_2_5·b_1_1·b_3_10
       + a_2_42·b_2_5·b_4_16 + a_2_42·b_2_52·b_1_12 + c_4_18·b_1_16
       + b_2_5·c_4_18·b_1_14 + b_2_52·c_4_17·b_1_12 + b_2_53·c_4_17
       + c_4_18·a_1_2·b_1_12·b_3_8 + b_2_5·a_4_11·c_4_17 + b_2_52·c_4_17·a_1_2·b_1_1
       + a_2_4·c_4_17·b_1_1·b_3_10 + a_2_4·b_2_52·c_4_17 + c_4_17·a_1_2·a_5_16
       + a_2_4·a_4_14·c_4_17 + a_2_42·b_2_5·c_4_17 + c_4_17·c_4_18·a_1_22
  97. a_4_14·a_6_26 + a_2_4·b_2_52·a_4_11 + a_2_42·b_2_5·b_1_1·b_3_8
       + a_2_42·b_2_52·b_1_12 + a_2_4·c_4_17·a_1_2·b_3_8 + a_2_4·a_4_14·c_4_17
       + a_2_4·a_4_11·c_4_17
  98. a_5_16·b_5_29 + b_2_5·b_4_16·a_1_2·b_1_13 + b_2_52·a_1_2·b_1_15 + b_2_52·a_6_26
       + b_2_53·a_1_2·b_3_8 + a_2_4·b_4_16·b_1_1·b_3_8 + a_2_4·b_2_5·b_1_1·b_5_29
       + a_2_4·b_2_5·b_4_16·b_1_12 + a_2_4·b_2_52·b_1_1·b_3_8 + a_2_4·b_2_52·b_1_14
       + a_2_4·b_2_52·b_4_16 + a_2_4·b_2_53·b_1_12 + a_2_4·b_2_5·a_1_2·b_5_29
       + a_2_4·b_2_5·a_6_26 + a_2_4·b_2_52·a_4_14 + a_2_4·b_2_52·a_4_11
       + a_2_42·b_1_1·b_5_29 + a_2_42·b_1_13·b_3_8 + a_2_42·b_4_16·b_1_12
       + a_2_42·b_2_5·b_1_14 + a_2_42·b_2_5·b_4_16 + a_2_42·b_2_52·b_1_12
       + a_2_42·b_2_53 + c_4_18·a_1_2·b_1_12·b_3_8 + b_2_5·c_4_18·a_1_2·b_1_13
       + b_2_5·a_4_14·c_4_17 + b_2_5·a_4_11·c_4_17 + a_2_4·c_4_18·b_1_1·b_3_8
       + a_2_4·c_4_18·b_1_14 + a_2_4·c_4_17·b_1_1·b_3_10 + a_2_4·b_2_52·c_4_17
       + c_4_17·a_1_2·a_5_16 + a_2_4·a_4_14·c_4_17 + a_2_42·c_4_18·b_1_12
       + a_2_42·b_2_5·c_4_17
  99. a_4_11·a_6_26 + a_2_42·b_1_13·b_3_8 + a_2_42·b_2_5·b_1_1·b_3_10
       + a_2_42·b_2_52·b_1_12 + a_2_42·b_2_53 + a_2_4·c_4_17·a_1_2·b_3_8
       + a_2_4·a_4_11·c_4_17 + a_2_42·b_2_5·c_4_17
  100. a_5_16·b_5_29 + b_4_16·a_6_26 + b_2_52·a_1_2·b_5_29 + b_2_52·a_1_2·b_1_12·b_3_8
       + b_2_52·a_1_2·b_1_15 + b_2_52·a_6_26 + b_2_52·b_4_16·a_1_2·b_1_1
       + b_2_54·a_1_2·b_1_1 + a_2_4·b_4_16·b_1_1·b_3_8 + a_2_4·b_4_16·b_1_14
       + a_2_4·b_2_5·b_1_1·b_5_29 + a_2_4·b_2_5·b_4_16·b_1_12 + a_2_4·b_2_52·b_1_1·b_3_8
       + a_2_4·b_2_52·b_1_14 + a_2_4·b_2_53·b_1_12 + a_2_4·b_2_5·a_1_2·b_5_29
       + a_2_4·b_2_52·a_4_14 + a_2_42·b_1_1·b_5_29 + a_2_42·b_1_13·b_3_8
       + a_2_42·b_4_16·b_1_12 + a_2_42·b_2_5·b_1_1·b_3_10 + c_4_18·a_1_2·b_1_12·b_3_8
       + c_4_18·a_1_2·b_1_15 + b_4_16·c_4_17·a_1_2·b_1_1 + b_2_5·a_4_11·c_4_17
       + b_2_52·c_4_17·a_1_2·b_1_1 + a_2_4·c_4_17·b_1_1·b_3_10 + a_2_4·b_4_16·c_4_17
       + c_4_17·a_1_2·a_5_16 + a_2_4·c_4_17·a_1_2·b_3_8 + a_2_4·a_4_11·c_4_17
       + a_2_42·c_4_18·b_1_12 + a_2_42·b_2_5·c_4_17
  101. a_6_26·b_5_29 + b_4_16·a_1_2·b_1_13·b_3_8 + b_2_5·b_4_16·a_1_2·b_1_1·b_3_8
       + b_2_5·b_4_16·a_1_2·b_1_14 + b_2_52·a_1_2·b_1_13·b_3_8
       + b_2_52·b_4_16·a_1_2·b_1_12 + b_2_53·a_1_2·b_1_1·b_3_8 + b_2_53·a_1_2·b_1_14
       + b_2_54·a_3_5 + a_2_4·b_1_16·b_3_8 + a_2_4·b_4_16·b_1_15
       + a_2_4·b_2_5·b_4_16·b_3_8 + a_2_4·b_2_5·b_4_16·b_1_13 + a_2_4·b_2_52·b_4_16·b_1_1
       + a_2_4·b_2_52·a_5_16 + a_2_4·b_2_53·a_3_9 + a_2_42·b_1_14·b_3_8
       + a_2_42·b_4_16·b_3_8 + a_2_42·b_4_16·b_1_13 + a_2_42·b_2_5·b_5_29
       + a_2_42·b_2_5·b_1_12·b_3_8 + a_2_42·b_2_5·b_1_15 + a_2_42·b_2_53·b_1_1
       + c_4_18·a_1_2·b_1_13·b_3_8 + c_4_18·a_1_2·b_1_16 + c_4_17·a_1_2·b_1_13·b_3_8
       + b_4_16·c_4_17·a_3_5 + b_4_16·c_4_17·a_1_2·b_1_12 + b_2_5·c_4_18·a_1_2·b_1_1·b_3_8
       + b_2_5·c_4_18·a_1_2·b_1_14 + b_2_5·c_4_17·a_1_2·b_1_1·b_3_8 + b_2_52·c_4_17·a_3_9
       + b_2_52·c_4_17·a_3_5 + b_2_53·c_4_17·a_1_2 + a_2_4·c_4_18·b_1_15
       + a_2_4·c_4_17·b_5_29 + a_2_4·b_4_16·c_4_17·b_1_1 + a_2_4·b_2_5·c_4_18·b_1_13
       + a_2_42·c_4_18·b_1_13 + a_2_42·c_4_17·b_3_8 + a_2_42·b_2_5·c_4_17·b_1_1
       + a_2_3·c_4_17·c_4_18·a_1_2
  102. a_6_26·a_5_16 + a_2_4·b_2_52·b_4_16·a_1_2 + a_2_4·b_2_54·a_1_2
       + a_2_42·b_4_16·b_3_8 + a_2_42·b_4_16·b_1_13 + a_2_42·b_2_5·b_1_12·b_3_8
       + a_2_42·b_2_5·b_1_15 + a_2_42·b_2_5·b_4_16·b_1_1 + a_2_42·b_2_52·b_3_8
       + a_2_4·c_4_17·a_5_16 + a_2_4·b_4_16·c_4_17·a_1_2 + a_2_4·b_2_52·c_4_17·a_1_2
       + a_2_42·b_2_5·c_4_17·b_1_1
  103. a_6_262 + a_2_42·b_1_18 + a_2_42·b_2_5·b_4_16·b_1_12
       + a_2_42·b_2_52·b_1_14 + a_2_42·b_2_52·b_4_16 + a_2_42·b_2_53·b_1_12
       + a_2_42·b_2_54 + a_2_4·b_2_5·a_4_14·c_4_17 + a_2_4·b_2_5·a_4_11·c_4_17
       + a_2_42·c_4_18·b_1_14 + a_2_42·c_4_17·b_1_1·b_3_10
       + a_2_42·b_2_5·c_4_18·b_1_12 + a_2_42·b_2_52·c_4_17 + a_2_42·c_4_172

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_17, a Duflot regular element of degree 4
    2. c_4_18, a Duflot regular element of degree 4
    3. b_1_14 + b_2_5·b_1_12 + b_2_52, an element of degree 4
    4. b_3_8 + b_2_5·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. a_3_5 → 0, an element of degree 3
  8. a_3_9 → 0, an element of degree 3
  9. b_3_8 → 0, an element of degree 3
  10. b_3_10 → 0, an element of degree 3
  11. a_4_11 → 0, an element of degree 4
  12. a_4_14 → 0, an element of degree 4
  13. b_4_16 → 0, an element of degree 4
  14. c_4_17 → c_1_04, an element of degree 4
  15. c_4_18 → c_1_14, an element of degree 4
  16. a_5_16 → 0, an element of degree 5
  17. b_5_29 → 0, an element of degree 5
  18. a_6_26 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. a_3_5 → 0, an element of degree 3
  8. a_3_9 → 0, an element of degree 3
  9. b_3_8 → c_1_22·c_1_3, an element of degree 3
  10. b_3_10 → c_1_33, an element of degree 3
  11. a_4_11 → 0, an element of degree 4
  12. a_4_14 → 0, an element of degree 4
  13. b_4_16 → c_1_12·c_1_22 + c_1_02·c_1_32, an element of degree 4
  14. c_4_17 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  15. c_4_18 → c_1_12·c_1_32 + c_1_14, an element of degree 4
  16. a_5_16 → 0, an element of degree 5
  17. b_5_29 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_22·c_1_3
       + c_1_12·c_1_23 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32, an element of degree 5
  18. a_6_26 → 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