Cohomology of group number 200 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 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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) · (t2  +  t  +  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-5,-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 15 minimal generators of maximal degree 6:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_7, a nilpotent element of degree 3
  7. b_3_8, an element of degree 3
  8. b_3_9, an element of degree 3
  9. b_4_13, an element of degree 4
  10. b_4_14, an element of degree 4
  11. c_4_15, a Duflot regular element of degree 4
  12. c_4_16, a Duflot regular element of degree 4
  13. b_5_25, an element of degree 5
  14. b_5_26, an element of degree 5
  15. b_6_38, an 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 65 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_22 + b_1_12·b_1_2
  4. b_1_1·b_1_22 + a_2_4·a_1_0
  5. b_1_1·b_1_22 + a_2_4·b_1_1
  6. b_1_1·b_1_22 + b_2_5·a_1_0
  7. b_2_5·b_1_1·b_1_2 + a_2_42
  8. a_1_0·a_3_7
  9. b_1_1·a_3_7
  10. a_1_0·b_3_8
  11. b_1_1·b_3_8
  12. a_1_0·b_3_9
  13. b_1_22·b_3_9 + b_2_5·b_3_8 + b_2_5·a_3_7 + a_2_4·b_3_8
  14. b_1_1·b_1_2·b_3_9 + a_2_4·a_3_7
  15. b_2_5·a_3_7 + a_2_4·b_3_9
  16. b_4_13·b_1_2 + b_2_5·b_3_8 + b_2_52·b_1_2 + b_1_22·a_3_7 + b_2_5·a_3_7 + a_2_4·b_1_23
  17. b_4_13·a_1_0
  18. b_1_22·a_3_7 + b_4_14·a_1_0 + a_2_4·b_3_8 + a_2_4·a_3_7
  19. b_4_14·b_1_1
  20. a_3_72 + a_2_42·b_2_5
  21. a_3_7·b_3_8 + a_2_4·b_2_5·b_1_22 + a_2_42·b_2_5
  22. b_3_82 + b_2_5·b_1_24 + a_2_42·b_2_5
  23. a_3_7·b_3_9 + a_2_4·b_2_52
  24. b_3_8·b_3_9 + b_2_52·b_1_22 + a_2_4·b_2_5·b_1_22 + a_2_4·b_2_52 + a_2_42·b_2_5
  25. a_2_4·b_1_2·b_3_9 + a_2_4·b_4_13 + a_2_4·b_2_52
  26. b_3_92 + b_4_13·b_1_12 + b_2_5·b_1_1·b_3_9 + b_2_53 + a_2_42·b_2_5
       + c_4_15·b_1_12
  27. b_1_2·b_5_25 + b_2_5·b_1_2·b_3_9 + b_2_5·b_1_2·b_3_8 + b_4_14·a_1_0·b_1_2
       + a_2_4·b_1_24 + a_2_4·b_4_14 + a_2_4·b_2_5·b_1_22
  28. a_1_0·b_5_25
  29. b_1_1·b_5_25 + b_2_5·b_1_2·b_3_9 + b_2_5·b_4_13 + b_2_53 + a_2_4·b_2_5·b_1_22
       + a_2_42·b_2_5
  30. a_1_0·b_5_26 + c_4_15·a_1_0·b_1_2
  31. b_1_1·b_5_26 + b_2_5·b_1_1·b_3_9 + c_4_15·b_1_1·b_1_2
  32. b_4_13·a_3_7 + a_2_4·b_2_5·b_3_9 + a_2_4·b_2_52·b_1_2
  33. b_4_13·b_3_8 + b_2_52·b_3_8 + b_2_52·b_1_23 + a_2_4·b_1_22·b_3_8
       + a_2_4·b_2_5·b_1_23 + a_2_4·b_2_52·b_1_2
  34. a_2_4·b_5_25 + a_2_4·b_2_5·b_3_9 + a_2_4·b_2_5·b_3_8
  35. b_4_14·b_3_9 + b_2_5·b_5_26 + b_2_52·b_3_9 + b_2_52·b_3_8 + b_4_14·a_3_7
       + b_4_14·a_1_0·b_1_22 + a_2_4·b_2_5·b_3_9 + a_2_4·b_2_5·b_1_23 + b_2_5·c_4_15·b_1_2
       + c_4_15·a_1_0·b_1_22 + a_2_4·c_4_15·a_1_0
  36. b_1_22·b_5_26 + b_4_14·b_3_8 + b_2_5·b_1_22·b_3_8 + b_2_52·b_3_8 + b_4_14·a_3_7
       + a_2_4·b_1_25 + a_2_4·b_2_5·b_3_9 + a_2_42·b_3_9 + c_4_15·b_1_23
       + c_4_15·a_1_0·b_1_22 + a_2_4·c_4_16·a_1_0
  37. b_4_14·a_3_7 + b_4_14·a_1_0·b_1_22 + a_2_4·b_5_26 + a_2_4·b_2_5·b_3_9
       + a_2_4·b_2_5·b_3_8 + a_2_42·b_3_9 + c_4_15·a_1_0·b_1_22 + a_2_4·c_4_15·b_1_2
       + a_2_4·c_4_15·a_1_0
  38. b_1_24·b_3_8 + b_6_38·b_1_2 + b_4_14·b_1_23 + b_2_5·b_1_25 + b_2_52·b_3_8
       + b_2_53·b_1_2 + b_4_14·a_3_7 + b_4_14·a_1_0·b_1_22 + a_2_4·b_1_25
       + a_2_4·b_2_5·b_3_9 + a_2_4·b_2_5·b_3_8 + a_2_4·b_2_5·b_1_23 + a_2_4·b_2_52·b_1_2
       + a_2_42·b_3_9 + c_4_16·b_1_23 + b_2_5·c_4_16·b_1_2 + b_2_5·c_4_15·b_1_2
       + a_2_4·c_4_16·a_1_0
  39. b_6_38·a_1_0 + b_4_14·a_1_0·b_1_22 + c_4_16·a_1_0·b_1_22 + a_2_4·c_4_16·a_1_0
       + a_2_4·c_4_15·a_1_0
  40. b_6_38·b_1_1 + b_4_13·b_3_9 + b_2_52·b_3_9 + b_2_53·b_1_2 + a_2_4·b_2_5·b_3_8
       + c_4_15·b_1_13 + b_2_5·c_4_16·b_1_1 + b_2_5·c_4_15·b_1_1 + a_2_4·c_4_15·a_1_0
  41. b_4_132 + b_2_5·b_1_13·b_3_9 + b_2_52·b_1_14 + b_2_53·b_1_22 + b_2_53·b_1_12
       + b_2_54 + c_4_16·b_1_14 + c_4_15·b_1_14
  42. b_4_14·b_1_24 + b_4_142 + b_2_5·b_1_23·b_3_8 + b_2_5·b_1_26
       + b_4_14·a_1_0·b_1_23 + a_2_4·b_1_23·b_3_8 + a_2_4·b_1_26 + a_2_4·b_4_14·b_1_22
       + a_2_4·b_2_5·b_1_2·b_3_8 + c_4_15·b_1_24 + a_2_42·c_4_16
  43. a_3_7·b_5_25 + a_2_4·b_2_52·b_1_22 + a_2_4·b_2_53 + a_2_42·b_2_52
  44. a_3_7·b_5_26 + a_2_4·b_2_5·b_4_14 + a_2_4·b_2_52·b_1_22 + a_2_4·b_2_53
       + c_4_15·b_1_2·a_3_7
  45. b_3_8·b_5_25 + b_4_13·b_4_14 + b_2_5·b_1_2·b_5_26 + b_2_52·b_1_2·b_3_9
       + b_2_52·b_1_2·b_3_8 + b_2_52·b_1_24 + b_2_52·b_4_14 + b_2_53·b_1_22
       + b_4_14·a_1_0·b_1_23 + a_2_4·b_1_23·b_3_8 + a_2_4·b_4_14·b_1_22
       + a_2_4·b_2_5·b_1_2·b_3_8 + a_2_4·b_2_5·b_1_24 + a_2_4·b_2_5·b_4_13
       + a_2_4·b_2_52·b_1_22 + b_2_5·c_4_15·b_1_22 + c_4_15·a_1_0·b_1_23
  46. b_3_8·b_5_25 + b_2_52·b_1_24 + b_2_53·b_1_22 + a_2_4·b_1_2·b_5_26
       + a_2_4·b_1_23·b_3_8 + a_2_4·b_2_5·b_4_13 + a_2_4·b_2_52·b_1_22
       + a_2_4·c_4_15·b_1_22
  47. b_3_8·b_5_26 + b_2_5·b_4_14·b_1_22 + b_2_52·b_1_24 + b_2_53·b_1_22
       + a_2_4·b_1_23·b_3_8 + a_2_4·b_2_5·b_4_14 + a_2_4·b_2_53 + a_2_42·b_2_52
       + c_4_15·b_1_2·b_3_8
  48. b_3_9·b_5_26 + b_2_5·b_4_13·b_1_12 + b_2_52·b_1_1·b_3_9 + b_2_52·b_4_14
       + b_2_53·b_1_22 + b_2_54 + a_2_4·b_2_5·b_1_2·b_3_8 + a_2_4·b_2_5·b_4_14
       + a_2_4·b_2_52·b_1_22 + c_4_15·b_1_2·b_3_9 + b_2_5·c_4_15·b_1_12
  49. b_3_9·b_5_25 + b_2_5·b_1_23·b_3_8 + b_2_5·b_6_38 + b_2_5·b_4_14·b_1_22
       + b_2_52·b_1_2·b_3_9 + b_2_52·b_1_24 + b_2_53·b_1_22 + a_2_4·b_2_5·b_1_2·b_3_8
       + a_2_4·b_2_5·b_1_24 + a_2_4·b_2_5·b_4_13 + a_2_4·b_2_53 + b_2_5·c_4_16·b_1_22
       + b_2_5·c_4_15·b_1_12 + b_2_52·c_4_16 + b_2_52·c_4_15 + a_2_42·c_4_16
  50. a_2_4·b_1_23·b_3_8 + a_2_4·b_6_38 + a_2_4·b_4_14·b_1_22 + a_2_4·b_2_5·b_1_24
       + a_2_4·b_2_5·b_4_13 + a_2_42·b_2_52 + a_2_4·c_4_16·b_1_22 + a_2_4·b_2_5·c_4_16
       + a_2_4·b_2_5·c_4_15
  51. b_4_14·b_5_25 + b_4_13·b_5_25 + b_2_5·b_4_14·b_3_8 + b_2_52·b_5_26 + b_2_52·b_5_25
       + b_2_52·b_1_12·b_3_9 + b_2_53·b_3_9 + b_2_53·b_3_8 + b_2_53·b_1_23
       + b_2_53·b_1_13 + b_2_54·b_1_2 + b_2_54·b_1_1 + b_4_142·a_1_0
       + a_2_4·b_2_5·b_1_22·b_3_8 + a_2_4·b_2_5·b_1_25 + a_2_4·b_2_5·b_4_14·b_1_2
       + a_2_4·b_2_52·b_3_9 + a_2_4·b_2_53·b_1_2 + b_2_5·c_4_16·b_1_13
       + b_2_5·c_4_15·b_1_13 + b_2_52·c_4_15·b_1_2 + a_2_4·c_4_15·b_1_23
  52. b_4_13·b_5_25 + b_2_52·b_5_25 + b_2_52·b_1_12·b_3_9 + b_2_53·b_1_23
       + b_2_53·b_1_13 + b_2_54·b_1_2 + b_2_54·b_1_1 + a_2_4·b_2_5·b_5_26
       + a_2_4·b_2_52·b_3_9 + a_2_4·b_2_52·b_3_8 + a_2_4·b_2_52·b_1_23
       + a_2_4·b_2_53·b_1_2 + a_2_42·b_2_5·b_3_9 + b_2_5·c_4_16·b_1_13
       + b_2_5·c_4_15·b_1_13 + a_2_4·b_2_5·c_4_15·b_1_2
  53. b_4_14·b_5_25 + b_4_13·b_5_26 + b_4_13·b_5_25 + b_2_5·b_4_14·b_3_8 + b_2_5·b_4_13·b_3_9
       + b_2_52·b_5_25 + b_2_52·b_1_12·b_3_9 + b_2_52·b_4_14·b_1_2 + b_2_53·b_3_8
       + b_2_53·b_1_13 + b_2_54·b_1_2 + b_2_54·b_1_1 + b_4_142·a_1_0 + a_2_4·b_4_14·b_3_8
       + a_2_4·b_2_5·b_1_22·b_3_8 + a_2_4·b_2_5·b_1_25 + a_2_4·b_2_52·b_3_9
       + a_2_4·b_2_52·b_1_23 + a_2_4·b_2_53·b_1_2 + b_2_5·c_4_16·b_1_13
       + b_2_5·c_4_15·b_3_8 + b_2_5·c_4_15·b_1_13 + b_2_52·c_4_15·b_1_2
       + b_4_14·c_4_15·a_1_0 + a_2_4·c_4_15·b_3_9 + a_2_4·c_4_15·b_3_8 + a_2_4·c_4_15·a_3_7
  54. b_6_38·a_3_7 + b_4_142·a_1_0 + a_2_4·b_4_14·b_3_8 + a_2_4·b_2_5·b_1_22·b_3_8
       + a_2_4·b_2_5·b_1_25 + a_2_4·b_2_52·b_3_9 + a_2_4·b_2_53·b_1_2
       + a_2_42·b_2_5·b_3_9 + b_4_14·c_4_16·a_1_0 + a_2_4·c_4_16·b_3_9 + a_2_4·c_4_16·b_3_8
       + a_2_4·c_4_15·b_3_9 + a_2_4·c_4_16·a_3_7
  55. b_4_14·b_5_26 + b_4_14·b_5_25 + b_4_14·b_1_22·b_3_8 + b_2_5·b_6_38·b_1_2
       + b_2_5·b_4_14·b_1_23 + b_2_53·b_3_8 + b_2_54·b_1_2 + a_2_4·b_6_38·b_1_2
       + a_2_4·b_2_5·b_4_14·b_1_2 + a_2_4·b_2_52·b_3_9 + a_2_4·b_2_52·b_1_23
       + c_4_15·b_1_22·b_3_8 + b_4_14·c_4_15·b_1_2 + b_2_5·c_4_16·b_1_23
       + b_2_52·c_4_16·b_1_2 + b_2_52·c_4_15·b_1_2 + a_2_4·c_4_16·b_1_23
       + a_2_4·c_4_15·b_3_8 + a_2_4·c_4_15·b_1_23 + a_2_4·b_2_5·c_4_16·b_1_2
       + a_2_4·b_2_5·c_4_15·b_1_2 + a_2_4·c_4_16·a_3_7 + a_2_4·c_4_15·a_3_7
  56. b_6_38·b_3_8 + b_4_14·b_5_26 + b_4_14·b_5_25 + b_4_13·b_5_25 + b_2_5·b_1_27
       + b_2_52·b_5_25 + b_2_52·b_1_25 + b_2_52·b_1_12·b_3_9 + b_2_53·b_3_8
       + b_2_53·b_1_13 + b_2_54·b_1_2 + b_2_54·b_1_1 + a_2_4·b_4_14·b_1_23
       + a_2_4·b_2_5·b_1_22·b_3_8 + a_2_4·b_2_52·b_3_8 + c_4_16·b_1_22·b_3_8
       + c_4_15·b_1_22·b_3_8 + b_4_14·c_4_15·b_1_2 + b_2_5·c_4_16·b_3_8
       + b_2_5·c_4_16·b_1_13 + b_2_5·c_4_15·b_3_8 + b_2_5·c_4_15·b_1_13
       + a_2_4·c_4_15·b_3_8 + a_2_4·c_4_15·b_1_23 + a_2_4·c_4_16·a_3_7 + a_2_4·c_4_15·a_3_7
  57. b_6_38·b_3_9 + b_4_13·b_5_25 + b_2_5·b_1_14·b_3_9 + b_2_5·b_4_14·b_3_8
       + b_2_5·b_4_13·b_3_9 + b_2_52·b_1_22·b_3_8 + b_2_52·b_1_25 + b_2_52·b_1_12·b_3_9
       + b_2_52·b_1_15 + b_2_52·b_4_13·b_1_1 + b_2_53·b_3_9 + b_2_53·b_3_8
       + b_2_53·b_1_23 + b_2_54·b_1_2 + a_2_4·b_4_14·b_3_8 + a_2_4·b_2_5·b_1_25
       + a_2_4·b_2_52·b_3_9 + a_2_4·b_2_52·b_3_8 + a_2_4·b_2_52·b_1_23
       + a_2_42·b_2_5·b_3_9 + c_4_16·b_1_15 + c_4_15·b_1_12·b_3_9 + c_4_15·b_1_15
       + b_4_13·c_4_15·b_1_1 + b_2_5·c_4_16·b_3_9 + b_2_5·c_4_16·b_3_8 + b_2_5·c_4_16·b_1_13
       + b_2_5·c_4_15·b_3_9 + b_2_5·c_4_15·b_1_13 + b_2_52·c_4_15·b_1_1
       + a_2_4·c_4_16·b_3_9 + a_2_4·c_4_16·b_3_8 + a_2_4·c_4_16·a_3_7 + a_2_4·c_4_15·a_3_7
  58. b_5_252 + b_2_53·b_1_24 + b_2_53·b_1_1·b_3_9 + b_2_54·b_1_12 + b_2_55
       + b_2_52·c_4_16·b_1_12 + b_2_52·c_4_15·b_1_12
  59. b_5_262 + b_2_5·b_4_142 + b_2_52·b_4_13·b_1_12 + b_2_53·b_1_24
       + b_2_53·b_1_1·b_3_9 + b_2_55 + a_2_42·b_2_53 + b_2_52·c_4_15·b_1_12
       + c_4_152·b_1_22
  60. b_5_25·b_5_26 + b_2_52·b_1_23·b_3_8 + b_2_52·b_6_38 + b_2_53·b_1_2·b_3_9
       + b_2_53·b_4_14 + a_2_4·b_2_5·b_6_38 + a_2_4·b_2_5·b_4_14·b_1_22
       + a_2_4·b_2_52·b_1_2·b_3_8 + a_2_4·b_2_52·b_1_24 + a_2_4·b_2_54
       + b_2_5·c_4_15·b_1_2·b_3_9 + b_2_5·c_4_15·b_1_2·b_3_8 + b_2_52·c_4_16·b_1_22
       + b_2_52·c_4_15·b_1_12 + b_2_53·c_4_16 + b_2_53·c_4_15 + b_4_14·c_4_15·a_1_0·b_1_2
       + a_2_4·c_4_15·b_1_2·b_3_8 + a_2_4·c_4_15·b_1_24 + a_2_4·b_4_14·c_4_15
       + a_2_4·b_2_5·c_4_16·b_1_22 + a_2_4·b_2_5·c_4_15·b_1_22 + a_2_4·b_2_52·c_4_16
       + a_2_4·b_2_52·c_4_15 + a_2_42·b_2_5·c_4_16
  61. b_4_13·b_6_38 + b_2_5·b_4_14·b_1_2·b_3_8 + b_2_5·b_4_13·b_1_14
       + b_2_52·b_1_23·b_3_8 + b_2_52·b_1_26 + b_2_52·b_6_38 + b_2_53·b_1_2·b_3_9
       + b_2_53·b_1_1·b_3_9 + b_2_54·b_1_22 + b_2_54·b_1_12 + b_4_142·a_1_0·b_1_2
       + a_2_4·b_6_38·b_1_22 + a_2_4·b_4_14·b_1_2·b_3_8 + a_2_4·b_2_5·b_1_2·b_5_26
       + a_2_4·b_2_5·b_1_26 + a_2_4·b_2_52·b_1_2·b_3_8 + a_2_4·b_2_52·b_1_24
       + a_2_4·b_2_52·b_4_14 + a_2_42·b_2_53 + c_4_16·b_1_13·b_3_9 + c_4_15·b_1_13·b_3_9
       + b_4_13·c_4_15·b_1_12 + b_2_5·c_4_16·b_1_2·b_3_8 + b_2_5·c_4_15·b_1_14
       + b_2_5·b_4_13·c_4_16 + b_2_5·b_4_13·c_4_15 + b_2_52·c_4_15·b_1_12 + b_2_53·c_4_16
       + b_2_53·c_4_15 + b_4_14·c_4_16·a_1_0·b_1_2 + a_2_4·c_4_16·b_1_2·b_3_8
       + a_2_4·b_4_13·c_4_16 + a_2_4·b_2_5·c_4_16·b_1_22 + a_2_4·b_2_52·c_4_16
  62. b_5_25·b_5_26 + b_4_14·b_1_23·b_3_8 + b_4_14·b_6_38 + b_4_142·b_1_22
       + b_2_5·b_4_142 + b_2_52·b_1_2·b_5_26 + b_2_52·b_1_26 + b_2_52·b_6_38
       + b_2_53·b_1_2·b_3_8 + a_2_4·b_4_14·b_1_2·b_3_8 + a_2_4·b_4_142
       + a_2_4·b_2_5·b_1_2·b_5_26 + a_2_4·b_2_52·b_1_2·b_3_8 + a_2_4·b_2_52·b_4_14
       + a_2_4·b_2_52·b_4_13 + a_2_4·b_2_54 + a_2_42·b_2_53 + b_4_14·c_4_16·b_1_22
       + b_2_5·c_4_15·b_1_2·b_3_9 + b_2_5·c_4_15·b_1_2·b_3_8 + b_2_5·c_4_15·b_1_24
       + b_2_5·b_4_14·c_4_16 + b_2_5·b_4_14·c_4_15 + b_2_52·c_4_16·b_1_22
       + b_2_52·c_4_15·b_1_22 + b_2_52·c_4_15·b_1_12 + b_2_53·c_4_16 + b_2_53·c_4_15
       + a_2_4·b_4_14·c_4_15
  63. b_6_38·b_5_26 + b_6_38·b_5_25 + b_4_142·b_3_8 + b_2_5·b_4_14·b_1_22·b_3_8
       + b_2_5·b_4_142·b_1_2 + b_2_52·b_1_27 + b_2_52·b_1_14·b_3_9
       + b_2_52·b_6_38·b_1_2 + b_2_52·b_4_14·b_1_23 + b_2_52·b_4_13·b_3_9
       + b_2_52·b_4_13·b_1_13 + b_2_53·b_5_26 + b_2_53·b_5_25 + b_2_53·b_1_25
       + b_2_53·b_1_15 + b_2_53·b_4_14·b_1_2 + b_2_53·b_4_13·b_1_1 + b_2_54·b_3_9
       + b_2_54·b_3_8 + b_2_54·b_1_13 + a_2_4·b_4_14·b_1_22·b_3_8 + a_2_4·b_4_142·b_1_2
       + a_2_4·b_2_5·b_1_27 + a_2_4·b_2_5·b_6_38·b_1_2 + a_2_4·b_2_5·b_4_14·b_3_8
       + a_2_4·b_2_52·b_5_26 + a_2_4·b_2_52·b_1_22·b_3_8 + a_2_4·b_2_52·b_4_14·b_1_2
       + a_2_4·b_2_53·b_3_8 + a_2_4·b_2_53·b_1_23 + a_2_4·b_2_54·b_1_2
       + a_2_42·b_2_52·b_3_9 + c_4_15·b_6_38·b_1_2 + b_4_14·c_4_16·b_3_8
       + b_2_5·c_4_16·b_5_26 + b_2_5·c_4_16·b_5_25 + b_2_5·c_4_16·b_1_12·b_3_9
       + b_2_5·c_4_16·b_1_15 + b_2_5·c_4_15·b_5_26 + b_2_5·c_4_15·b_5_25
       + b_2_5·c_4_15·b_1_25 + b_2_5·c_4_15·b_1_15 + b_2_52·c_4_16·b_1_23
       + b_2_52·c_4_15·b_1_13 + b_2_53·c_4_16·b_1_2 + a_2_4·c_4_16·b_5_26
       + a_2_4·c_4_15·b_1_22·b_3_8 + a_2_4·b_4_14·c_4_16·b_1_2 + a_2_4·b_2_5·c_4_16·b_3_9
       + a_2_4·b_2_5·c_4_15·b_1_23 + a_2_4·b_2_52·c_4_16·b_1_2
       + b_2_5·c_4_15·c_4_16·b_1_2 + b_2_5·c_4_152·b_1_2 + a_2_4·c_4_15·c_4_16·b_1_2
       + a_2_4·c_4_162·a_1_0 + a_2_4·c_4_15·c_4_16·a_1_0
  64. b_6_38·b_5_26 + b_4_142·b_3_8 + b_2_5·b_4_142·b_1_2 + b_2_52·b_1_14·b_3_9
       + b_2_52·b_4_14·b_3_8 + b_2_52·b_4_13·b_3_9 + b_2_53·b_5_26 + b_2_53·b_5_25
       + b_2_53·b_1_22·b_3_8 + b_2_53·b_1_25 + b_2_53·b_1_15 + b_2_53·b_4_14·b_1_2
       + b_2_53·b_4_13·b_1_1 + b_2_54·b_3_8 + b_2_54·b_1_23 + b_2_54·b_1_13
       + b_2_55·b_1_1 + b_4_142·a_1_0·b_1_22 + a_2_4·b_6_38·b_1_23
       + a_2_4·b_2_5·b_1_27 + a_2_4·b_2_5·b_6_38·b_1_2 + a_2_4·b_2_52·b_1_25
       + a_2_4·b_2_53·b_3_9 + a_2_4·b_2_53·b_3_8 + a_2_42·b_2_52·b_3_9
       + c_4_15·b_6_38·b_1_2 + b_4_14·c_4_16·b_3_8 + b_2_5·c_4_16·b_5_26
       + b_2_5·c_4_16·b_1_22·b_3_8 + b_2_5·c_4_16·b_1_15 + b_2_5·c_4_15·b_5_26
       + b_2_5·c_4_15·b_1_25 + b_2_5·c_4_15·b_1_12·b_3_9 + b_2_5·c_4_15·b_1_15
       + b_2_5·b_4_13·c_4_15·b_1_1 + b_2_52·c_4_16·b_3_8 + b_2_53·c_4_15·b_1_2
       + b_2_53·c_4_15·b_1_1 + b_4_14·c_4_16·a_1_0·b_1_22 + a_2_4·c_4_16·b_5_26
       + a_2_4·c_4_15·b_1_22·b_3_8 + a_2_4·b_2_5·c_4_16·b_3_8 + a_2_4·b_2_52·c_4_16·b_1_2
       + a_2_4·b_2_52·c_4_15·b_1_2 + a_2_42·c_4_16·b_3_9 + a_2_42·c_4_15·b_3_9
       + b_2_5·c_4_15·c_4_16·b_1_2 + b_2_5·c_4_152·b_1_2 + a_2_4·c_4_15·c_4_16·b_1_2
       + a_2_4·c_4_162·a_1_0 + a_2_4·c_4_15·c_4_16·a_1_0
  65. b_6_382 + b_4_143 + b_2_5·b_1_210 + b_2_5·b_4_14·b_6_38
       + b_2_5·b_4_13·b_1_13·b_3_9 + b_2_52·b_6_38·b_1_22 + b_2_53·b_1_2·b_5_26
       + b_2_53·b_1_26 + b_2_53·b_4_13·b_1_12 + b_2_54·b_1_2·b_3_9 + b_2_54·b_4_14
       + b_2_56 + b_4_142·a_1_0·b_1_23 + a_2_4·b_4_14·b_6_38 + a_2_4·b_4_142·b_1_22
       + a_2_4·b_2_5·b_4_142 + a_2_4·b_2_52·b_1_2·b_5_26 + a_2_4·b_2_52·b_6_38
       + a_2_4·b_2_54·b_1_22 + a_2_4·b_2_55 + b_4_142·c_4_15 + b_4_13·c_4_16·b_1_14
       + b_4_13·c_4_15·b_1_14 + b_2_5·c_4_16·b_1_13·b_3_9 + b_2_5·c_4_15·b_1_23·b_3_8
       + b_2_5·b_4_14·c_4_16·b_1_22 + b_2_52·c_4_16·b_1_24 + b_2_52·b_4_14·c_4_16
       + b_2_52·b_4_14·c_4_15 + b_2_53·c_4_16·b_1_22 + b_2_53·c_4_16·b_1_12
       + b_4_14·c_4_15·a_1_0·b_1_23 + a_2_4·c_4_15·b_6_38 + a_2_4·b_4_14·c_4_16·b_1_22
       + a_2_4·b_2_5·b_4_14·c_4_16 + a_2_4·b_2_5·b_4_14·c_4_15 + a_2_4·b_2_5·b_4_13·c_4_15
       + a_2_4·b_2_52·c_4_16·b_1_22 + a_2_4·b_2_52·c_4_15·b_1_22
       + a_2_4·b_2_53·c_4_16 + a_2_4·b_2_53·c_4_15 + a_2_42·b_2_52·c_4_15
       + c_4_162·b_1_24 + c_4_15·c_4_16·b_1_14 + c_4_152·b_1_24 + b_2_52·c_4_162
       + b_2_52·c_4_152 + a_2_4·c_4_15·c_4_16·b_1_22 + a_2_4·b_2_5·c_4_15·c_4_16
       + a_2_4·b_2_5·c_4_152 + a_2_42·c_4_15·c_4_16

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_15, a Duflot regular element of degree 4
    2. c_4_16, a Duflot regular element of degree 4
    3. b_1_24 + b_1_14 + b_2_5·b_1_22 + b_2_52, an element of degree 4
    4. b_3_8 + b_2_5·b_1_2 + b_2_5·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 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. b_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. b_2_5 → 0, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. b_3_8 → 0, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. b_4_13 → 0, an element of degree 4
  10. b_4_14 → 0, an element of degree 4
  11. c_4_15 → c_1_04, an element of degree 4
  12. c_4_16 → c_1_14, an element of degree 4
  13. b_5_25 → 0, an element of degree 5
  14. b_5_26 → 0, an element of degree 5
  15. b_6_38 → 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. b_1_1 → c_1_2, 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. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. b_3_8 → 0, an element of degree 3
  8. b_3_9 → c_1_33 + c_1_22·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. b_4_13 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + 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
  10. b_4_14 → 0, an element of degree 4
  11. c_4_15 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_23
       + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  12. c_4_16 → c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·c_1_23 + c_1_14 + c_1_0·c_1_23
       + c_1_02·c_1_22, an element of degree 4
  13. b_5_25 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_1·c_1_22·c_1_32
       + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_2·c_1_32
       + c_1_02·c_1_22·c_1_3, an element of degree 5
  14. b_5_26 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_0·c_1_22·c_1_32
       + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3, an element of degree 5
  15. b_6_38 → c_1_2·c_1_35 + c_1_25·c_1_3 + c_1_1·c_1_22·c_1_33 + c_1_1·c_1_24·c_1_3
       + c_1_1·c_1_25 + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32 + c_1_12·c_1_24
       + c_1_14·c_1_32 + c_1_14·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_34 + c_1_0·c_1_24·c_1_3
       + c_1_0·c_1_25 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_34
       + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23
       + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3, 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. b_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. b_2_5 → c_1_32, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. b_3_8 → c_1_22·c_1_3, an element of degree 3
  8. b_3_9 → c_1_33, an element of degree 3
  9. b_4_13 → c_1_34 + c_1_2·c_1_33, an element of degree 4
  10. b_4_14 → c_1_22·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22, an element of degree 4
  11. c_4_15 → c_1_34 + c_1_2·c_1_33 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_22
       + c_1_04, an element of degree 4
  12. c_4_16 → c_1_2·c_1_33 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  13. b_5_25 → c_1_35 + c_1_22·c_1_33, an element of degree 5
  14. b_5_26 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_0·c_1_22·c_1_32
       + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3
       + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  15. b_6_38 → c_1_2·c_1_35 + c_1_23·c_1_33 + c_1_25·c_1_3 + c_1_12·c_1_22·c_1_32
       + c_1_12·c_1_24 + c_1_14·c_1_32 + c_1_14·c_1_22 + c_1_0·c_1_22·c_1_33
       + c_1_0·c_1_24·c_1_3 + c_1_02·c_1_34 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_24
       + c_1_04·c_1_32, 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