Cohomology of group number 239 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 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    t4  +  t3  +  t2  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-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 5:

  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. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_6, a nilpotent element of degree 3
  7. a_3_2, a nilpotent element of degree 3
  8. b_3_8, an element of degree 3
  9. b_3_10, an element of degree 3
  10. b_4_16, an element of degree 4
  11. b_4_17, an element of degree 4
  12. c_4_18, a Duflot regular element of degree 4
  13. c_4_19, a Duflot regular element of degree 4
  14. a_5_12, a nilpotent element of degree 5
  15. b_5_31, an element of degree 5

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 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_22
  4. b_1_1·b_1_22 + b_2_4·b_1_1
  5. b_2_5·a_1_0
  6. b_2_5·b_1_1
  7. b_1_24 + b_2_52 + b_2_4·b_1_22
  8. a_1_0·a_3_6
  9. b_1_1·a_3_6
  10. b_2_4·a_1_0·b_1_2 + a_1_0·a_3_2
  11. b_1_1·a_3_2
  12. a_1_0·b_3_8
  13. a_1_0·b_3_10
  14. b_1_22·a_3_2 + b_2_5·a_3_6
  15. b_2_5·b_3_8 + b_2_52·b_1_2 + b_1_22·a_3_6 + b_2_5·a_3_6
  16. b_1_22·b_3_8 + b_2_5·b_1_23 + b_2_4·b_3_8 + b_2_4·b_2_5·b_1_2 + b_1_22·a_3_6
       + b_2_5·a_3_6 + b_2_4·a_3_6
  17. b_2_5·b_3_10 + b_2_5·b_1_23 + b_2_52·b_1_2 + b_2_4·b_2_5·b_1_2 + b_1_22·a_3_6
       + b_2_5·a_3_2 + b_2_5·a_3_6
  18. b_1_22·b_3_10 + b_2_5·b_1_23 + b_2_52·b_1_2 + b_2_4·b_3_10 + b_2_4·b_1_23
       + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_5·a_3_2 + b_2_4·a_3_2
  19. b_4_16·a_1_0
  20. b_1_22·b_3_10 + b_1_22·b_3_8 + b_4_16·b_1_1 + b_2_52·b_1_2 + b_2_4·b_1_12·b_1_2
       + b_2_4·b_1_13 + b_2_42·b_1_1 + b_2_5·a_3_6
  21. b_1_22·a_3_6 + b_4_17·a_1_0 + b_2_5·a_3_2 + b_2_4·a_3_6
  22. b_1_22·b_3_10 + b_1_22·b_3_8 + b_1_1·b_1_2·b_3_8 + b_4_17·b_1_1 + b_2_52·b_1_2
       + b_2_4·b_1_13 + b_2_5·a_3_6
  23. a_3_62
  24. a_3_22
  25. a_3_2·b_3_8 + b_2_5·b_1_2·a_3_2 + a_3_6·a_3_2
  26. a_3_6·b_3_8 + b_2_5·b_1_2·a_3_6
  27. a_3_2·b_3_10 + b_2_5·b_1_2·a_3_2 + b_2_5·b_1_2·a_3_6 + b_2_4·b_1_2·a_3_2
  28. a_3_6·b_3_10 + b_2_5·b_1_2·a_3_2 + b_2_5·b_1_2·a_3_6
  29. b_3_102 + b_1_12·b_1_2·b_3_10 + b_1_13·b_3_8 + b_1_15·b_1_2 + b_2_4·b_1_1·b_3_10
       + b_2_4·b_1_13·b_1_2 + b_2_4·b_2_52 + c_4_18·b_1_12
  30. b_4_16·b_1_22 + b_2_5·b_4_17 + b_2_5·b_4_16 + b_2_52·b_1_22 + b_2_4·b_4_16
       + b_2_4·b_2_5·b_1_22 + b_2_4·b_2_52 + b_2_4·b_1_2·a_3_6 + a_3_6·a_3_2
  31. b_4_17·b_1_22 + b_4_16·b_1_22 + b_2_5·b_4_16 + b_2_53 + b_2_4·b_1_2·b_3_8
       + b_2_4·b_2_5·b_1_22 + b_2_4·b_2_52 + b_2_42·b_1_22 + b_2_42·b_1_1·b_1_2
       + b_2_4·b_1_2·a_3_6
  32. b_4_17·a_1_0·b_1_2 + a_3_6·a_3_2
  33. b_3_82 + b_1_12·b_1_2·b_3_10 + b_4_17·b_1_12 + b_2_52·b_1_22 + b_2_4·b_1_1·b_3_8
       + b_2_4·b_1_14 + c_4_19·b_1_12
  34. a_3_6·a_3_2 + a_1_0·a_5_12
  35. b_1_1·a_5_12
  36. a_1_0·b_5_31
  37. b_3_102 + b_3_8·b_3_10 + b_1_1·b_5_31 + b_1_13·b_3_8 + b_2_52·b_1_22 + b_2_53
       + b_2_4·b_1_2·b_3_10 + b_2_4·b_1_2·b_3_8 + b_2_4·b_1_1·b_3_8 + b_2_4·b_1_13·b_1_2
       + b_2_42·b_1_1·b_1_2 + b_2_42·b_1_12 + b_2_5·b_1_2·a_3_6 + b_2_4·b_1_2·a_3_2
       + a_3_6·a_3_2 + c_4_19·b_1_12
  38. b_4_16·b_3_10 + b_4_16·b_3_8 + b_2_5·b_4_17·b_1_2 + b_2_5·b_4_16·b_1_2
       + b_2_52·b_1_23 + b_2_4·b_1_1·b_1_2·b_3_10 + b_2_4·b_1_12·b_3_10
       + b_2_4·b_1_14·b_1_2 + b_2_4·b_2_5·b_1_23 + b_2_4·b_2_52·b_1_2 + b_2_42·b_3_10
       + b_2_42·b_3_8 + b_2_42·b_1_23 + b_2_42·b_1_12·b_1_2 + b_2_43·b_1_2 + b_4_16·a_3_2
       + b_2_4·b_2_5·a_3_2 + b_2_42·a_3_2 + b_2_42·a_3_6 + b_2_4·c_4_19·b_1_1
       + b_2_4·c_4_18·b_1_1
  39. b_4_17·a_3_6 + b_4_16·a_3_2 + b_4_16·a_3_6 + b_2_52·a_3_2 + b_2_4·b_4_17·a_1_0
       + b_2_4·b_2_5·a_3_2 + b_2_42·a_3_6 + b_2_4·c_4_19·a_1_0
  40. b_4_17·b_3_8 + b_4_16·b_3_10 + b_2_52·b_1_23 + b_2_4·b_1_12·b_3_8
       + b_2_4·b_1_14·b_1_2 + b_2_4·b_4_17·b_1_1 + b_2_4·b_2_5·b_1_23 + b_2_4·b_2_52·b_1_2
       + b_2_42·b_3_8 + b_2_42·b_1_12·b_1_2 + b_2_42·b_1_13 + b_2_42·b_2_5·b_1_2
       + b_4_16·a_3_6 + b_2_52·a_3_2 + b_2_52·a_3_6 + b_2_4·b_2_5·a_3_6 + b_2_42·a_3_6
       + c_4_19·b_1_12·b_1_2 + b_2_4·c_4_19·b_1_1 + b_2_4·c_4_18·b_1_1 + b_2_4·c_4_19·a_1_0
  41. b_4_16·a_3_2 + b_2_5·a_5_12 + b_2_52·a_3_2 + b_2_4·b_2_5·a_3_6
  42. b_1_22·a_5_12 + b_4_17·a_3_6 + b_4_16·a_3_2 + b_2_52·a_3_2 + b_2_52·a_3_6
       + b_2_4·c_4_19·a_1_0
  43. b_4_17·a_3_2 + b_4_17·a_3_6 + b_2_52·a_3_2 + b_2_52·a_3_6 + b_2_4·a_5_12
       + b_2_4·b_2_5·a_3_2 + b_2_4·b_2_5·a_3_6 + b_2_4·c_4_19·a_1_0
  44. b_2_5·b_5_31 + b_2_5·b_4_16·b_1_2 + b_2_52·b_1_23 + b_2_53·b_1_2
       + b_2_4·b_2_52·b_1_2 + b_4_16·a_3_6 + b_2_52·a_3_6 + b_2_4·b_2_5·a_3_6
  45. b_1_22·b_5_31 + b_4_16·b_3_10 + b_2_5·b_4_16·b_1_2 + b_2_52·b_1_23 + b_2_53·b_1_2
       + b_2_4·b_1_1·b_1_2·b_3_10 + b_2_4·b_1_12·b_3_10 + b_2_4·b_1_12·b_3_8
       + b_2_4·b_4_16·b_1_2 + b_2_42·b_3_10 + b_2_42·b_3_8 + b_2_42·b_1_23 + b_2_43·b_1_2
       + b_4_16·a_3_2 + b_4_16·a_3_6 + b_2_52·a_3_2 + b_2_4·b_2_5·a_3_2 + b_2_4·b_2_5·a_3_6
       + b_2_42·a_3_2 + b_2_42·a_3_6 + b_2_4·c_4_19·b_1_1
  46. b_1_1·b_1_2·b_5_31 + b_4_17·b_3_10 + b_4_16·b_3_8 + b_2_5·b_4_16·b_1_2
       + b_2_52·b_1_23 + b_2_53·b_1_2 + b_2_4·b_1_1·b_1_2·b_3_10 + b_2_4·b_1_14·b_1_2
       + b_2_4·b_1_15 + b_2_4·b_4_17·b_1_2 + b_2_4·b_4_17·b_1_1 + b_2_4·b_4_16·b_1_2
       + b_2_4·b_2_5·b_1_23 + b_2_42·b_1_23 + b_2_42·b_1_13 + b_4_17·a_3_2
       + b_4_16·a_3_2 + b_4_16·a_3_6 + b_2_52·a_3_2 + b_2_52·a_3_6 + c_4_19·b_1_12·b_1_2
       + c_4_18·b_1_12·b_1_2 + b_2_4·c_4_19·b_1_1 + b_2_4·c_4_18·b_1_1
  47. b_4_16·b_3_8 + b_2_5·b_4_16·b_1_2 + b_2_4·b_5_31 + b_2_4·b_1_12·b_3_8
       + b_2_4·b_1_14·b_1_2 + b_2_4·b_4_16·b_1_2 + b_2_4·b_2_5·b_1_23 + b_2_4·b_2_52·b_1_2
       + b_2_42·b_1_12·b_1_2 + b_2_42·b_2_5·b_1_2 + b_4_17·a_3_6 + b_2_52·a_3_2
       + b_2_4·b_2_5·a_3_2 + b_2_4·b_2_5·a_3_6 + b_2_4·c_4_18·b_1_1 + b_2_4·c_4_18·a_1_0
  48. b_4_162 + b_2_52·b_4_17 + b_2_54 + b_2_4·b_2_5·b_4_16 + b_2_4·b_2_52·b_1_22
       + b_2_42·b_1_2·b_3_8 + b_2_42·b_1_1·b_3_8 + b_2_42·b_1_13·b_1_2 + b_2_42·b_1_14
       + b_2_42·b_2_52 + b_2_43·b_1_22 + b_2_43·b_1_1·b_1_2 + b_2_43·b_1_12
       + b_2_52·b_1_2·a_3_6 + b_2_42·b_1_2·a_3_6 + b_2_52·c_4_18 + b_2_4·c_4_19·b_1_22
       + b_2_4·c_4_18·b_1_22
  49. a_3_2·a_5_12
  50. a_3_6·a_5_12 + c_4_19·a_1_0·a_3_2
  51. b_4_172 + b_2_54 + b_2_4·b_1_12·b_1_2·b_3_10 + b_2_4·b_4_17·b_1_12
       + b_2_4·b_2_5·b_4_17 + b_2_4·b_2_52·b_1_22 + b_2_4·b_2_53 + b_2_42·b_1_2·b_3_8
       + b_2_42·b_1_13·b_1_2 + b_2_42·b_2_5·b_1_22 + b_2_43·b_1_1·b_1_2 + b_3_10·a_5_12
       + b_4_16·b_1_2·a_3_6 + b_2_5·b_1_2·a_5_12 + b_2_52·b_1_2·a_3_6 + b_2_4·b_1_2·a_5_12
       + b_2_42·b_1_2·a_3_6 + b_2_4·c_4_19·b_1_12 + b_2_4·c_4_18·b_1_22 + b_2_42·c_4_19
  52. b_4_172 + b_2_54 + b_2_4·b_1_12·b_1_2·b_3_10 + b_2_4·b_4_17·b_1_12
       + b_2_4·b_2_5·b_4_17 + b_2_4·b_2_52·b_1_22 + b_2_4·b_2_53 + b_2_42·b_1_2·b_3_8
       + b_2_42·b_1_13·b_1_2 + b_2_42·b_2_5·b_1_22 + b_2_43·b_1_1·b_1_2
       + b_2_4·b_2_5·b_1_2·a_3_2 + b_2_4·a_1_0·a_5_12 + b_2_4·c_4_19·b_1_12
       + b_2_4·c_4_18·b_1_22 + b_2_42·c_4_19
  53. b_3_8·a_5_12 + b_2_5·b_1_2·a_5_12 + c_4_19·a_1_0·a_3_2
  54. a_3_2·b_5_31 + b_2_5·b_1_2·a_5_12 + b_2_52·b_1_2·a_3_6 + b_2_4·b_2_5·b_1_2·a_3_2
       + b_2_4·b_2_5·b_1_2·a_3_6 + c_4_19·a_1_0·a_3_2 + c_4_18·a_1_0·a_3_2
  55. a_3_6·b_5_31 + b_4_16·b_1_2·a_3_6 + b_2_52·b_1_2·a_3_2 + b_2_52·b_1_2·a_3_6
  56. b_4_172 + b_4_16·b_4_17 + b_4_162 + b_2_52·b_4_16 + b_2_53·b_1_22
       + b_2_4·b_1_2·b_5_31 + b_2_4·b_1_12·b_1_2·b_3_10 + b_2_4·b_2_5·b_4_17
       + b_2_4·b_2_52·b_1_22 + b_2_4·b_2_53 + b_2_42·b_1_2·b_3_10 + b_2_42·b_1_1·b_3_10
       + b_2_42·b_1_1·b_3_8 + b_2_42·b_2_52 + b_2_52·b_1_2·a_3_2
       + b_2_4·b_2_5·b_1_2·a_3_6 + b_2_42·b_1_2·a_3_2 + b_2_5·c_4_18·b_1_22
       + b_2_4·c_4_19·b_1_12 + b_2_4·c_4_18·b_1_22 + b_2_4·c_4_18·b_1_1·b_1_2
       + b_2_4·b_2_5·c_4_19 + b_2_42·c_4_19 + c_4_19·a_1_0·a_3_2 + c_4_18·a_1_0·a_3_2
  57. b_3_10·b_5_31 + b_4_17·b_1_1·b_3_10 + b_4_17·b_1_14 + b_4_172 + b_4_16·b_4_17
       + b_2_52·b_4_16 + b_2_54 + b_2_4·b_1_1·b_5_31 + b_2_4·b_1_12·b_1_2·b_3_10
       + b_2_4·b_1_13·b_3_8 + b_2_4·b_1_15·b_1_2 + b_2_4·b_2_5·b_4_16
       + b_2_4·b_2_52·b_1_22 + b_2_42·b_1_2·b_3_10 + b_2_42·b_1_2·b_3_8
       + b_2_42·b_1_1·b_3_10 + b_2_42·b_1_1·b_3_8 + b_2_42·b_1_13·b_1_2 + b_2_42·b_4_16
       + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52 + b_2_43·b_1_22 + b_2_5·b_1_2·a_5_12
       + b_2_52·b_1_2·a_3_6 + b_2_42·b_1_2·a_3_2 + b_2_42·b_1_2·a_3_6 + c_4_19·b_1_1·b_3_10
       + c_4_19·b_1_14 + c_4_18·b_1_1·b_3_10 + c_4_18·b_1_1·b_3_8 + c_4_18·b_1_13·b_1_2
       + b_2_5·c_4_18·b_1_22 + b_2_52·c_4_18 + b_2_4·c_4_19·b_1_22
       + b_2_4·c_4_19·b_1_1·b_1_2 + b_2_4·c_4_18·b_1_1·b_1_2 + b_2_4·c_4_18·b_1_12
       + b_2_4·b_2_5·c_4_19 + b_2_42·c_4_19
  58. b_3_10·b_5_31 + b_3_8·b_5_31 + b_4_17·b_1_1·b_3_10 + b_4_17·b_1_14 + b_4_172
       + b_2_52·b_4_16 + b_2_53·b_1_22 + b_2_4·b_1_12·b_1_2·b_3_10 + b_2_4·b_1_13·b_3_10
       + b_2_4·b_1_15·b_1_2 + b_2_4·b_1_16 + b_2_4·b_2_5·b_4_17 + b_2_4·b_2_52·b_1_22
       + b_2_42·b_1_2·b_3_10 + b_2_42·b_1_1·b_3_8 + b_2_42·b_2_5·b_1_22
       + b_2_42·b_2_52 + b_2_43·b_1_1·b_1_2 + b_2_43·b_1_12 + b_2_52·b_1_2·a_3_6
       + b_2_4·b_2_5·b_1_2·a_3_2 + b_2_42·b_1_2·a_3_2 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_14
       + c_4_18·b_1_1·b_3_10 + b_2_4·c_4_19·b_1_1·b_1_2 + b_2_4·c_4_18·b_1_22
       + b_2_4·c_4_18·b_1_1·b_1_2 + b_2_4·c_4_18·b_1_12 + b_2_42·c_4_19
  59. b_4_17·a_5_12 + b_2_52·a_5_12 + b_2_53·a_3_2 + b_2_53·a_3_6 + b_2_4·b_2_52·a_3_6
       + b_4_17·c_4_19·a_1_0 + b_4_17·c_4_18·a_1_0 + b_2_5·c_4_18·a_3_2 + b_2_5·c_4_18·a_3_6
       + b_2_4·c_4_19·a_3_2 + b_2_4·c_4_19·a_3_6 + b_2_4·c_4_18·a_3_6
  60. b_4_16·a_5_12 + b_2_5·b_4_16·a_3_6 + b_2_53·a_3_2 + b_2_53·a_3_6 + b_2_4·b_4_16·a_3_6
       + b_2_4·b_2_52·a_3_2 + b_2_42·b_4_17·a_1_0 + b_2_42·b_2_5·a_3_2
       + b_2_42·b_2_5·a_3_6 + b_2_43·a_3_6 + b_4_17·c_4_19·a_1_0 + b_4_17·c_4_18·a_1_0
       + b_2_5·c_4_18·a_3_2 + b_2_4·c_4_19·a_3_6 + b_2_4·c_4_18·a_3_6
  61. b_4_17·b_5_31 + b_2_52·b_4_17·b_1_2 + b_2_52·b_4_16·b_1_2 + b_2_54·b_1_2
       + b_2_4·b_1_12·b_5_31 + b_2_4·b_1_13·b_1_2·b_3_10 + b_2_4·b_1_16·b_1_2
       + b_2_4·b_2_5·b_4_16·b_1_2 + b_2_4·b_2_53·b_1_2 + b_2_42·b_1_12·b_3_8
       + b_2_42·b_1_14·b_1_2 + b_2_42·b_1_15 + b_2_42·b_4_17·b_1_1
       + b_2_42·b_4_16·b_1_2 + b_2_43·b_1_23 + b_2_43·b_1_12·b_1_2 + b_2_43·b_1_13
       + b_2_5·b_4_16·a_3_6 + b_2_53·a_3_6 + b_2_4·b_2_5·a_5_12 + b_2_4·b_2_52·a_3_2
       + b_2_4·b_2_52·a_3_6 + c_4_19·b_1_1·b_1_2·b_3_10 + b_4_17·c_4_19·b_1_1
       + b_4_17·c_4_18·b_1_1 + b_2_5·c_4_18·b_1_23 + b_2_52·c_4_18·b_1_2
       + b_2_4·c_4_19·b_3_10 + b_2_4·c_4_19·b_1_12·b_1_2 + b_2_4·c_4_18·b_3_8
       + b_2_4·c_4_18·b_1_23 + b_2_4·c_4_18·b_1_12·b_1_2 + b_2_4·b_2_5·c_4_18·b_1_2
       + b_2_42·c_4_19·b_1_2 + b_4_17·c_4_19·a_1_0 + b_2_5·c_4_18·a_3_2 + b_2_5·c_4_18·a_3_6
       + b_2_4·c_4_19·a_3_2
  62. b_4_16·b_5_31 + b_2_53·b_1_23 + b_2_54·b_1_2 + b_2_4·b_1_12·b_5_31
       + b_2_4·b_4_17·b_1_13 + b_2_4·b_2_5·b_4_16·b_1_2 + b_2_4·b_2_53·b_1_2
       + b_2_42·b_1_1·b_1_2·b_3_10 + b_2_42·b_1_12·b_3_10 + b_2_42·b_1_12·b_3_8
       + b_2_42·b_1_14·b_1_2 + b_2_42·b_2_5·b_1_23 + b_2_42·b_2_52·b_1_2
       + b_2_43·b_1_23 + b_2_43·b_1_12·b_1_2 + b_2_43·b_1_13 + b_2_44·b_1_1
       + b_2_52·a_5_12 + b_2_53·a_3_2 + b_2_53·a_3_6 + b_2_4·b_2_52·a_3_2
       + b_2_42·b_4_17·a_1_0 + b_2_42·b_2_5·a_3_2 + b_2_42·b_2_5·a_3_6 + b_2_43·a_3_6
       + b_2_52·c_4_18·b_1_2 + b_2_4·c_4_19·b_3_8 + b_2_4·c_4_19·b_1_23
       + b_2_4·c_4_19·b_1_12·b_1_2 + b_2_4·c_4_19·b_1_13 + b_2_4·c_4_18·b_3_10
       + b_2_4·c_4_18·b_1_12·b_1_2 + b_2_4·b_2_5·c_4_19·b_1_2 + b_2_4·b_2_5·c_4_18·b_1_2
       + b_2_42·c_4_18·b_1_2 + b_2_42·c_4_18·b_1_1 + b_4_17·c_4_18·a_1_0
       + b_2_5·c_4_18·a_3_6 + b_2_4·c_4_19·a_3_6 + b_2_4·c_4_18·a_3_2 + b_2_4·c_4_18·a_3_6
  63. a_5_122
  64. b_5_312 + b_4_17·b_1_13·b_3_10 + b_2_53·b_4_17 + b_2_55 + b_2_4·b_1_13·b_5_31
       + b_2_4·b_1_15·b_3_10 + b_2_4·b_1_15·b_3_8 + b_2_4·b_1_18
       + b_2_4·b_4_17·b_1_1·b_3_10 + b_2_4·b_2_52·b_4_17 + b_2_42·b_1_1·b_5_31
       + b_2_42·b_1_12·b_1_2·b_3_10 + b_2_42·b_1_13·b_3_8 + b_2_42·b_1_15·b_1_2
       + b_2_42·b_2_5·b_4_16 + b_2_43·b_1_2·b_3_10 + b_2_43·b_1_1·b_3_10
       + b_2_43·b_1_13·b_1_2 + b_2_43·b_1_14 + b_2_44·b_1_22 + b_2_53·b_1_2·a_3_2
       + b_2_4·b_2_52·b_1_2·a_3_2 + b_2_42·b_2_5·b_1_2·a_3_6 + b_2_43·b_1_2·a_3_2
       + b_2_43·b_1_2·a_3_6 + b_2_42·a_1_0·a_5_12 + c_4_19·b_1_12·b_1_2·b_3_10
       + c_4_19·b_1_13·b_3_8 + c_4_18·b_1_12·b_1_2·b_3_10 + b_4_17·c_4_18·b_1_12
       + b_2_52·c_4_18·b_1_22 + b_2_4·c_4_19·b_1_1·b_3_10 + b_2_4·c_4_19·b_1_13·b_1_2
       + b_2_4·c_4_19·b_1_14 + b_2_4·c_4_18·b_1_1·b_3_8 + b_2_4·c_4_18·b_1_14
       + b_2_4·b_2_52·c_4_19 + b_2_4·b_2_52·c_4_18 + b_2_42·c_4_19·b_1_22
       + b_2_42·c_4_18·b_1_22 + c_4_192·b_1_12 + c_4_18·c_4_19·b_1_12
       + c_4_182·b_1_12
  65. a_5_12·b_5_31 + b_2_52·b_1_2·a_5_12 + b_2_53·b_1_2·a_3_2 + b_2_4·b_4_16·b_1_2·a_3_6
       + b_2_4·b_2_5·b_1_2·a_5_12 + b_2_42·b_2_5·b_1_2·a_3_2 + b_2_43·b_1_2·a_3_6
       + b_2_42·a_1_0·a_5_12 + b_2_5·c_4_18·b_1_2·a_3_2 + b_2_4·c_4_19·b_1_2·a_3_6
       + b_2_4·c_4_18·b_1_2·a_3_6

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 10.
  • 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_18, a Duflot regular element of degree 4
    2. c_4_19, a Duflot regular element of degree 4
    3. b_1_1·b_1_2 + b_1_12 + b_2_5 + b_2_4, an element of degree 2
    4. b_1_23 + b_1_12·b_1_2 + b_2_5·b_1_2 + b_2_4·b_1_2 + b_2_4·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 9].
  • 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. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. a_3_6 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. b_3_8 → 0, an element of degree 3
  9. b_3_10 → 0, an element of degree 3
  10. b_4_16 → 0, an element of degree 4
  11. b_4_17 → 0, an element of degree 4
  12. c_4_18 → c_1_04, an element of degree 4
  13. c_4_19 → c_1_14 + c_1_04, an element of degree 4
  14. a_5_12 → 0, an element of degree 5
  15. b_5_31 → 0, an element of degree 5

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 → c_1_3, an element of degree 1
  4. b_2_4 → c_1_32, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. a_3_6 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22
       + c_1_02·c_1_2, an element of degree 3
  9. b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  10. b_4_16 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32, an element of degree 4
  11. b_4_17 → c_1_2·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_2·c_1_3, an element of degree 4
  12. c_4_18 → c_1_2·c_1_33 + c_1_22·c_1_32 + 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
  13. c_4_19 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32
       + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. a_5_12 → 0, an element of degree 5
  15. b_5_31 → c_1_35 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_1·c_1_2·c_1_33 + c_1_1·c_1_24
       + c_1_12·c_1_33 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_0·c_1_1·c_1_23
       + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2
       + c_1_04·c_1_2, an element of degree 5

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_3, an element of degree 1
  4. b_2_4 → c_1_22, 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_6 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. b_3_8 → c_1_33 + c_1_2·c_1_32, an element of degree 3
  9. b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  10. b_4_16 → c_1_34 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3, an element of degree 4
  11. b_4_17 → c_1_34 + c_1_23·c_1_3 + 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_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  12. c_4_18 → c_1_34 + c_1_23·c_1_3 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. c_4_19 → c_1_34 + c_1_2·c_1_33 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_14
       + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  14. a_5_12 → 0, an element of degree 5
  15. b_5_31 → c_1_35 + c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_0·c_1_34
       + c_1_0·c_1_2·c_1_33 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32, an element of degree 5

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