Cohomology of group number 260 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 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t7  +  t5  +  t4  +  t2  +  1)
    (t  −  1)3 · (t2  +  1)2 · (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 18 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. b_1_2, an element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. a_3_6, a nilpotent element of degree 3
  7. a_4_7, a nilpotent element of degree 4
  8. a_4_8, a nilpotent element of degree 4
  9. c_4_9, a Duflot regular element of degree 4
  10. a_5_12, a nilpotent element of degree 5
  11. a_5_11, a nilpotent element of degree 5
  12. a_5_14, a nilpotent element of degree 5
  13. a_6_19, a nilpotent element of degree 6
  14. a_7_22, a nilpotent element of degree 7
  15. a_7_23, a nilpotent element of degree 7
  16. a_8_26, a nilpotent element of degree 8
  17. a_8_27, a nilpotent element of degree 8
  18. c_8_29, 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 119 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_0·b_1_22 + a_1_13
  4. a_2_4·a_1_1 + a_1_13
  5. a_2_4·a_1_0
  6. a_2_42
  7. a_2_4·b_1_22 + a_1_1·a_3_5
  8. a_1_0·a_3_5
  9. a_1_1·a_3_6 + a_1_13·b_1_2
  10. a_1_0·a_3_6
  11. a_1_13·b_1_22
  12. a_2_4·a_3_5
  13. a_2_4·a_3_6
  14. b_1_22·a_3_6 + a_1_1·b_1_2·a_3_5 + a_4_7·a_1_1
  15. a_4_7·a_1_0
  16. a_4_8·a_1_0
  17. a_3_52 + a_1_1·b_1_22·a_3_5 + a_1_12·b_1_24
  18. a_3_62
  19. a_3_5·a_3_6 + a_2_4·a_4_7
  20. a_3_5·a_3_6 + a_4_8·a_1_12
  21. a_2_4·a_4_8
  22. a_3_5·a_3_6 + a_1_1·a_5_12 + a_1_1·b_1_22·a_3_5 + a_4_7·a_1_1·b_1_2 + c_4_9·a_1_12
  23. a_1_0·a_5_12
  24. a_1_0·a_5_11
  25. a_4_8·b_1_22 + a_4_7·b_1_22 + a_1_1·a_5_14 + a_1_1·a_5_11 + c_4_9·a_1_12
  26. a_3_5·a_3_6 + a_1_0·a_5_14
  27. a_4_7·a_3_6 + a_4_8·a_1_12·b_1_2
  28. a_4_8·a_3_5 + a_4_7·a_1_1·b_1_22
  29. a_4_8·a_3_6
  30. a_4_7·a_3_6 + a_2_4·a_5_12 + c_4_9·a_1_13
  31. b_1_22·a_5_12 + b_1_24·a_3_5 + a_4_7·b_1_23 + a_1_1·b_1_23·a_3_5 + a_1_12·b_1_25
       + a_4_7·a_3_5 + a_4_7·a_1_1·b_1_22 + a_1_12·a_5_11 + c_4_9·a_1_1·b_1_22
       + c_4_9·a_1_13
  32. b_1_22·a_5_12 + b_1_24·a_3_5 + a_4_7·b_1_23 + a_1_1·b_1_23·a_3_5 + a_1_12·b_1_25
       + a_4_7·a_3_6 + a_4_7·a_3_5 + a_2_4·a_5_11 + c_4_9·a_1_1·b_1_22 + c_4_9·a_1_13
  33. b_1_22·a_5_12 + b_1_24·a_3_5 + a_4_7·b_1_23 + a_1_1·b_1_23·a_3_5 + a_1_12·b_1_25
       + a_4_7·a_1_1·b_1_22 + a_1_12·a_5_14 + c_4_9·a_1_1·b_1_22
  34. b_1_22·a_5_12 + b_1_24·a_3_5 + a_4_7·b_1_23 + a_1_1·b_1_23·a_3_5 + a_1_12·b_1_25
       + a_4_7·a_3_6 + a_4_7·a_1_1·b_1_22 + a_2_4·a_5_14 + c_4_9·a_1_1·b_1_22
  35. b_1_22·a_5_12 + b_1_24·a_3_5 + a_4_7·b_1_23 + a_1_12·b_1_25 + a_6_19·a_1_1
       + a_4_7·a_1_1·b_1_22 + c_4_9·a_1_1·b_1_22 + c_4_9·a_1_12·b_1_2
  36. a_6_19·a_1_0
  37. a_4_72
  38. a_4_82
  39. a_4_7·a_4_8
  40. a_3_6·a_5_11 + a_2_4·b_1_2·a_5_11
  41. a_3_6·a_5_12 + a_1_13·a_5_14 + c_4_9·a_1_13·b_1_2
  42. a_4_7·b_1_24 + a_3_6·a_5_12 + a_3_5·a_5_14 + a_3_5·a_5_11 + a_3_5·a_5_12
       + a_1_1·b_1_22·a_5_14 + a_1_1·b_1_22·a_5_11 + a_2_4·b_1_2·a_5_11
       + a_1_12·b_1_2·a_5_14 + c_4_9·a_1_12·b_1_22
  43. a_3_6·a_5_14 + a_3_6·a_5_12 + a_1_12·b_1_2·a_5_14 + c_4_9·a_1_13·b_1_2
  44. b_1_25·a_3_5 + a_6_19·b_1_22 + a_4_7·b_1_24 + a_3_5·a_5_11 + a_1_1·b_1_22·a_5_14
       + a_1_1·b_1_22·a_5_11 + a_2_4·b_1_2·a_5_11 + a_1_12·b_1_2·a_5_14
       + c_4_9·a_1_1·b_1_23
  45. a_3_5·a_5_12 + a_1_12·b_1_26 + a_6_19·a_1_1·b_1_2 + a_1_12·b_1_2·a_5_11
       + c_4_9·a_1_1·a_3_5 + c_4_9·a_1_12·b_1_22 + c_4_9·a_1_13·b_1_2
  46. a_2_4·a_6_19 + c_4_9·a_1_13·b_1_2
  47. a_4_7·b_1_24 + a_3_6·a_5_12 + a_3_5·a_5_11 + a_3_5·a_5_12 + a_1_1·a_7_22
       + a_1_1·b_1_22·a_5_11 + a_1_12·b_1_26 + a_1_12·b_1_2·a_5_14 + a_1_12·b_1_2·a_5_11
  48. a_1_0·a_7_22
  49. a_3_6·a_5_12 + a_3_5·a_5_12 + a_1_1·a_7_23 + a_1_1·b_1_22·a_5_14 + a_1_1·b_1_22·a_5_11
       + a_1_12·b_1_26 + c_4_9·a_1_12·b_1_22
  50. a_1_0·a_7_23
  51. a_4_8·a_5_12 + a_1_1·a_3_5·a_5_11 + a_1_12·b_1_22·a_5_11 + a_4_8·c_4_9·a_1_1
  52. a_4_8·a_5_14 + a_4_8·a_5_11 + a_4_7·a_5_12 + a_1_12·b_1_22·a_5_14 + a_4_8·c_4_9·a_1_1
       + a_4_7·c_4_9·a_1_1
  53. a_4_8·a_5_14 + a_4_8·a_5_11 + a_1_12·b_1_22·a_5_11 + a_1_13·b_1_2·a_5_14
       + a_4_8·c_4_9·a_1_1
  54. a_4_7·a_5_14 + a_4_7·a_5_11 + a_4_7·c_4_9·a_1_1
  55. a_1_12·b_1_27 + a_6_19·a_3_5 + a_6_19·a_1_1·b_1_22 + a_4_7·a_5_12
       + a_1_12·b_1_22·a_5_11 + c_4_9·a_1_1·b_1_2·a_3_5 + c_4_9·a_1_12·b_1_23
       + a_4_7·c_4_9·a_1_1
  56. a_6_19·a_3_6 + a_4_8·a_5_14 + a_4_8·a_5_11 + a_1_12·b_1_22·a_5_11 + a_4_8·c_4_9·a_1_1
  57. a_4_8·a_5_14 + a_4_8·a_5_11 + a_4_7·a_5_12 + a_2_4·a_7_22 + a_4_8·c_4_9·a_1_1
       + a_4_7·c_4_9·a_1_1
  58. b_1_22·a_7_23 + b_1_24·a_5_14 + b_1_24·a_5_11 + a_6_19·b_1_23 + a_1_1·b_1_2·a_7_22
       + a_1_1·b_1_23·a_5_11 + a_4_8·a_5_12 + a_4_7·a_5_11 + a_4_7·a_5_12
       + a_1_12·b_1_22·a_5_11 + c_4_9·b_1_22·a_3_5 + c_4_9·a_1_12·b_1_23
       + a_4_8·c_4_9·a_1_1 + a_4_7·c_4_9·a_1_1
  59. a_4_8·a_5_14 + a_4_8·a_5_11 + a_4_8·a_5_12 + a_4_7·a_5_12 + a_2_4·a_7_23
       + a_1_12·b_1_22·a_5_11 + a_4_7·c_4_9·a_1_1
  60. a_1_1·b_1_2·a_7_22 + a_8_26·a_1_1 + a_6_19·a_1_1·b_1_22 + a_4_8·a_5_14 + a_4_8·a_5_11
       + a_4_8·a_5_12 + a_4_7·a_5_11 + a_4_7·a_5_12 + c_4_9·a_1_1·b_1_2·a_3_5
       + c_4_9·a_1_12·b_1_23 + a_4_8·c_4_9·a_1_1
  61. a_8_26·a_1_0
  62. a_1_1·b_1_23·a_5_11 + a_8_27·a_1_1 + a_6_19·a_1_1·b_1_22 + a_4_8·a_5_14
       + a_4_8·a_5_12 + a_4_7·a_5_12 + a_1_12·b_1_22·a_5_11 + c_4_9·a_1_1·b_1_2·a_3_5
  63. a_8_27·a_1_0 + a_4_8·a_5_14 + a_4_8·a_5_11 + a_1_12·b_1_22·a_5_11 + a_4_8·c_4_9·a_1_1
  64. a_5_12·a_5_11 + b_1_22·a_3_5·a_5_11 + a_4_8·a_6_19 + a_4_7·b_1_2·a_5_11
       + a_4_8·a_1_1·a_5_11 + c_4_9·a_1_1·a_5_11 + a_4_8·c_4_9·a_1_1·b_1_2
       + a_4_8·c_4_9·a_1_12
  65. a_5_122 + a_6_19·b_1_2·a_3_5 + c_4_9·a_1_1·b_1_22·a_3_5 + c_4_92·a_1_12
  66. a_5_12·a_5_11 + b_1_22·a_3_5·a_5_11 + a_4_7·b_1_2·a_5_11 + a_4_7·a_6_19
       + a_1_1·b_1_2·a_3_5·a_5_11 + a_1_12·b_1_23·a_5_14 + c_4_9·a_1_1·a_5_11
       + a_4_7·c_4_9·a_1_1·b_1_2
  67. a_5_12·a_5_11 + a_3_5·a_7_22 + b_1_22·a_3_5·a_5_11 + a_1_1·b_1_24·a_5_14
       + a_6_19·a_1_1·b_1_23 + a_4_7·b_1_2·a_5_11 + a_1_1·b_1_2·a_3_5·a_5_11
       + a_1_12·b_1_23·a_5_14 + a_4_8·a_1_1·a_5_11 + c_4_9·a_1_1·a_5_11
       + c_4_9·a_1_1·b_1_22·a_3_5 + c_4_9·a_1_12·b_1_24 + a_4_8·c_4_9·a_1_12
  68. a_3_6·a_7_22 + a_1_12·b_1_23·a_5_14 + a_4_8·a_1_1·a_5_11 + a_4_8·c_4_9·a_1_12
  69. a_5_12·a_5_14 + a_5_122 + a_3_5·a_7_23 + b_1_22·a_3_5·a_5_11 + a_4_7·b_1_2·a_5_11
       + a_1_1·b_1_2·a_3_5·a_5_11 + a_1_12·b_1_23·a_5_14 + a_4_8·a_1_1·a_5_11
       + c_4_9·a_1_1·a_5_14 + c_4_9·a_1_12·b_1_24 + a_4_7·c_4_9·a_1_1·b_1_2
       + c_4_92·a_1_12
  70. a_3_6·a_7_23 + a_1_1·b_1_2·a_3_5·a_5_11 + a_1_12·b_1_23·a_5_14
       + a_4_8·c_4_9·a_1_12
  71. a_5_142 + a_5_12·a_5_11 + a_5_122 + b_1_22·a_3_5·a_5_11 + a_4_7·b_1_2·a_5_11
       + a_1_1·b_1_2·a_3_5·a_5_11 + a_4_8·a_1_1·a_5_11 + c_8_29·a_1_12 + c_4_9·a_1_1·a_5_11
       + c_4_9·a_1_1·b_1_22·a_3_5 + a_4_8·c_4_9·a_1_12 + c_4_92·a_1_12
  72. a_5_12·a_5_14 + a_5_122 + a_1_1·b_1_24·a_5_14 + a_8_26·a_1_1·b_1_2
       + c_4_9·a_1_1·a_5_14 + c_4_9·a_1_1·b_1_22·a_3_5 + c_4_9·a_1_12·b_1_24
       + a_4_8·c_4_9·a_1_1·b_1_2 + c_4_92·a_1_12
  73. a_2_4·a_8_26 + a_1_12·b_1_23·a_5_14 + a_4_8·a_1_1·a_5_11 + a_4_8·c_4_9·a_1_12
  74. b_1_23·a_7_22 + b_1_25·a_5_11 + a_8_27·b_1_22 + a_8_26·b_1_22 + a_5_142
       + a_5_11·a_5_14 + a_5_12·a_5_14 + a_1_1·b_1_24·a_5_14 + a_4_8·a_1_1·a_5_11
       + c_4_9·a_1_1·b_1_25 + a_4_7·c_4_9·b_1_22 + c_4_9·a_1_1·b_1_22·a_3_5
       + c_4_9·a_1_12·b_1_24 + a_4_8·c_4_9·a_1_12
  75. a_5_142 + a_5_112 + a_5_12·a_5_11 + b_1_22·a_3_5·a_5_11 + a_8_27·a_1_1·b_1_2
       + a_6_19·a_1_1·b_1_23 + a_4_8·b_1_2·a_5_11 + a_4_7·b_1_2·a_5_11
       + a_1_1·b_1_2·a_3_5·a_5_11 + a_1_12·b_1_23·a_5_14 + a_4_8·a_1_1·a_5_11
       + c_4_9·a_1_1·a_5_11 + c_4_9·a_1_12·b_1_24 + a_4_7·c_4_9·a_1_1·b_1_2
       + c_4_92·a_1_12
  76. a_5_12·a_5_11 + b_1_22·a_3_5·a_5_11 + a_4_7·b_1_2·a_5_11 + a_1_1·b_1_2·a_3_5·a_5_11
       + a_8_27·a_1_12 + c_4_9·a_1_1·a_5_11
  77. a_2_4·a_8_27 + a_1_1·b_1_2·a_3_5·a_5_11 + a_4_8·c_4_9·a_1_12
  78. b_1_23·a_3_5·a_5_11 + a_6_19·a_5_11 + a_6_19·a_5_12 + a_6_19·b_1_22·a_3_5
       + a_4_7·b_1_22·a_5_11 + a_1_1·a_5_11·a_5_14 + a_1_1·b_1_22·a_3_5·a_5_11
       + a_1_12·b_1_24·a_5_14 + c_4_9·a_1_1·b_1_2·a_5_11 + c_4_9·a_6_19·a_1_1
       + a_2_4·c_4_9·a_5_11 + c_4_9·a_1_12·a_5_11
  79. b_1_23·a_3_5·a_5_11 + a_6_19·a_5_11 + a_4_8·a_7_22 + a_4_7·a_7_22 + a_1_1·a_5_11·a_5_14
       + a_1_1·b_1_22·a_3_5·a_5_11 + a_1_12·b_1_24·a_5_14 + a_4_8·a_1_1·b_1_2·a_5_11
       + c_4_9·a_1_1·b_1_2·a_5_11 + a_2_4·c_4_9·a_5_11 + c_4_9·a_1_12·a_5_14
       + a_4_8·c_4_9·a_1_12·b_1_2 + c_4_92·a_1_13
  80. a_4_8·a_7_23 + a_1_1·b_1_22·a_3_5·a_5_11 + a_2_4·c_4_9·a_5_11 + c_4_9·a_1_12·a_5_11
       + a_4_8·c_4_9·a_1_12·b_1_2
  81. b_1_23·a_3_5·a_5_11 + a_6_19·a_5_11 + a_4_7·a_7_23 + a_4_7·b_1_22·a_5_11
       + a_1_1·a_5_11·a_5_14 + a_1_1·b_1_22·a_3_5·a_5_11 + a_1_12·b_1_24·a_5_14
       + a_4_8·a_1_1·b_1_2·a_5_11 + c_4_9·a_1_1·b_1_2·a_5_11 + c_4_9·a_1_12·a_5_14
       + c_4_9·a_1_12·a_5_11 + a_4_8·c_4_9·a_1_12·b_1_2 + c_4_92·a_1_13
  82. a_4_8·a_7_22 + a_1_1·a_5_11·a_5_14 + a_1_1·b_1_22·a_3_5·a_5_11 + a_2_4·c_4_9·a_5_11
       + c_8_29·a_1_13 + a_4_8·c_4_9·a_1_12·b_1_2 + c_4_92·a_1_13
  83. a_1_1·b_1_25·a_5_14 + a_8_26·a_1_1·b_1_22 + a_6_19·a_5_14 + a_6_19·b_1_22·a_3_5
       + a_4_8·a_7_22 + a_4_7·b_1_22·a_5_11 + a_1_1·a_5_11·a_5_14
       + a_1_1·b_1_22·a_3_5·a_5_11 + a_4_8·a_1_1·b_1_2·a_5_11 + c_4_9·a_1_1·b_1_2·a_5_14
       + c_4_9·a_1_12·b_1_25 + a_2_4·c_4_9·a_5_11 + c_4_9·a_1_12·a_5_14
       + c_4_9·a_1_12·a_5_11
  84. a_1_1·b_1_25·a_5_14 + a_8_26·a_3_5 + a_6_19·b_1_22·a_3_5 + a_6_19·a_1_1·b_1_24
       + a_4_8·a_7_22 + a_1_1·b_1_22·a_3_5·a_5_11 + a_1_12·b_1_24·a_5_14
       + c_4_9·a_6_19·a_1_1 + c_4_9·a_1_12·a_5_11 + c_4_92·a_1_12·b_1_2 + c_4_92·a_1_13
  85. a_8_26·a_3_6 + a_1_12·b_1_24·a_5_14 + a_4_8·c_4_9·a_1_12·b_1_2
  86. b_1_23·a_3_5·a_5_11 + a_6_19·a_5_11 + a_4_7·b_1_22·a_5_11 + a_1_1·a_5_11·a_5_14
       + a_1_1·b_1_22·a_3_5·a_5_11 + a_8_27·a_1_12·b_1_2 + c_4_9·a_1_1·b_1_2·a_5_11
  87. a_8_27·a_3_5 + a_6_19·a_5_11 + a_6_19·b_1_22·a_3_5 + a_1_1·a_5_11·a_5_14
       + a_1_1·b_1_22·a_3_5·a_5_11 + a_1_12·b_1_24·a_5_14 + c_4_9·a_1_1·b_1_2·a_5_11
       + c_4_9·a_1_12·b_1_25 + c_4_9·a_6_19·a_1_1 + c_4_9·a_1_12·a_5_11
       + c_4_92·a_1_12·b_1_2 + c_4_92·a_1_13
  88. a_8_27·a_3_6 + a_1_1·b_1_22·a_3_5·a_5_11
  89. a_6_19·b_1_23·a_3_5 + a_6_192 + c_4_9·a_6_19·a_1_1·b_1_2
       + c_4_9·a_1_12·b_1_2·a_5_14
  90. a_5_14·a_7_23 + a_5_12·a_7_23 + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_11
       + a_4_7·b_1_23·a_5_11 + a_6_19·a_1_1·a_5_14 + a_6_19·a_1_1·a_5_11
       + a_1_12·a_5_11·a_5_14 + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22
       + c_4_9·a_1_1·b_1_22·a_5_14 + a_2_4·c_4_9·b_1_2·a_5_11 + c_8_29·a_1_13·b_1_2
       + c_4_9·a_1_12·b_1_2·a_5_14 + c_4_92·a_1_1·a_3_5 + c_4_92·a_1_13·b_1_2
  91. a_5_14·a_7_22 + a_5_12·a_7_23 + a_5_12·a_7_22 + a_6_19·b_1_2·a_5_14
       + a_6_19·b_1_2·a_5_11 + a_6_19·b_1_23·a_3_5 + a_4_7·b_1_23·a_5_11
       + a_6_19·a_1_1·a_5_14 + a_6_19·a_1_1·a_5_11 + c_8_29·a_1_1·a_3_5
       + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22 + c_4_9·a_1_1·b_1_22·a_5_14
       + c_4_9·a_6_19·a_1_1·b_1_2 + a_2_4·c_4_9·b_1_2·a_5_11 + c_4_9·a_1_12·b_1_2·a_5_14
       + c_4_92·a_1_1·a_3_5 + c_4_92·a_1_13·b_1_2
  92. a_5_12·a_7_23 + a_6_19·b_1_2·a_5_14 + a_6_19·b_1_2·a_5_11 + a_6_19·b_1_23·a_3_5
       + a_4_8·a_8_26 + a_4_7·b_1_23·a_5_11 + a_6_19·a_1_1·a_5_11 + c_4_9·a_1_12·b_1_26
       + c_4_92·a_1_1·a_3_5 + c_4_92·a_1_12·b_1_22
  93. a_5_12·a_7_22 + a_8_26·a_1_1·b_1_23 + a_6_19·b_1_2·a_5_14 + a_6_19·b_1_23·a_3_5
       + a_6_19·a_1_1·b_1_25 + a_1_1·b_1_2·a_5_11·a_5_14 + a_6_19·a_1_1·a_5_14
       + a_6_19·a_1_1·a_5_11 + a_1_12·a_5_11·a_5_14 + c_4_9·a_1_1·a_7_22
       + c_4_9·a_1_1·b_1_22·a_5_14 + c_4_9·a_6_19·a_1_1·b_1_2 + c_4_9·a_1_13·a_5_14
       + c_4_92·a_1_12·b_1_22 + c_4_92·a_1_13·b_1_2
  94. a_5_14·a_7_23 + a_5_12·a_7_23 + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_11
       + a_4_7·a_8_26 + a_1_1·b_1_2·a_5_11·a_5_14 + a_6_19·a_1_1·a_5_14
       + a_1_12·a_5_11·a_5_14 + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22
       + c_4_9·a_1_1·b_1_22·a_5_14 + a_2_4·c_4_9·b_1_2·a_5_11 + c_4_9·a_1_12·b_1_2·a_5_14
       + c_4_92·a_1_1·a_3_5
  95. a_5_14·a_7_23 + a_5_12·a_7_23 + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_11
       + a_4_8·a_8_27 + a_1_1·b_1_2·a_5_11·a_5_14 + a_6_19·a_1_1·a_5_14
       + a_1_12·a_5_11·a_5_14 + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22
       + c_4_9·a_1_1·b_1_22·a_5_14 + a_2_4·c_4_9·b_1_2·a_5_11 + c_4_9·a_1_12·b_1_2·a_5_11
       + c_4_92·a_1_1·a_3_5 + c_4_92·a_1_13·b_1_2
  96. b_1_27·a_5_11 + a_8_27·b_1_24 + a_6_19·b_1_26 + a_5_14·a_7_22 + a_5_11·a_7_22
       + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_11 + a_4_7·b_1_23·a_5_11
       + a_1_1·b_1_2·a_5_11·a_5_14 + a_1_12·a_5_11·a_5_14 + c_4_9·a_6_19·b_1_22
       + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·b_1_22·a_5_11 + c_4_9·a_1_12·b_1_26
       + c_4_9·a_1_12·b_1_2·a_5_14 + c_4_9·a_1_12·b_1_2·a_5_11 + c_4_92·a_1_1·b_1_23
       + c_4_92·a_1_1·a_3_5
  97. a_5_14·a_7_23 + a_5_11·a_7_23 + a_8_27·a_1_1·b_1_23 + a_6_19·b_1_2·a_5_14
       + a_6_19·b_1_2·a_5_11 + a_6_19·a_1_1·b_1_25 + a_1_12·a_5_11·a_5_14
       + c_4_9·a_3_5·a_5_11 + c_4_9·a_1_1·a_7_22 + c_4_9·a_1_1·b_1_22·a_5_14
       + a_2_4·c_4_9·b_1_2·a_5_11
  98. a_5_14·a_7_23 + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_14 + a_6_19·b_1_23·a_3_5
       + a_1_1·b_1_2·a_5_11·a_5_14 + a_8_27·a_1_12·b_1_22 + a_6_19·a_1_1·a_5_14
       + a_1_12·a_5_11·a_5_14 + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22
       + c_4_9·a_1_1·b_1_22·a_5_14 + c_4_9·a_1_12·b_1_26 + a_2_4·c_4_9·b_1_2·a_5_11
       + c_4_9·a_1_12·b_1_2·a_5_14 + c_4_9·a_1_12·b_1_2·a_5_11 + c_4_9·a_1_13·a_5_14
       + c_4_92·a_1_12·b_1_22
  99. a_5_14·a_7_23 + b_1_22·a_5_11·a_5_14 + a_6_19·b_1_2·a_5_14 + a_6_19·b_1_23·a_3_5
       + a_4_7·b_1_23·a_5_11 + a_4_7·a_8_27 + a_1_1·b_1_2·a_5_11·a_5_14
       + c_8_29·a_1_12·b_1_22 + c_4_9·a_1_1·a_7_22 + c_4_9·a_1_1·b_1_22·a_5_14
       + c_4_9·a_1_12·b_1_26 + c_4_9·a_1_12·b_1_2·a_5_14 + c_4_9·a_1_12·b_1_2·a_5_11
       + c_4_9·a_1_13·a_5_14 + c_4_92·a_1_12·b_1_22 + c_4_92·a_1_13·b_1_2
  100. a_6_19·a_7_23 + a_6_19·b_1_22·a_5_14 + a_6_19·b_1_22·a_5_11 + a_6_192·b_1_2
       + a_6_19·a_1_1·b_1_2·a_5_14 + a_6_19·a_1_1·b_1_2·a_5_11 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_4_9·a_6_19·a_3_5 + c_4_9·a_1_12·b_1_22·a_5_14 + c_4_9·a_1_13·b_1_2·a_5_14
  101. a_8_26·a_1_1·b_1_24 + a_6_19·a_7_22 + a_6_19·b_1_22·a_5_14 + a_6_19·a_1_1·b_1_26
       + a_6_192·b_1_2 + a_1_1·a_5_11·a_7_22 + a_6_19·a_1_1·b_1_2·a_5_14
       + c_4_9·a_1_1·b_1_23·a_5_14 + c_4_9·a_8_26·a_1_1 + c_4_9·a_6_19·a_1_1·b_1_22
       + a_4_7·c_4_9·a_5_11 + c_4_9·a_1_12·b_1_22·a_5_14 + c_4_92·a_1_1·b_1_2·a_3_5
       + a_4_8·c_4_92·a_1_1 + a_4_7·c_4_92·a_1_1
  102. a_8_26·a_5_12 + a_6_19·a_7_22 + a_6_192·b_1_2 + a_1_1·a_5_11·a_7_22
       + a_6_19·a_1_1·b_1_2·a_5_14 + a_6_19·a_1_1·b_1_2·a_5_11 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_4_9·a_6_19·a_3_5 + c_4_9·a_6_19·a_1_1·b_1_22 + a_4_7·c_4_9·a_5_11
       + c_4_9·a_1_1·a_3_5·a_5_11 + c_4_9·a_1_12·b_1_22·a_5_14
       + c_4_9·a_1_13·b_1_2·a_5_14 + a_4_8·c_4_92·a_1_1 + a_4_7·c_4_92·a_1_1
  103. a_8_26·a_5_14 + a_6_19·a_7_22 + a_6_19·b_1_22·a_5_14 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_8_29·a_1_1·b_1_2·a_3_5 + c_8_29·a_1_12·b_1_23 + c_4_9·a_1_1·b_1_23·a_5_14
       + c_4_9·a_6_19·a_1_1·b_1_22 + a_4_8·c_4_9·a_5_11 + a_4_7·c_8_29·a_1_1
       + c_4_9·a_1_1·a_3_5·a_5_11 + c_4_9·a_1_12·b_1_22·a_5_11 + a_4_8·c_4_92·a_1_1
  104. b_1_2·a_5_11·a_7_22 + a_8_26·a_5_11 + a_6_19·b_1_22·a_5_11
       + a_1_1·b_1_22·a_5_11·a_5_14 + a_6_19·a_1_1·b_1_2·a_5_14 + a_6_19·a_1_1·b_1_2·a_5_11
       + a_1_12·b_1_2·a_5_11·a_5_14 + c_4_9·b_1_2·a_3_5·a_5_11 + c_4_9·a_8_27·a_1_1
       + c_4_9·a_6_19·a_1_1·b_1_22 + a_4_7·c_8_29·a_1_1 + a_4_7·c_4_9·a_5_11
       + c_4_9·a_1_12·b_1_22·a_5_11 + c_4_9·a_1_13·b_1_2·a_5_14
       + c_4_92·a_1_1·b_1_2·a_3_5
  105. a_8_27·a_5_11 + a_8_27·a_1_1·b_1_24 + a_6_19·b_1_22·a_5_11 + a_6_19·a_1_1·b_1_26
       + a_6_192·b_1_2 + a_1_1·b_1_22·a_5_11·a_5_14 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_8_29·a_1_12·b_1_23 + c_4_9·b_1_2·a_3_5·a_5_11 + c_4_9·a_6_19·a_3_5
       + a_4_8·c_8_29·a_1_1 + a_4_7·c_4_9·a_5_11 + c_4_9·a_1_1·a_3_5·a_5_11
       + c_4_9·a_1_12·b_1_22·a_5_14 + c_4_9·a_1_12·b_1_22·a_5_11
       + c_4_92·a_1_1·b_1_2·a_3_5 + a_4_8·c_4_92·a_1_1
  106. a_8_27·a_5_12 + a_6_19·b_1_22·a_5_11 + a_6_192·b_1_2 + a_1_1·a_5_11·a_7_22
       + a_1_1·b_1_22·a_5_11·a_5_14 + a_8_27·a_1_12·b_1_23 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_4_9·a_6_19·a_3_5 + a_4_8·c_4_9·a_5_11 + c_4_9·a_1_1·a_3_5·a_5_11
       + c_4_9·a_1_12·b_1_22·a_5_11 + a_4_7·c_4_92·a_1_1
  107. b_1_23·a_5_11·a_5_14 + a_8_27·a_5_14 + a_8_26·a_1_1·b_1_24 + a_6_19·a_7_22
       + a_6_19·a_1_1·b_1_26 + a_6_192·b_1_2 + a_1_12·b_1_2·a_5_11·a_5_14
       + c_4_9·a_6_19·a_3_5 + c_4_9·a_6_19·a_1_1·b_1_22 + a_4_8·c_8_29·a_1_1
       + a_4_8·c_4_9·a_5_11 + a_4_7·c_4_9·a_5_11 + c_4_9·a_1_1·a_3_5·a_5_11
       + c_4_9·a_1_12·b_1_22·a_5_11 + c_4_92·a_1_1·b_1_2·a_3_5 + c_4_92·a_1_12·b_1_23
       + a_4_8·c_4_92·a_1_1 + a_4_7·c_4_92·a_1_1
  108. a_7_232 + a_7_222 + a_8_27·a_1_1·b_1_25 + a_6_19·b_1_2·a_7_22
       + a_6_19·a_1_1·b_1_27 + a_6_19·a_8_26 + a_8_26·a_1_1·a_5_11
       + c_8_29·a_1_1·b_1_22·a_3_5 + c_4_9·a_6_19·b_1_2·a_3_5 + a_4_7·c_4_9·b_1_2·a_5_11
       + c_4_9·a_1_1·b_1_2·a_3_5·a_5_11 + c_4_9·a_1_12·b_1_23·a_5_14
       + a_4_8·c_8_29·a_1_12 + c_4_92·a_1_1·b_1_22·a_3_5 + a_4_8·c_4_92·a_1_1·b_1_2
       + a_4_7·c_4_92·a_1_1·b_1_2
  109. a_7_22·a_7_23 + a_7_222 + a_8_26·b_1_2·a_5_11 + a_6_19·b_1_2·a_7_22
       + a_6_19·b_1_23·a_5_11 + a_6_19·a_8_26 + a_6_192·b_1_22
       + a_6_19·a_1_1·b_1_22·a_5_14 + a_6_19·a_1_1·b_1_22·a_5_11
       + a_1_12·b_1_22·a_5_11·a_5_14 + c_8_29·a_1_12·b_1_24
       + c_4_9·b_1_22·a_3_5·a_5_11 + c_4_9·a_8_27·a_1_1·b_1_2 + c_4_9·a_8_26·a_1_1·b_1_2
       + c_4_9·a_6_19·b_1_2·a_3_5 + a_4_7·c_8_29·a_1_1·b_1_2 + a_4_7·c_4_9·b_1_2·a_5_11
       + c_4_9·a_1_1·b_1_2·a_3_5·a_5_11 + c_4_92·a_1_1·b_1_22·a_3_5
       + c_4_92·a_1_12·b_1_24
  110. a_7_222 + a_6_19·a_1_1·b_1_22·a_5_14 + a_6_19·a_1_1·b_1_22·a_5_11
       + a_1_12·b_1_22·a_5_11·a_5_14 + c_8_29·a_1_1·b_1_22·a_3_5
       + c_4_9·a_6_19·b_1_2·a_3_5 + c_4_9·a_1_12·b_1_23·a_5_14 + c_4_9·a_8_27·a_1_12
       + a_4_8·c_4_9·a_1_1·a_5_11 + c_4_92·a_1_1·b_1_22·a_3_5 + c_4_92·a_1_12·b_1_24
  111. a_6_19·b_1_23·a_5_11 + a_6_19·a_8_27 + a_6_192·b_1_22 + a_8_26·a_1_1·a_5_11
       + a_6_19·a_1_1·b_1_22·a_5_14 + a_6_19·a_1_1·b_1_22·a_5_11
       + c_4_9·a_6_19·b_1_2·a_3_5 + a_4_8·c_4_9·b_1_2·a_5_11
       + c_4_9·a_1_1·b_1_2·a_3_5·a_5_11 + c_4_9·a_1_12·b_1_23·a_5_14
       + a_4_8·c_4_9·a_1_1·a_5_11 + a_4_7·c_4_92·a_1_1·b_1_2
  112. a_7_222 + a_6_19·b_1_2·a_7_22 + a_6_19·a_8_26 + a_6_192·b_1_22
       + a_8_27·a_1_1·a_5_14 + a_6_19·a_3_5·a_5_11 + c_8_29·a_1_1·b_1_22·a_3_5
       + c_4_9·a_6_19·a_1_1·b_1_23 + a_4_7·c_4_9·b_1_2·a_5_11
       + c_4_9·a_1_1·b_1_2·a_3_5·a_5_11 + c_4_92·a_1_1·b_1_22·a_3_5
       + c_4_92·a_1_12·b_1_24 + a_4_8·c_4_92·a_1_1·b_1_2 + a_4_7·c_4_92·a_1_1·b_1_2
       + a_4_8·c_4_92·a_1_12
  113. a_8_27·a_7_22 + a_8_26·b_1_22·a_5_11 + a_6_19·a_8_27·b_1_2 + a_6_19·a_8_26·b_1_2
       + a_8_26·a_1_1·b_1_2·a_5_11 + a_6_19·b_1_2·a_3_5·a_5_11 + a_6_19·a_8_27·a_1_1
       + c_4_9·a_8_27·a_1_1·b_1_22 + c_4_9·a_8_26·a_1_1·b_1_22 + c_4_9·a_6_19·a_5_14
       + c_4_9·a_6_19·a_5_11 + c_4_9·a_6_19·b_1_22·a_3_5 + c_4_9·a_6_19·a_1_1·b_1_24
       + a_4_7·c_4_9·b_1_22·a_5_11 + a_2_4·c_8_29·a_5_11 + c_8_29·a_1_12·a_5_14
       + c_4_9·a_1_1·a_5_11·a_5_14 + c_4_9·a_1_1·b_1_22·a_3_5·a_5_11
       + c_4_9·a_8_27·a_1_12·b_1_2 + a_4_8·c_8_29·a_1_12·b_1_2
       + a_4_8·c_4_9·a_1_1·b_1_2·a_5_11 + c_4_92·a_1_1·b_1_2·a_5_14
       + c_4_92·a_1_1·b_1_2·a_5_11 + a_2_4·c_4_92·a_5_11 + c_4_92·a_1_12·a_5_11
  114. a_8_27·a_7_23 + a_8_27·b_1_22·a_5_14 + a_8_27·a_1_1·b_1_26 + a_8_26·a_7_23
       + a_8_26·a_7_22 + a_8_26·b_1_22·a_5_11 + a_6_19·b_1_24·a_5_14 + a_6_19·a_1_1·b_1_28
       + a_6_19·a_8_26·b_1_2 + a_8_26·a_1_1·b_1_2·a_5_11 + a_6_19·b_1_2·a_3_5·a_5_11
       + a_6_19·a_8_26·a_1_1 + c_4_9·a_6_19·a_5_14 + a_4_7·c_4_9·b_1_22·a_5_11
       + a_2_4·c_8_29·a_5_11 + c_8_29·a_1_12·a_5_11 + c_4_9·a_1_12·b_1_24·a_5_14
       + a_4_8·c_8_29·a_1_12·b_1_2 + a_4_8·c_4_9·a_1_1·b_1_2·a_5_11
       + c_4_92·a_1_1·b_1_2·a_5_14 + c_4_92·a_1_12·b_1_25 + c_4_9·c_8_29·a_1_13
       + a_4_8·c_4_92·a_1_12·b_1_2
  115. a_8_26·a_7_23 + a_8_26·a_7_22 + a_8_26·b_1_22·a_5_11 + a_6_19·b_1_24·a_5_14
       + a_6_19·a_8_26·b_1_2 + a_8_27·a_1_1·b_1_2·a_5_14 + a_8_26·a_1_1·b_1_2·a_5_11
       + c_8_29·a_1_12·b_1_25 + c_4_9·a_6_19·a_5_14 + c_4_9·a_6_19·b_1_22·a_3_5
       + c_8_29·a_1_12·a_5_14 + c_8_29·a_1_12·a_5_11 + c_4_9·a_1_1·a_5_11·a_5_14
       + c_4_9·a_1_1·b_1_22·a_3_5·a_5_11 + a_4_8·c_8_29·a_1_12·b_1_2
       + c_4_92·a_1_1·b_1_2·a_5_14 + c_4_9·c_8_29·a_1_13 + c_4_92·a_1_12·a_5_11
       + c_4_93·a_1_13
  116. a_8_26·a_7_22 + a_6_19·a_8_26·b_1_2 + a_6_192·b_1_23 + a_6_19·b_1_2·a_3_5·a_5_11
       + a_6_19·a_8_27·a_1_1 + a_8_27·a_1_12·a_5_14 + a_6_19·c_8_29·a_1_1
       + c_4_9·a_6_19·a_5_14 + c_4_9·a_6_19·b_1_22·a_3_5 + c_4_9·a_6_19·a_1_1·b_1_24
       + a_2_4·c_8_29·a_5_11 + c_4_9·a_1_1·a_5_11·a_5_14 + c_4_9·a_1_1·b_1_22·a_3_5·a_5_11
       + c_4_9·a_8_27·a_1_12·b_1_2 + a_4_8·c_4_9·a_1_1·b_1_2·a_5_11
       + c_4_9·c_8_29·a_1_12·b_1_2 + c_4_92·a_1_1·b_1_2·a_5_14 + c_4_92·a_6_19·a_1_1
       + c_4_9·c_8_29·a_1_13 + a_4_8·c_4_92·a_1_12·b_1_2 + c_4_93·a_1_12·b_1_2
  117. a_8_27·a_1_1·b_1_27 + a_8_272 + a_8_262 + a_6_19·a_1_1·b_1_29
       + a_6_192·b_1_24 + a_6_19·a_5_11·a_5_14 + a_6_19·b_1_22·a_3_5·a_5_11
       + c_8_29·a_1_12·b_1_26 + a_6_19·c_8_29·a_1_1·b_1_2 + c_8_29·a_1_12·b_1_2·a_5_14
       + c_4_9·a_1_1·b_1_2·a_5_11·a_5_14 + c_4_9·a_8_27·a_1_12·b_1_22
       + c_4_9·a_6_19·a_1_1·a_5_11 + c_4_9·c_8_29·a_1_12·b_1_22
       + c_4_92·a_1_12·b_1_2·a_5_11
  118. a_8_27·a_1_1·b_1_27 + a_8_272 + a_6_19·a_1_1·b_1_29 + a_6_19·a_5_11·a_5_14
       + a_6_19·b_1_22·a_3_5·a_5_11 + a_6_19·a_8_27·a_1_1·b_1_2 + a_6_19·a_8_26·a_1_1·b_1_2
       + c_8_29·a_1_12·b_1_26 + c_4_9·a_6_192 + c_4_9·a_1_1·b_1_2·a_5_11·a_5_14
       + c_4_9·a_6_19·a_1_1·a_5_11 + c_4_92·a_1_12·b_1_26 + c_4_92·a_6_19·a_1_1·b_1_2
       + c_4_92·a_1_12·b_1_2·a_5_14 + c_4_92·a_1_12·b_1_2·a_5_11
       + c_4_92·a_1_13·a_5_14
  119. a_8_26·b_1_23·a_5_11 + a_8_26·a_8_27 + a_8_262 + a_6_19·a_8_26·b_1_22
       + a_6_192·b_1_24 + a_8_26·a_1_1·b_1_22·a_5_11 + a_6_19·c_8_29·a_1_1·b_1_2
       + c_4_9·a_8_26·a_1_1·b_1_23 + c_4_9·a_6_19·b_1_2·a_5_14
       + c_4_9·a_6_19·a_1_1·b_1_25 + c_4_9·a_6_192 + c_8_29·a_1_12·b_1_2·a_5_11
       + c_4_9·a_6_19·a_1_1·a_5_14 + c_8_29·a_1_13·a_5_14 + c_4_9·a_1_12·a_5_11·a_5_14
       + c_4_9·c_8_29·a_1_12·b_1_22 + c_4_92·a_1_1·b_1_22·a_5_14
       + a_2_4·c_4_92·b_1_2·a_5_11 + c_4_92·a_1_13·a_5_14 + c_4_93·a_1_12·b_1_22
       + c_4_93·a_1_13·b_1_2

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 16.
  • 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_9, a Duflot regular element of degree 4
    2. c_8_29, a Duflot regular element of degree 8
    3. b_1_22, 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 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. a_4_7 → 0, an element of degree 4
  8. a_4_8 → 0, an element of degree 4
  9. c_4_9 → c_1_14, an element of degree 4
  10. a_5_12 → 0, an element of degree 5
  11. a_5_11 → 0, an element of degree 5
  12. a_5_14 → 0, an element of degree 5
  13. a_6_19 → 0, an element of degree 6
  14. a_7_22 → 0, an element of degree 7
  15. a_7_23 → 0, an element of degree 7
  16. a_8_26 → 0, an element of degree 8
  17. a_8_27 → 0, an element of degree 8
  18. c_8_29 → c_1_18 + 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. b_1_2 → c_1_2, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. a_4_7 → 0, an element of degree 4
  8. a_4_8 → 0, an element of degree 4
  9. c_4_9 → c_1_14, an element of degree 4
  10. a_5_12 → 0, an element of degree 5
  11. a_5_11 → 0, an element of degree 5
  12. a_5_14 → 0, an element of degree 5
  13. a_6_19 → 0, an element of degree 6
  14. a_7_22 → 0, an element of degree 7
  15. a_7_23 → 0, an element of degree 7
  16. a_8_26 → 0, an element of degree 8
  17. a_8_27 → 0, an element of degree 8
  18. c_8_29 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + 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