Cohomology of group number 1543 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 4.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 3.
  • 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 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t8  +  3·t7  +  4·t6  +  5·t5  +  8·t4  +  5·t3  +  4·t2  +  3·t  +  1)
    (t  +  1)2 · (t  −  1)3 · (t2  +  1)3
  • The a-invariants are -∞,-∞,-∞,-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 17 minimal generators of maximal degree 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. a_1_3, a nilpotent element of degree 1
  5. a_3_5, a nilpotent element of degree 3
  6. a_3_6, a nilpotent element of degree 3
  7. a_3_7, a nilpotent element of degree 3
  8. a_3_8, a nilpotent element of degree 3
  9. a_4_10, a nilpotent element of degree 4
  10. a_4_11, a nilpotent element of degree 4
  11. a_4_12, a nilpotent element of degree 4
  12. c_4_13, a Duflot regular element of degree 4
  13. c_4_14, a Duflot regular element of degree 4
  14. c_4_15, a Duflot regular element of degree 4
  15. a_6_27, a nilpotent element of degree 6
  16. a_6_28, a nilpotent element of degree 6
  17. a_6_29, 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 87 minimal relations of maximal degree 12:

  1. a_1_22 + a_1_1·a_1_2 + a_1_12
  2. a_1_1·a_1_3 + a_1_1·a_1_2 + a_1_0·a_1_2 + a_1_0·a_1_1
  3. a_1_32 + a_1_1·a_1_2 + a_1_12 + a_1_0·a_1_3 + a_1_0·a_1_2 + a_1_02
  4. a_1_03
  5. a_1_13
  6. a_1_0·a_1_12 + a_1_02·a_1_1
  7. a_1_2·a_3_5 + a_1_1·a_3_6 + a_1_1·a_3_5
  8. a_1_3·a_3_5 + a_1_2·a_3_5 + a_1_0·a_3_6
  9. a_1_3·a_3_6 + a_1_3·a_3_5 + a_1_2·a_3_5 + a_1_1·a_3_5 + a_1_0·a_3_5
  10. a_1_2·a_3_6 + a_1_1·a_3_5
  11. a_1_2·a_3_7 + a_1_1·a_3_8 + a_1_1·a_3_5
  12. a_1_3·a_3_7 + a_1_2·a_3_7 + a_1_0·a_3_8 + a_1_0·a_3_7 + a_1_0·a_3_5
  13. a_1_12·a_3_7 + a_1_12·a_3_5 + a_1_0·a_1_1·a_3_7 + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_7
       + a_1_02·a_3_5
  14. a_1_2·a_1_3·a_3_8 + a_1_12·a_3_7 + a_1_12·a_3_5 + a_1_0·a_1_1·a_3_8
       + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_6 + a_1_02·a_3_5
  15. a_1_12·a_3_5 + a_1_0·a_1_2·a_3_8 + a_1_0·a_1_1·a_3_8 + a_1_0·a_1_1·a_3_7
       + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_5
  16. a_4_10·a_1_1 + a_1_12·a_3_6 + a_1_12·a_3_5 + a_1_0·a_1_1·a_3_8 + a_1_0·a_1_1·a_3_7
       + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_5
  17. a_4_10·a_1_0 + a_1_12·a_3_5 + a_1_0·a_1_1·a_3_7 + a_1_0·a_1_1·a_3_6 + a_1_02·a_3_8
  18. a_4_11·a_1_1 + a_4_10·a_1_2 + a_1_12·a_3_8 + a_1_12·a_3_5 + a_1_0·a_1_1·a_3_5
       + a_1_02·a_3_6
  19. a_4_11·a_1_0 + a_4_10·a_1_3 + a_4_10·a_1_2 + a_1_12·a_3_7 + a_1_12·a_3_6
       + a_1_0·a_1_1·a_3_8 + a_1_0·a_1_1·a_3_7 + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_1·a_3_5
       + a_1_02·a_3_8 + a_1_02·a_3_5
  20. a_4_11·a_1_3 + a_4_10·a_1_3 + a_4_10·a_1_2 + a_1_12·a_3_8 + a_1_12·a_3_7 + a_1_12·a_3_6
       + a_1_0·a_1_1·a_3_8 + a_1_0·a_1_1·a_3_6 + a_1_02·a_3_6
  21. a_4_11·a_1_2 + a_1_12·a_3_8 + a_1_12·a_3_7 + a_1_12·a_3_6 + a_1_0·a_1_1·a_3_8
       + a_1_0·a_1_1·a_3_5
  22. a_4_12·a_1_1 + a_1_12·a_3_8 + a_1_12·a_3_7 + a_1_12·a_3_6 + a_1_12·a_3_5
       + a_1_0·a_1_1·a_3_7
  23. a_4_12·a_1_0 + a_1_0·a_1_1·a_3_8 + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_1·a_3_5
  24. a_4_12·a_1_3 + a_4_10·a_1_3 + a_1_0·a_1_1·a_3_5
  25. a_4_12·a_1_2 + a_4_10·a_1_2 + a_1_12·a_3_7 + a_1_12·a_3_6 + a_1_0·a_1_1·a_3_8
       + a_1_0·a_1_1·a_3_7 + a_1_02·a_3_6
  26. a_3_62 + a_3_5·a_3_6 + a_3_52 + a_1_02·a_1_1·a_3_5
  27. a_3_52 + a_1_02·a_1_1·a_3_7 + a_1_02·a_1_1·a_3_5
  28. a_3_62 + a_1_02·a_1_1·a_3_8 + a_1_02·a_1_1·a_3_6 + a_1_02·a_1_1·a_3_5
  29. a_3_82 + a_3_7·a_3_8 + a_3_72 + a_3_62 + a_3_5·a_3_7 + a_1_02·a_1_1·a_3_6
       + a_1_02·a_1_1·a_3_5
  30. a_3_6·a_3_8 + a_3_62 + a_3_5·a_3_7
  31. a_3_6·a_3_7 + a_3_5·a_3_8 + a_3_5·a_3_7 + a_3_52 + a_1_02·a_1_1·a_3_6
       + a_1_02·a_1_1·a_3_5
  32. a_3_72 + a_3_62 + c_4_15·a_1_02 + c_4_14·a_1_12
  33. a_3_82 + a_3_52 + a_1_02·a_1_1·a_3_6 + a_1_02·a_1_1·a_3_5 + c_4_15·a_1_0·a_1_3
       + c_4_15·a_1_0·a_1_2 + c_4_14·a_1_1·a_1_2 + c_4_14·a_1_12
  34. a_4_10·a_3_7 + a_4_10·a_3_5 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_8
       + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_2 + c_4_15·a_1_02·a_1_1
       + c_4_14·a_1_0·a_1_1·a_1_2 + c_4_14·a_1_02·a_1_1
  35. a_4_11·a_3_8 + a_4_10·a_3_6 + a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7
       + c_4_15·a_1_02·a_1_3 + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_0·a_1_1·a_1_2
  36. a_4_11·a_3_7 + a_4_10·a_3_8 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_7
       + c_4_15·a_1_02·a_1_2 + c_4_15·a_1_02·a_1_1 + c_4_14·a_1_12·a_1_2
  37. a_4_11·a_3_6 + a_4_10·a_3_7 + a_4_10·a_3_6 + a_1_1·a_3_5·a_3_8 + c_4_15·a_1_02·a_1_3
       + c_4_15·a_1_02·a_1_2 + c_4_15·a_1_02·a_1_1 + c_4_14·a_1_0·a_1_1·a_1_2
       + c_4_14·a_1_02·a_1_1
  38. a_4_11·a_3_5 + a_4_10·a_3_6 + a_1_1·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7
  39. a_4_12·a_3_8 + a_4_10·a_3_8 + a_4_10·a_3_7 + a_4_10·a_3_6 + a_1_0·a_3_5·a_3_8
       + a_1_0·a_3_5·a_3_7
  40. a_4_12·a_3_7 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + c_4_15·a_1_02·a_1_2
       + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_02·a_1_1
  41. a_4_12·a_3_6 + a_4_10·a_3_6 + a_1_1·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7
  42. a_4_12·a_3_5 + a_4_10·a_3_7 + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_2
       + c_4_15·a_1_02·a_1_1 + c_4_14·a_1_0·a_1_1·a_1_2 + c_4_14·a_1_02·a_1_1
  43. a_6_27·a_1_1 + a_4_10·a_3_7 + a_1_1·a_3_5·a_3_7 + c_4_15·a_1_12·a_1_2
       + c_4_15·a_1_02·a_1_3 + c_4_14·a_1_12·a_1_2 + c_4_13·a_1_12·a_1_2
       + c_4_13·a_1_0·a_1_1·a_1_2 + c_4_13·a_1_02·a_1_2 + c_4_13·a_1_02·a_1_1
  44. a_6_27·a_1_0 + a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7 + c_4_15·a_1_0·a_1_1·a_1_2
       + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_2 + c_4_14·a_1_0·a_1_1·a_1_2
       + c_4_14·a_1_02·a_1_2 + c_4_13·a_1_0·a_1_1·a_1_2 + c_4_13·a_1_02·a_1_3
  45. a_6_27·a_1_3 + a_4_10·a_3_7 + a_4_10·a_3_6 + a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7
       + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_2 + c_4_15·a_1_02·a_1_1
       + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_02·a_1_3 + c_4_13·a_1_12·a_1_2
       + c_4_13·a_1_0·a_1_1·a_1_2 + c_4_13·a_1_02·a_1_3 + c_4_13·a_1_02·a_1_2
  46. a_6_27·a_1_2 + a_4_10·a_3_7 + a_4_10·a_3_6 + a_1_0·a_3_5·a_3_7
       + c_4_15·a_1_0·a_1_1·a_1_2 + c_4_15·a_1_02·a_1_3 + c_4_14·a_1_12·a_1_2
       + c_4_14·a_1_02·a_1_2 + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_0·a_1_1·a_1_2
  47. a_6_28·a_1_1 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + c_4_15·a_1_0·a_1_1·a_1_2
       + c_4_14·a_1_12·a_1_2 + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_02·a_1_1
  48. a_6_28·a_1_0 + a_4_10·a_3_7 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_8
       + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_1 + c_4_13·a_1_0·a_1_1·a_1_2
  49. a_6_28·a_1_3 + a_4_10·a_3_6 + a_1_1·a_3_5·a_3_7 + c_4_15·a_1_0·a_1_1·a_1_2
       + c_4_14·a_1_0·a_1_1·a_1_2 + c_4_13·a_1_02·a_1_2 + c_4_13·a_1_02·a_1_1
  50. a_6_28·a_1_2 + a_1_1·a_3_5·a_3_7 + c_4_15·a_1_0·a_1_1·a_1_2 + c_4_15·a_1_02·a_1_1
       + c_4_13·a_1_0·a_1_1·a_1_2
  51. a_6_29·a_1_1 + a_1_0·a_3_5·a_3_7 + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_0·a_1_1·a_1_2
       + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_02·a_1_1
  52. a_6_29·a_1_0 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_7 + c_4_15·a_1_02·a_1_1
       + c_4_14·a_1_0·a_1_1·a_1_2 + c_4_14·a_1_02·a_1_2 + c_4_14·a_1_02·a_1_1
       + c_4_13·a_1_0·a_1_1·a_1_2 + c_4_13·a_1_02·a_1_1
  53. a_6_29·a_1_3 + a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_7
       + c_4_15·a_1_0·a_1_1·a_1_2 + c_4_15·a_1_02·a_1_3 + c_4_15·a_1_02·a_1_2
       + c_4_15·a_1_02·a_1_1 + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_02·a_1_3
       + c_4_14·a_1_02·a_1_2 + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_0·a_1_1·a_1_2
       + c_4_13·a_1_02·a_1_2
  54. a_6_29·a_1_2 + a_1_0·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_1_2 + c_4_15·a_1_02·a_1_2
       + c_4_14·a_1_12·a_1_2 + c_4_14·a_1_02·a_1_2 + c_4_14·a_1_02·a_1_1
       + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_0·a_1_1·a_1_2
  55. a_4_102
  56. a_4_10·a_4_11 + a_1_0·a_1_1·a_3_5·a_3_8 + a_1_0·a_1_1·a_3_5·a_3_7
  57. a_4_112
  58. a_4_122
  59. a_4_10·a_4_12 + a_1_02·a_3_5·a_3_8
  60. a_4_11·a_4_12 + a_1_0·a_1_1·a_3_5·a_3_7
  61. a_6_27·a_3_8 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_8
       + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_6 + c_4_15·a_1_02·a_3_8
       + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_12·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8
       + c_4_14·a_1_0·a_1_1·a_3_5 + c_4_14·a_1_02·a_3_8 + c_4_14·a_1_02·a_3_7
       + c_4_13·a_1_12·a_3_8 + c_4_13·a_1_12·a_3_6 + c_4_13·a_1_0·a_1_1·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_6 + c_4_13·a_1_02·a_3_6
  62. a_6_27·a_3_7 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_12·a_3_8
       + c_4_15·a_1_0·a_1_2·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_7
       + c_4_15·a_1_0·a_1_1·a_3_6 + c_4_15·a_1_0·a_1_1·a_3_5 + c_4_15·a_1_02·a_3_8
       + c_4_15·a_1_02·a_3_7 + c_4_15·a_1_02·a_3_6 + c_4_14·a_1_12·a_3_8
       + c_4_14·a_1_12·a_3_6 + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_02·a_3_7
       + c_4_13·a_1_12·a_3_8 + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7
       + c_4_13·a_1_0·a_1_1·a_3_6 + c_4_13·a_1_02·a_3_8 + c_4_13·a_1_02·a_3_7
  63. a_6_27·a_3_6 + c_4_15·a_1_12·a_3_6 + c_4_15·a_1_0·a_1_2·a_3_8
       + c_4_15·a_1_0·a_1_1·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_7
       + c_4_14·a_1_0·a_1_1·a_3_6 + c_4_14·a_1_02·a_3_6 + c_4_14·a_1_02·a_3_5
       + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7
       + c_4_13·a_1_0·a_1_1·a_3_6 + c_4_13·a_1_02·a_3_6
  64. a_6_27·a_3_5 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_12·a_3_6
       + c_4_15·a_1_0·a_1_1·a_3_5 + c_4_15·a_1_02·a_3_6 + c_4_15·a_1_02·a_3_5
       + c_4_14·a_1_12·a_3_6 + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8
       + c_4_14·a_1_0·a_1_1·a_3_7 + c_4_13·a_1_12·a_3_6 + c_4_13·a_1_0·a_1_2·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7 + c_4_13·a_1_02·a_3_6
       + c_4_13·a_1_02·a_3_5
  65. a_6_28·a_3_8 + c_4_15·a_1_12·a_3_6 + c_4_15·a_1_0·a_1_1·a_3_8
       + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_6 + c_4_15·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_02·a_3_7 + c_4_14·a_1_02·a_3_6 + c_4_14·a_1_02·a_3_5
       + c_4_13·a_1_12·a_3_6 + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_6
       + c_4_13·a_1_02·a_3_7
  66. a_6_28·a_3_7 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_2·a_3_8
       + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_6 + c_4_15·a_1_02·a_3_6
       + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_12·a_3_8 + c_4_14·a_1_12·a_3_6
       + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_6
       + c_4_14·a_1_0·a_1_1·a_3_5 + c_4_14·a_1_02·a_3_7 + c_4_14·a_1_02·a_3_5
       + c_4_13·a_1_12·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_5 + c_4_13·a_1_02·a_3_7
       + c_4_13·a_1_02·a_3_5
  67. a_6_28·a_3_6 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_12·a_3_6 + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8
       + c_4_14·a_1_0·a_1_1·a_3_7 + c_4_14·a_1_0·a_1_1·a_3_6 + c_4_14·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_02·a_3_5 + c_4_13·a_1_12·a_3_6 + c_4_13·a_1_0·a_1_2·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7 + c_4_13·a_1_0·a_1_1·a_3_5
       + c_4_13·a_1_02·a_3_5
  68. a_6_28·a_3_5 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_6
       + c_4_15·a_1_0·a_1_1·a_3_5 + c_4_14·a_1_12·a_3_6 + c_4_13·a_1_12·a_3_6
       + c_4_13·a_1_0·a_1_1·a_3_5
  69. a_6_29·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_6
       + c_4_15·a_1_0·a_1_1·a_3_5 + c_4_15·a_1_02·a_3_8 + c_4_15·a_1_02·a_3_6
       + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_12·a_3_8 + c_4_14·a_1_12·a_3_6
       + c_4_14·a_1_0·a_1_1·a_3_5 + c_4_14·a_1_02·a_3_8 + c_4_14·a_1_02·a_3_7
       + c_4_13·a_1_12·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7
       + c_4_13·a_1_0·a_1_1·a_3_6 + c_4_13·a_1_02·a_3_7 + c_4_13·a_1_02·a_3_6
  70. a_6_29·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_0·a_1_1·a_3_5
       + c_4_15·a_1_02·a_3_7 + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_12·a_3_8
       + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_6 + c_4_14·a_1_0·a_1_1·a_3_5
       + c_4_14·a_1_02·a_3_7 + c_4_14·a_1_02·a_3_5 + c_4_13·a_1_12·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_7 + c_4_13·a_1_0·a_1_1·a_3_5 + c_4_13·a_1_02·a_3_5
  71. a_6_29·a_3_6 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_6
       + c_4_15·a_1_02·a_3_6 + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_8
       + c_4_14·a_1_0·a_1_1·a_3_7 + c_4_14·a_1_02·a_3_6 + c_4_14·a_1_02·a_3_5
       + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7
       + c_4_13·a_1_0·a_1_1·a_3_5 + c_4_13·a_1_02·a_3_5
  72. a_6_29·a_3_5 + a_1_02·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_0·a_1_1·a_3_5
       + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_12·a_3_6 + c_4_14·a_1_0·a_1_2·a_3_8
       + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_7 + c_4_14·a_1_0·a_1_1·a_3_5
       + c_4_13·a_1_12·a_3_6 + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_7 + c_4_13·a_1_0·a_1_1·a_3_6 + c_4_13·a_1_02·a_3_5
  73. a_4_12·a_6_27 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_15·a_1_02·a_1_1·a_3_6
       + c_4_14·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_7
       + c_4_14·a_1_02·a_1_1·a_3_6 + c_4_13·a_1_02·a_1_1·a_3_6
  74. a_4_10·a_6_27 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_15·a_1_02·a_1_1·a_3_7
       + c_4_15·a_1_02·a_1_1·a_3_5 + c_4_14·a_1_02·a_1_1·a_3_7
       + c_4_14·a_1_02·a_1_1·a_3_5 + c_4_13·a_1_02·a_1_1·a_3_8
       + c_4_13·a_1_02·a_1_1·a_3_7 + c_4_13·a_1_02·a_1_1·a_3_5
  75. a_4_11·a_6_27 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_15·a_1_02·a_1_1·a_3_5
       + c_4_14·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_6
       + c_4_13·a_1_02·a_1_1·a_3_5
  76. a_4_12·a_6_28 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_15·a_1_02·a_1_1·a_3_7
       + c_4_14·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_7
       + c_4_14·a_1_02·a_1_1·a_3_6 + c_4_13·a_1_02·a_1_1·a_3_7
       + c_4_13·a_1_02·a_1_1·a_3_6
  77. a_4_10·a_6_28 + c_4_15·a_1_02·a_1_1·a_3_7 + c_4_15·a_1_02·a_1_1·a_3_6
       + c_4_14·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_5
       + c_4_13·a_1_02·a_1_1·a_3_7 + c_4_13·a_1_02·a_1_1·a_3_5
  78. a_4_11·a_6_28 + c_4_15·a_1_02·a_1_1·a_3_7 + c_4_14·a_1_02·a_1_1·a_3_8
       + c_4_14·a_1_02·a_1_1·a_3_7 + c_4_14·a_1_02·a_1_1·a_3_5
       + c_4_13·a_1_02·a_1_1·a_3_6 + c_4_13·a_1_02·a_1_1·a_3_5
  79. a_4_12·a_6_29 + c_4_14·a_1_02·a_1_1·a_3_7 + c_4_14·a_1_02·a_1_1·a_3_6
       + c_4_13·a_1_02·a_1_1·a_3_6 + c_4_13·a_1_02·a_1_1·a_3_5
  80. a_4_10·a_6_29 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_15·a_1_02·a_1_1·a_3_6
       + c_4_14·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_6
       + c_4_13·a_1_02·a_1_1·a_3_8
  81. a_4_11·a_6_29 + c_4_15·a_1_02·a_1_1·a_3_8 + c_4_14·a_1_02·a_1_1·a_3_7
       + c_4_14·a_1_02·a_1_1·a_3_6 + c_4_14·a_1_02·a_1_1·a_3_5
       + c_4_13·a_1_02·a_1_1·a_3_7 + c_4_13·a_1_02·a_1_1·a_3_5
  82. a_6_272
  83. a_6_282
  84. a_6_27·a_6_28 + c_4_15·a_1_0·a_1_1·a_3_5·a_3_8 + c_4_15·a_1_02·a_3_5·a_3_7
       + c_4_14·a_1_0·a_1_1·a_3_5·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_5·a_3_7
       + c_4_14·a_1_02·a_3_5·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_5·a_3_8
       + c_4_13·a_1_02·a_3_5·a_3_7
  85. a_6_292
  86. a_6_28·a_6_29 + c_4_15·a_1_0·a_1_1·a_3_5·a_3_8 + c_4_14·a_1_0·a_1_1·a_3_5·a_3_7
       + c_4_14·a_1_02·a_3_5·a_3_8 + c_4_14·a_1_02·a_3_5·a_3_7
       + c_4_13·a_1_0·a_1_1·a_3_5·a_3_7
  87. a_6_27·a_6_29 + c_4_15·a_1_0·a_1_1·a_3_5·a_3_8 + c_4_14·a_1_02·a_3_5·a_3_8
       + c_4_14·a_1_02·a_3_5·a_3_7 + c_4_13·a_1_0·a_1_1·a_3_5·a_3_8
       + c_4_13·a_1_0·a_1_1·a_3_5·a_3_7 + c_4_13·a_1_02·a_3_5·a_3_8

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_13, a Duflot regular element of degree 4
    2. c_4_14, a Duflot regular element of degree 4
    3. c_4_15, a Duflot regular element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9].
  • 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 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. a_1_3 → 0, an element of degree 1
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. a_3_7 → 0, an element of degree 3
  8. a_3_8 → 0, an element of degree 3
  9. a_4_10 → 0, an element of degree 4
  10. a_4_11 → 0, an element of degree 4
  11. a_4_12 → 0, an element of degree 4
  12. c_4_13 → c_1_24 + c_1_14 + c_1_04, an element of degree 4
  13. c_4_14 → c_1_24 + c_1_04, an element of degree 4
  14. c_4_15 → c_1_14 + c_1_04, an element of degree 4
  15. a_6_27 → 0, an element of degree 6
  16. a_6_28 → 0, an element of degree 6
  17. a_6_29 → 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