Cohomology of group number 1964 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 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t5  +  t4  +  2·t2  +  2·t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-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 16 minimal generators of maximal degree 8:

  1. a_1_1, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_0, an element of degree 1
  4. b_1_3, an element of degree 1
  5. a_4_8, a nilpotent element of degree 4
  6. a_4_9, a nilpotent element of degree 4
  7. a_4_11, a nilpotent element of degree 4
  8. b_4_10, an element of degree 4
  9. c_4_12, a Duflot regular element of degree 4
  10. a_5_14, a nilpotent element of degree 5
  11. a_5_20, a nilpotent element of degree 5
  12. b_5_19, an element of degree 5
  13. a_8_30, a nilpotent element of degree 8
  14. a_8_26, a nilpotent element of degree 8
  15. b_8_36, an element of degree 8
  16. c_8_38, 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 82 minimal relations of maximal degree 16:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_3 + a_1_1·b_1_0 + a_1_22 + a_1_1·a_1_2 + a_1_12
  3. a_1_12·b_1_0 + a_1_12·a_1_2
  4. a_1_22·b_1_3 + a_1_1·a_1_2·b_1_3 + a_1_12·b_1_3 + a_1_12·a_1_2 + a_1_13
  5. a_1_13·b_1_32
  6. a_1_1·b_1_04 + a_4_8·b_1_0
  7. a_4_9·a_1_1 + a_4_8·a_1_2
  8. a_1_1·b_1_04 + a_4_9·b_1_0
  9. a_4_9·a_1_2 + a_4_8·a_1_2 + a_4_8·a_1_1
  10. a_4_8·b_1_3 + a_4_11·a_1_1
  11. a_4_9·b_1_3 + a_1_12·b_1_33 + a_4_11·a_1_2 + a_4_8·a_1_2
  12. a_1_1·b_1_04 + b_4_10·a_1_1 + a_4_11·b_1_0 + a_4_8·a_1_2
  13. b_4_10·b_1_3 + a_1_1·b_1_04 + a_4_11·b_1_0 + a_4_9·b_1_3 + a_1_12·b_1_33
       + a_4_8·a_1_1
  14. b_4_10·a_1_2 + a_4_8·a_1_2 + a_4_8·a_1_1
  15. a_1_2·a_5_14 + a_1_1·a_5_20 + a_4_11·a_1_2·b_1_3 + c_4_12·a_1_22 + c_4_12·a_1_1·a_1_2
       + c_4_12·a_1_12
  16. b_1_0·a_5_20 + a_4_8·a_1_12 + c_4_12·a_1_1·b_1_0
  17. b_1_0·a_5_14 + a_1_1·b_5_19 + a_4_11·b_1_02 + a_4_8·b_1_02 + a_1_1·a_5_14
       + a_4_11·a_1_1·a_1_2
  18. b_1_3·b_5_19 + b_1_3·a_5_14 + b_1_0·a_5_14 + a_4_11·b_1_02 + a_4_8·b_1_02
       + a_1_2·a_5_20 + a_1_2·a_5_14 + a_1_1·a_5_14 + a_4_11·a_1_1·b_1_3 + a_4_11·a_1_1·a_1_2
       + a_4_11·a_1_12
  19. a_1_2·b_5_19 + a_1_2·a_5_14 + a_4_11·a_1_1·a_1_2 + a_4_11·a_1_12 + a_4_8·a_1_12
  20. a_1_2·b_1_3·a_5_20 + a_1_1·b_1_3·a_5_20 + a_1_1·b_1_3·a_5_14 + a_4_11·a_1_1·b_1_32
       + a_1_12·a_5_20 + a_1_12·a_5_14 + a_4_11·a_1_12·b_1_3 + a_4_11·a_1_13
       + c_4_12·a_1_12·a_1_2
  21. a_1_12·b_5_19 + a_1_12·a_5_20 + a_1_12·a_5_14 + c_4_12·a_1_12·a_1_2
       + c_4_12·a_1_13
  22. a_4_82
  23. a_4_92
  24. a_4_9·a_4_11 + a_4_8·a_4_11 + c_4_12·a_1_13·b_1_3
  25. a_4_8·a_4_9 + a_4_11·a_1_13·b_1_3
  26. a_4_11·b_1_04 + a_4_8·b_1_04 + a_4_8·b_4_10
  27. a_4_11·b_1_04 + a_4_9·b_4_10 + a_4_8·b_1_04
  28. a_4_8·a_4_11 + a_4_8·a_4_9 + a_1_12·b_1_3·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_32
       + a_4_11·a_1_12·b_1_32
  29. a_1_1·b_1_32·a_5_20 + a_1_1·b_1_32·a_5_14 + a_1_12·b_1_36 + a_4_11·a_1_2·b_1_33
       + a_4_11·a_1_1·b_1_33 + a_4_112 + a_4_8·a_4_11 + a_4_8·a_4_9
       + a_4_11·a_1_1·a_1_2·b_1_32 + a_4_11·a_1_12·b_1_32 + c_4_12·a_1_1·a_1_2·b_1_32
       + c_4_12·a_1_12·b_1_32 + c_4_12·a_1_13·b_1_3
  30. a_1_1·b_1_02·b_5_19 + a_4_11·b_1_04 + a_4_11·b_4_10 + a_4_8·b_1_04 + a_4_8·a_4_11
       + a_4_11·a_1_12·b_1_32 + c_4_12·a_1_13·b_1_3
  31. b_1_03·b_5_19 + b_4_102 + a_4_11·b_1_04 + a_4_11·b_4_10 + a_4_8·a_4_11
       + a_4_11·a_1_12·b_1_32 + c_4_12·a_1_13·b_1_3
  32. a_4_9·a_5_14 + a_4_8·a_5_20 + a_4_11·a_1_1·a_1_2·b_1_33
  33. a_4_9·a_5_20 + a_4_9·a_5_14 + a_4_8·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_33
       + a_4_112·a_1_1
  34. b_4_10·a_5_20 + a_4_9·a_5_14 + a_4_8·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_33
       + a_4_112·a_1_1 + a_4_11·c_4_12·b_1_0 + a_4_8·c_4_12·b_1_0
  35. a_4_11·b_4_10·b_1_0 + a_4_8·b_5_19 + a_4_8·b_4_10·b_1_0 + a_4_8·a_5_14
  36. a_4_11·b_4_10·b_1_0 + a_4_9·b_5_19 + a_4_8·b_4_10·b_1_0 + a_4_9·a_5_14
  37. b_4_10·a_5_14 + a_4_11·b_5_19 + a_4_11·a_5_14 + a_4_9·a_5_14
       + a_4_11·a_1_1·a_1_2·b_1_33
  38. a_1_1·b_1_33·a_5_14 + a_8_30·a_1_1 + a_4_11·a_1_1·b_1_34 + a_4_8·a_5_14
       + a_4_11·a_1_1·a_1_2·b_1_33 + a_4_112·a_1_1 + c_4_12·a_1_12·b_1_33
       + a_4_11·c_4_12·a_1_1 + a_4_8·c_4_12·a_1_2
  39. b_1_34·a_5_14 + a_8_30·b_1_3 + a_4_11·b_1_35 + a_1_1·b_1_33·a_5_14
       + a_1_12·b_1_37 + a_4_11·a_5_14 + a_4_11·a_1_1·b_1_34
       + a_4_11·a_1_1·a_1_2·b_1_33 + a_4_112·a_1_2 + c_4_12·a_1_1·b_1_34
       + a_4_11·c_4_12·b_1_3 + c_4_12·a_1_1·a_1_2·b_1_33 + c_4_12·a_1_12·b_1_33
       + a_4_11·c_4_12·a_1_2
  40. a_8_30·b_1_0 + a_4_11·b_4_10·b_1_0 + a_4_11·c_4_12·b_1_0
  41. a_1_1·b_1_33·a_5_14 + a_1_12·b_1_37 + a_8_30·a_1_2 + a_4_11·a_1_2·b_1_34
       + a_4_11·a_1_1·b_1_34 + a_4_112·b_1_3 + a_4_9·a_5_14 + c_4_12·a_1_12·b_1_33
       + a_4_11·c_4_12·a_1_2 + a_4_8·c_4_12·a_1_2 + a_4_8·c_4_12·a_1_1
  42. a_1_1·b_1_33·a_5_14 + a_8_26·a_1_1 + a_4_9·a_5_14 + a_4_112·a_1_1
       + c_4_12·a_1_12·b_1_33 + a_4_11·c_4_12·a_1_1 + a_4_8·c_4_12·a_1_2
  43. b_1_34·a_5_14 + a_8_26·b_1_3 + a_1_1·b_1_33·a_5_14 + a_4_11·a_5_20
       + a_4_11·a_1_1·b_1_34 + a_4_9·a_5_14 + a_4_112·a_1_1 + c_4_12·a_1_1·b_1_34
       + a_4_11·c_4_12·b_1_3 + c_4_12·a_1_1·a_1_2·b_1_33 + c_4_12·a_1_12·b_1_33
       + a_4_11·c_4_12·a_1_2 + a_4_8·c_4_12·a_1_2 + a_4_8·c_4_12·a_1_1
  44. a_8_26·b_1_0 + a_4_11·b_4_10·b_1_0 + a_4_8·b_4_10·b_1_0 + a_4_11·c_4_12·b_1_0
       + a_4_8·c_4_12·b_1_0
  45. a_1_1·b_1_33·a_5_14 + a_1_12·b_1_37 + a_8_26·a_1_2 + a_4_11·a_1_1·b_1_34
       + a_4_112·b_1_3 + a_4_9·a_5_14 + a_4_8·a_5_14 + a_4_112·a_1_2 + a_4_112·a_1_1
       + c_4_12·a_1_12·b_1_33 + a_4_11·c_4_12·a_1_2 + a_4_8·c_4_12·a_1_2
       + a_4_8·c_4_12·a_1_1
  46. b_8_36·a_1_1 + b_4_10·a_5_14 + a_4_11·b_4_10·b_1_0 + a_4_8·b_4_10·b_1_0
       + a_1_1·b_1_33·a_5_14 + a_1_12·b_1_37 + a_4_11·a_1_2·b_1_34 + a_4_112·b_1_3
       + a_4_8·a_5_14 + a_4_112·a_1_2 + a_4_11·c_4_12·b_1_0 + a_4_8·c_4_12·b_1_0
       + c_4_12·a_1_1·a_1_2·b_1_33 + c_4_12·a_1_12·b_1_33 + a_4_11·c_4_12·a_1_1
       + a_4_8·c_4_12·a_1_1
  47. b_8_36·b_1_3 + b_1_34·a_5_20 + b_4_10·a_5_14 + a_4_11·b_1_35 + a_4_11·b_4_10·b_1_0
       + a_4_8·b_4_10·b_1_0 + a_4_11·a_5_20 + a_4_11·a_5_14 + a_4_11·a_1_2·b_1_34
       + a_4_11·a_1_1·b_1_34 + a_4_112·b_1_3 + a_4_8·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_33
       + a_4_112·a_1_2 + a_4_11·c_4_12·b_1_3 + a_4_11·c_4_12·b_1_0 + a_4_8·c_4_12·b_1_0
       + c_4_12·a_1_1·a_1_2·b_1_33 + a_4_11·c_4_12·a_1_1
  48. b_8_36·b_1_0 + b_4_10·b_5_19 + b_4_10·a_5_14 + a_4_11·b_4_10·b_1_0 + a_4_8·b_1_05
       + a_4_8·b_4_10·b_1_0 + b_4_10·c_4_12·b_1_0 + a_4_11·c_4_12·b_1_0 + a_4_8·c_4_12·b_1_0
  49. b_8_36·a_1_2 + a_1_12·b_1_37 + a_4_112·b_1_3 + a_4_8·a_5_14 + a_4_112·a_1_1
       + c_4_12·a_1_1·a_1_2·b_1_33 + c_4_12·a_1_12·b_1_33 + a_4_11·c_4_12·a_1_2
       + a_4_8·c_4_12·a_1_2
  50. a_5_20·b_5_19 + a_5_14·a_5_20 + a_4_11·a_1_1·a_5_20 + a_4_11·a_1_1·a_5_14
       + a_4_112·a_1_1·b_1_3 + c_4_12·a_1_1·b_5_19 + c_4_12·a_1_1·a_5_14
       + a_4_11·c_4_12·a_1_1·a_1_2 + a_4_8·c_4_12·a_1_12
  51. a_5_142 + a_4_112·b_1_32 + a_4_112·a_1_1·b_1_3 + a_4_8·a_1_1·a_5_14
       + c_8_38·a_1_12 + c_4_12·a_1_1·a_1_2·b_1_34 + c_4_122·a_1_22
  52. a_5_14·b_5_19 + a_4_8·b_1_0·b_5_19 + a_5_142 + a_4_11·a_1_1·a_5_20
       + a_4_112·a_1_2·b_1_3 + c_8_38·a_1_1·b_1_0 + a_4_8·c_4_12·a_1_12
  53. b_5_192 + b_4_10·b_1_0·b_5_19 + b_4_102·b_1_02 + a_4_11·b_1_0·b_5_19
       + a_4_8·b_1_06 + a_4_8·b_4_10·b_1_02 + a_5_142 + c_8_38·b_1_02
       + a_4_8·c_4_12·b_1_02
  54. a_5_14·a_5_20 + a_4_11·b_1_3·a_5_20 + a_4_11·a_1_1·a_1_2·b_1_34
       + a_4_112·a_1_2·b_1_3 + a_4_112·a_1_1·b_1_3 + a_4_8·a_1_1·a_5_14 + c_8_38·a_1_1·a_1_2
       + c_4_12·a_1_2·a_5_20 + c_4_12·a_1_1·a_5_20 + c_4_12·a_1_1·a_5_14
       + c_4_12·a_1_1·a_1_2·b_1_34 + c_4_12·a_1_12·b_1_34 + a_4_11·c_4_12·a_1_1·b_1_3
       + a_4_8·c_4_12·a_1_12 + c_4_122·a_1_12
  55. a_5_202 + a_4_11·a_1_1·a_1_2·b_1_34 + a_4_112·a_1_2·b_1_3 + c_8_38·a_1_22
       + c_4_12·a_1_12·b_1_34 + a_4_8·c_4_12·a_1_12 + c_4_122·a_1_22
       + c_4_122·a_1_12
  56. a_4_8·a_8_30 + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_112·a_1_1·b_1_33
       + c_4_12·a_1_12·b_1_3·a_5_14 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32
       + a_4_11·c_4_12·a_1_13·b_1_3
  57. a_4_9·a_8_30 + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_36
       + a_4_112·a_1_1·b_1_33 + a_4_113 + c_4_12·a_1_12·b_1_3·a_5_14
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32 + a_4_11·c_4_12·a_1_13·b_1_3
       + c_4_122·a_1_13·b_1_3
  58. b_4_10·a_8_30 + a_4_11·b_4_102 + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_113
       + a_4_11·b_4_10·c_4_12 + a_4_11·c_4_12·a_1_12·b_1_32 + a_4_11·c_4_12·a_1_13·b_1_3
  59. a_4_11·b_1_33·a_5_14 + a_4_11·a_8_30 + a_4_112·b_1_34
       + a_4_11·a_1_1·a_1_2·b_1_36 + a_4_112·a_1_2·b_1_33 + a_4_112·a_1_1·b_1_33
       + a_4_11·a_1_12·b_1_3·a_5_14 + a_4_11·c_4_12·a_1_1·b_1_33 + a_4_112·c_4_12
       + c_8_38·a_1_13·b_1_3 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32
       + a_4_11·c_4_12·a_1_12·b_1_32 + a_4_11·c_4_12·a_1_13·b_1_3
  60. a_4_8·a_8_26 + a_4_11·a_1_1·b_1_32·a_5_14 + c_4_12·a_1_12·b_1_3·a_5_14
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32
  61. a_4_9·a_8_26 + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_36 + a_4_113
       + c_4_12·a_1_12·b_1_3·a_5_14 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32
       + c_4_122·a_1_13·b_1_3
  62. b_4_10·a_8_26 + a_4_11·b_4_102 + a_4_8·b_4_102 + a_4_11·a_1_1·b_1_32·a_5_14
       + a_4_112·a_1_2·b_1_33 + a_4_113 + a_4_11·a_1_12·b_1_3·a_5_14
       + a_4_11·b_4_10·c_4_12 + a_4_8·b_4_10·c_4_12 + a_4_11·c_4_12·a_1_12·b_1_32
       + a_4_11·c_4_12·a_1_13·b_1_3
  63. a_4_11·b_1_33·a_5_14 + a_4_11·a_8_26 + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_113
       + a_4_11·c_4_12·a_1_1·b_1_33 + a_4_112·c_4_12 + c_8_38·a_1_13·b_1_3
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32 + a_4_11·c_4_12·a_1_12·b_1_32
       + a_4_11·c_4_12·a_1_13·b_1_3
  64. a_4_11·b_4_102 + a_4_8·b_8_36 + a_4_8·b_4_102 + a_4_11·a_1_1·b_1_32·a_5_14
       + a_4_112·a_1_1·b_1_33 + a_4_113 + a_4_8·b_4_10·c_4_12
       + c_4_12·a_1_12·b_1_3·a_5_14 + a_4_11·c_4_12·a_1_13·b_1_3
  65. a_4_11·b_4_102 + a_4_9·b_8_36 + a_4_8·b_4_102 + a_4_113 + a_4_8·b_4_10·c_4_12
       + c_4_12·a_1_12·b_1_3·a_5_14 + c_4_122·a_1_13·b_1_3
  66. b_4_10·b_8_36 + b_4_102·b_1_04 + b_4_103 + a_4_8·b_1_08 + a_4_8·b_4_10·b_1_04
       + a_4_112·a_1_2·b_1_33 + a_4_112·a_1_1·b_1_33 + a_4_113
       + a_4_11·a_1_12·b_1_3·a_5_14 + c_8_38·b_1_04 + b_4_102·c_4_12
       + a_4_11·b_4_10·c_4_12 + a_4_8·c_4_12·b_1_04 + a_4_8·b_4_10·c_4_12
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32 + a_4_11·c_4_12·a_1_12·b_1_32
  67. a_4_11·b_8_36 + a_4_8·b_4_102 + a_4_11·b_1_33·a_5_20 + a_4_112·b_1_34
       + a_4_11·a_1_1·b_1_32·a_5_14 + a_4_11·a_1_1·a_1_2·b_1_36 + a_4_112·a_1_2·b_1_33
       + a_4_112·a_1_1·b_1_33 + c_8_38·a_1_1·b_1_03 + a_4_11·b_4_10·c_4_12
       + a_4_112·c_4_12 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_32
       + a_4_11·c_4_12·a_1_12·b_1_32 + a_4_11·c_4_12·a_1_13·b_1_3
       + c_4_122·a_1_13·b_1_3
  68. a_8_30·a_5_20 + a_4_11·a_8_30·a_1_1 + a_4_112·a_5_14 + a_4_112·a_1_2·b_1_34
       + a_4_113·b_1_3 + a_4_113·a_1_1 + c_8_38·a_1_1·a_1_2·b_1_33
       + c_4_12·a_1_1·a_1_2·b_1_37 + c_4_12·a_8_30·a_1_1 + a_4_11·c_4_12·a_5_20
       + a_4_11·c_4_12·a_1_2·b_1_34 + a_4_112·c_4_12·b_1_3 + a_4_8·c_8_38·a_1_2
       + a_4_8·c_4_12·a_5_20 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_33 + a_4_112·c_4_12·a_1_2
       + a_4_112·c_4_12·a_1_1 + c_4_122·a_1_1·a_1_2·b_1_33 + c_4_122·a_1_12·b_1_33
       + a_4_11·c_4_122·a_1_1 + a_4_8·c_4_122·a_1_2 + a_4_8·c_4_122·a_1_1
  69. a_8_30·a_5_20 + a_8_30·a_5_14 + a_4_11·a_8_30·b_1_3 + a_4_112·a_5_14
       + a_4_112·a_1_2·b_1_34 + c_8_38·a_1_1·a_1_2·b_1_33 + c_8_38·a_1_12·b_1_33
       + a_4_11·c_4_12·a_5_20 + a_4_11·c_4_12·a_5_14 + a_4_11·c_4_12·a_1_2·b_1_34
       + a_4_8·c_8_38·a_1_2 + a_4_8·c_8_38·a_1_1 + a_4_8·c_4_12·a_5_14 + a_4_112·c_4_12·a_1_2
       + a_4_112·c_4_12·a_1_1 + c_4_122·a_1_12·b_1_33 + a_4_8·c_4_122·a_1_2
  70. a_8_30·b_5_19 + a_4_8·b_4_102·b_1_0 + a_4_11·a_8_30·b_1_3 + a_4_112·a_1_1·b_1_34
       + a_4_113·a_1_1 + a_4_11·c_4_12·b_5_19 + a_4_8·c_8_38·b_1_0 + c_8_38·a_1_12·b_1_33
       + c_4_12·a_1_1·a_1_2·b_1_37 + c_4_12·a_8_30·a_1_1 + a_4_112·c_4_12·b_1_3
       + a_4_8·c_8_38·a_1_1 + a_4_8·c_4_12·a_5_20 + a_4_8·c_4_12·a_5_14
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_33 + a_4_112·c_4_12·a_1_1
       + c_4_122·a_1_1·a_1_2·b_1_33 + a_4_11·c_4_122·a_1_1 + a_4_8·c_4_122·a_1_1
  71. a_8_26·a_5_20 + a_8_30·a_5_20 + a_4_11·b_1_34·a_5_20 + a_4_112·a_1_1·b_1_34
       + a_4_113·b_1_3 + a_4_113·a_1_1 + a_4_8·c_8_38·a_1_1
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_33 + a_4_112·c_4_12·a_1_2 + a_4_8·c_4_122·a_1_2
       + a_4_8·c_4_122·a_1_1
  72. a_8_26·a_5_14 + a_8_30·a_5_20 + a_4_112·b_1_35 + a_4_112·a_5_14
       + a_4_112·a_1_2·b_1_34 + c_8_38·a_1_1·a_1_2·b_1_33 + c_8_38·a_1_12·b_1_33
       + a_4_11·c_4_12·a_5_20 + a_4_11·c_4_12·a_5_14 + a_4_11·c_4_12·a_1_2·b_1_34
       + a_4_11·c_4_12·a_1_1·b_1_34 + a_4_112·c_4_12·b_1_3 + a_4_8·c_4_12·a_5_14
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_33 + a_4_112·c_4_12·a_1_1
       + c_4_122·a_1_12·b_1_33
  73. a_8_26·b_5_19 + a_4_8·b_4_10·b_5_19 + a_4_8·b_4_102·b_1_0 + a_4_112·b_1_35
       + a_4_112·a_1_2·b_1_34 + a_4_112·a_1_1·b_1_34 + a_4_113·a_1_1
       + a_4_11·c_4_12·b_5_19 + a_4_8·c_8_38·b_1_0 + a_4_8·c_4_12·b_5_19
       + c_8_38·a_1_12·b_1_33 + c_4_12·a_1_1·a_1_2·b_1_37 + c_4_12·a_8_30·a_1_1
       + a_4_11·c_4_12·a_1_1·b_1_34 + a_4_8·c_8_38·a_1_2 + a_4_8·c_4_12·a_5_20
       + a_4_112·c_4_12·a_1_2 + a_4_112·c_4_12·a_1_1 + c_4_122·a_1_1·a_1_2·b_1_33
       + a_4_11·c_4_122·a_1_1 + a_4_8·c_4_122·a_1_2 + a_4_8·c_4_122·a_1_1
  74. b_8_36·a_5_20 + a_4_11·b_1_34·a_5_20 + a_4_112·a_5_14 + a_4_112·a_1_2·b_1_34
       + a_4_11·c_4_12·b_5_19 + a_4_8·c_4_12·b_5_19 + c_8_38·a_1_1·a_1_2·b_1_33
       + c_8_38·a_1_12·b_1_33 + c_4_12·a_8_30·a_1_2 + c_4_12·a_8_30·a_1_1
       + a_4_11·c_4_12·a_5_20 + a_4_11·c_4_12·a_5_14 + a_4_11·c_4_12·a_1_2·b_1_34
       + a_4_112·c_4_12·b_1_3 + a_4_8·c_8_38·a_1_1 + a_4_11·c_4_122·b_1_0
       + a_4_8·c_4_122·b_1_0 + c_4_122·a_1_1·a_1_2·b_1_33 + a_4_11·c_4_122·a_1_2
       + a_4_11·c_4_122·a_1_1 + a_4_8·c_4_122·a_1_2
  75. b_8_36·a_5_14 + a_4_8·b_4_10·b_5_19 + a_4_11·b_1_34·a_5_20 + a_4_11·a_8_30·b_1_3
       + a_4_112·b_1_35 + a_4_112·a_5_14 + a_4_112·a_1_2·b_1_34
       + a_4_112·a_1_1·b_1_34 + a_4_113·b_1_3 + a_4_11·c_8_38·b_1_0 + a_4_11·c_4_12·b_5_19
       + a_4_8·c_8_38·b_1_0 + c_8_38·a_1_1·a_1_2·b_1_33 + c_4_12·a_1_1·a_1_2·b_1_37
       + c_4_12·a_8_30·a_1_2 + c_4_12·a_8_30·a_1_1 + a_4_11·c_4_12·a_1_2·b_1_34
       + a_4_11·c_4_12·a_1_1·b_1_34 + a_4_8·c_8_38·a_1_2 + a_4_8·c_8_38·a_1_1
       + a_4_8·c_4_12·a_5_20 + a_4_112·c_4_12·a_1_2 + a_4_112·c_4_12·a_1_1
       + c_4_122·a_1_12·b_1_33 + a_4_11·c_4_122·a_1_2 + a_4_11·c_4_122·a_1_1
       + a_4_8·c_4_122·a_1_2 + a_4_8·c_4_122·a_1_1
  76. b_8_36·b_5_19 + b_4_102·b_1_05 + a_4_8·b_1_09 + a_4_8·b_4_10·b_5_19
       + a_4_8·b_4_10·b_1_05 + a_4_8·b_4_102·b_1_0 + a_4_11·b_1_34·a_5_20
       + a_4_11·a_8_30·b_1_3 + a_4_112·b_1_35 + a_4_112·a_5_14 + c_8_38·b_1_05
       + b_4_10·c_8_38·b_1_0 + b_4_10·c_4_12·b_5_19 + a_4_11·c_8_38·b_1_0
       + a_4_11·c_4_12·b_5_19 + a_4_8·c_4_12·b_5_19 + a_4_8·c_4_12·b_1_05
       + a_4_8·b_4_10·c_4_12·b_1_0 + c_8_38·a_1_1·a_1_2·b_1_33
       + c_4_12·a_1_1·a_1_2·b_1_37 + c_4_12·a_8_30·a_1_2 + c_4_12·a_8_30·a_1_1
       + a_4_11·c_4_12·a_1_2·b_1_34 + a_4_11·c_4_12·a_1_1·b_1_34 + a_4_8·c_8_38·a_1_2
       + a_4_8·c_8_38·a_1_1 + a_4_8·c_4_12·a_5_14 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_33
       + c_4_122·a_1_12·b_1_33 + a_4_11·c_4_122·a_1_2 + a_4_11·c_4_122·a_1_1
       + a_4_8·c_4_122·a_1_2 + a_4_8·c_4_122·a_1_1
  77. a_8_302 + a_4_112·a_1_1·b_1_37 + a_4_113·a_1_1·b_1_33
       + c_8_38·a_1_12·b_1_36 + c_4_12·a_1_1·a_1_2·b_1_310
       + c_4_122·a_1_1·a_1_2·b_1_36 + a_4_112·c_4_122
  78. a_8_30·a_8_26 + a_4_11·a_8_30·b_1_34 + a_4_11·a_8_30·a_1_1·b_1_33
       + a_4_113·b_1_34 + c_8_38·a_1_12·b_1_36 + c_4_12·a_1_1·a_1_2·b_1_310
       + a_4_11·c_8_38·a_1_1·a_1_2·b_1_32 + a_4_11·c_8_38·a_1_12·b_1_32
       + a_4_11·c_4_12·a_1_1·b_1_32·a_5_14 + a_4_11·c_4_12·a_1_1·a_1_2·b_1_36
       + c_4_122·a_1_1·a_1_2·b_1_36 + a_4_112·c_4_122
       + a_4_11·c_4_122·a_1_12·b_1_32 + a_4_11·c_4_122·a_1_13·b_1_3
  79. a_8_262 + a_4_112·b_1_38 + a_4_112·a_1_1·b_1_37 + a_4_113·a_1_1·b_1_33
       + c_8_38·a_1_12·b_1_36 + c_4_12·a_1_1·a_1_2·b_1_310
       + c_4_122·a_1_1·a_1_2·b_1_36 + a_4_112·c_4_122
  80. a_8_30·b_8_36 + a_4_8·b_4_103 + a_4_11·a_8_30·b_1_34
       + a_4_11·a_8_30·a_1_1·b_1_33 + a_4_112·a_1_2·b_1_37 + a_4_112·a_1_1·b_1_37
       + a_4_112·a_8_30 + a_4_113·b_1_34 + a_4_113·a_1_1·b_1_33 + a_4_8·c_4_12·b_8_36
       + a_4_8·b_4_10·c_8_38 + c_8_38·a_1_1·a_1_2·b_1_36 + c_4_12·a_1_1·a_1_2·b_1_310
       + c_4_12·a_8_30·a_1_1·b_1_33 + a_4_11·c_4_12·b_1_33·a_5_20
       + a_4_11·c_4_12·a_1_2·b_1_37 + a_4_11·c_4_12·a_8_30 + a_4_112·c_4_12·b_1_34
       + a_4_11·c_8_38·a_1_12·b_1_32 + a_4_112·c_4_12·a_1_2·b_1_33
       + a_4_112·c_4_12·a_1_1·b_1_33 + a_4_11·c_8_38·a_1_13·b_1_3
       + c_4_12·c_8_38·a_1_1·b_1_03 + a_4_11·b_4_10·c_4_122 + a_4_8·b_4_10·c_4_122
       + c_4_122·a_1_1·a_1_2·b_1_36 + c_4_122·a_1_12·b_1_36
       + a_4_11·c_4_122·a_1_1·b_1_33 + c_4_122·a_1_12·b_1_3·a_5_14
       + a_4_11·c_4_122·a_1_1·a_1_2·b_1_32 + a_4_11·c_4_122·a_1_12·b_1_32
       + a_4_11·c_4_122·a_1_13·b_1_3 + c_4_123·a_1_13·b_1_3
  81. b_8_362 + b_4_103·b_1_04 + a_4_8·b_4_10·b_1_08 + a_4_112·b_1_38
       + a_4_11·a_8_30·a_1_1·b_1_33 + a_4_112·a_1_2·b_1_37 + a_4_112·a_8_30
       + a_4_113·b_1_34 + b_4_10·c_8_38·b_1_04 + b_4_102·c_8_38 + a_4_8·c_8_38·b_1_04
       + a_4_8·b_4_10·c_4_12·b_1_04 + a_4_8·b_4_102·c_4_12 + c_8_38·a_1_1·a_1_2·b_1_36
       + c_8_38·a_1_12·b_1_36 + c_4_12·a_8_30·a_1_2·b_1_33
       + c_4_12·a_8_30·a_1_1·b_1_33 + a_4_11·c_4_12·a_1_2·b_1_37
       + a_4_112·c_4_12·b_1_34 + a_4_11·c_8_38·a_1_13·b_1_3
       + a_4_11·c_4_12·a_1_12·b_1_3·a_5_14 + b_4_102·c_4_122
       + c_4_122·a_1_1·a_1_2·b_1_36 + a_4_11·c_4_122·a_1_2·b_1_33
       + a_4_11·c_4_122·a_1_1·b_1_33 + a_4_112·c_4_122
       + a_4_11·c_4_122·a_1_1·a_1_2·b_1_32 + a_4_11·c_4_122·a_1_13·b_1_3
  82. a_8_26·b_8_36 + a_4_8·b_4_102·b_1_04 + a_4_11·b_1_37·a_5_20
       + a_4_11·a_8_30·b_1_34 + a_4_112·b_1_38 + a_4_11·a_8_30·a_1_1·b_1_33
       + a_4_112·a_1_2·b_1_37 + a_4_112·a_8_30 + a_4_113·b_1_34 + a_4_8·c_8_38·b_1_04
       + a_4_8·b_4_10·c_8_38 + a_4_8·b_4_102·c_4_12 + c_8_38·a_1_1·a_1_2·b_1_36
       + c_4_12·a_1_1·a_1_2·b_1_310 + c_4_12·a_8_30·a_1_1·b_1_33
       + a_4_11·c_4_12·b_1_33·a_5_20 + a_4_11·c_4_12·a_1_2·b_1_37 + a_4_11·c_4_12·a_8_30
       + a_4_11·c_4_12·a_1_1·a_1_2·b_1_36 + a_4_112·c_4_12·a_1_2·b_1_33
       + a_4_11·c_8_38·a_1_13·b_1_3 + c_4_12·c_8_38·a_1_1·b_1_03
       + a_4_11·b_4_10·c_4_122 + a_4_8·b_4_10·c_4_122 + c_4_122·a_1_1·a_1_2·b_1_36
       + c_4_122·a_1_12·b_1_36 + a_4_11·c_4_122·a_1_1·b_1_33
       + a_4_11·c_4_122·a_1_13·b_1_3 + c_4_123·a_1_13·b_1_3

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_12, a Duflot regular element of degree 4
    2. c_8_38, a Duflot regular element of degree 8
    3. b_1_32 + b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 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_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_4_8 → 0, an element of degree 4
  6. a_4_9 → 0, an element of degree 4
  7. a_4_11 → 0, an element of degree 4
  8. b_4_10 → 0, an element of degree 4
  9. c_4_12 → c_1_14, an element of degree 4
  10. a_5_14 → 0, an element of degree 5
  11. a_5_20 → 0, an element of degree 5
  12. b_5_19 → 0, an element of degree 5
  13. a_8_30 → 0, an element of degree 8
  14. a_8_26 → 0, an element of degree 8
  15. b_8_36 → 0, an element of degree 8
  16. c_8_38 → 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_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_4_8 → 0, an element of degree 4
  6. a_4_9 → 0, an element of degree 4
  7. a_4_11 → 0, an element of degree 4
  8. b_4_10 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  9. c_4_12 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  10. a_5_14 → 0, an element of degree 5
  11. a_5_20 → 0, an element of degree 5
  12. b_5_19 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  13. a_8_30 → 0, an element of degree 8
  14. a_8_26 → 0, an element of degree 8
  15. b_8_36 → c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24 + c_1_03·c_1_25
       + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22
       + c_1_05·c_1_23 + c_1_06·c_1_22, an element of degree 8
  16. c_8_38 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_16·c_1_22 + c_1_18 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22
       + c_1_03·c_1_25 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23
       + c_1_04·c_1_12·c_1_22 + c_1_05·c_1_23 + c_1_06·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_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. a_4_8 → 0, an element of degree 4
  6. a_4_9 → 0, an element of degree 4
  7. a_4_11 → 0, an element of degree 4
  8. b_4_10 → 0, an element of degree 4
  9. c_4_12 → c_1_14, an element of degree 4
  10. a_5_14 → 0, an element of degree 5
  11. a_5_20 → 0, an element of degree 5
  12. b_5_19 → 0, an element of degree 5
  13. a_8_30 → 0, an element of degree 8
  14. a_8_26 → 0, an element of degree 8
  15. b_8_36 → 0, an element of degree 8
  16. c_8_38 → 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