Cohomology of group number 332 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 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 4 and 4, respectively.

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
    (2) · (t4  +  1/2·t3  +  1/2·t2  +  1/2·t  +  1/2)
    (t  +  1)2 · (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 6:

  1. b_1_0, an 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. b_3_7, an 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. b_4_15, an element of degree 4
  12. c_4_16, a Duflot regular element of degree 4
  13. c_4_17, a Duflot regular element of degree 4
  14. b_5_27, an element of degree 5
  15. b_6_39, 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. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_1_1·b_1_22
  4. b_1_23 + b_2_4·b_1_2
  5. b_2_4·b_1_1
  6. b_2_5·b_1_1
  7. b_1_2·b_3_7 + b_2_4·b_1_22
  8. b_1_1·b_3_7
  9. b_1_2·b_3_8 + b_2_5·b_1_22 + b_2_4·b_1_22
  10. b_1_0·b_3_8 + b_1_0·b_3_7 + b_2_4·b_1_22 + b_2_42
  11. b_1_1·b_3_8
  12. b_1_0·b_3_9
  13. b_1_22·b_3_9 + b_2_4·b_3_9
  14. b_1_22·b_3_9 + b_4_13·b_1_2 + b_2_42·b_1_2
  15. b_1_02·b_3_7 + b_4_13·b_1_0 + b_2_4·b_3_7 + b_2_4·b_1_03 + b_2_42·b_1_2
       + b_2_42·b_1_0
  16. b_4_13·b_1_1
  17. b_4_14·b_1_0 + b_2_5·b_3_7 + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_0
  18. b_1_12·b_3_9 + b_4_14·b_1_1
  19. b_4_15·b_1_2 + b_4_14·b_1_2 + b_2_5·b_3_9 + b_2_42·b_1_2
  20. b_1_02·b_3_7 + b_4_15·b_1_0 + b_2_4·b_3_8 + b_2_4·b_3_7 + b_2_4·b_2_5·b_1_2
       + b_2_4·b_2_5·b_1_0
  21. b_1_1·b_1_2·b_3_9 + b_1_12·b_3_9 + b_4_15·b_1_1
  22. b_3_7·b_3_9 + b_2_4·b_1_2·b_3_9
  23. b_3_8·b_3_9 + b_2_5·b_1_2·b_3_9 + b_2_4·b_1_2·b_3_9
  24. b_3_82 + b_3_72 + b_2_52·b_1_22 + b_2_4·b_2_5·b_1_22 + b_2_4·b_2_5·b_1_02
       + b_2_42·b_1_02 + b_2_42·b_2_5 + c_4_16·b_1_02
  25. b_3_7·b_3_8 + b_3_72 + b_2_4·b_1_2·b_3_9 + b_2_4·b_1_0·b_3_7 + b_2_4·b_4_13
       + b_2_4·b_2_5·b_1_22 + b_2_42·b_1_02 + b_2_43
  26. b_3_82 + b_4_13·b_1_02 + b_2_5·b_1_0·b_3_7 + b_2_52·b_1_22 + b_2_4·b_2_5·b_1_22
       + b_2_42·b_1_02 + b_2_42·b_2_5 + b_2_43 + c_4_17·b_1_02
  27. b_4_14·b_1_22 + b_2_5·b_1_2·b_3_9 + b_2_5·b_1_0·b_3_7 + b_2_5·b_4_13 + b_2_4·b_4_14
       + b_2_4·b_2_5·b_1_02 + b_2_42·b_1_22 + b_2_42·b_2_5 + b_2_43
  28. b_3_92 + b_4_14·b_1_1·b_1_2 + b_4_14·b_1_12 + b_2_5·b_1_2·b_3_9
       + b_2_4·b_2_5·b_1_22 + b_2_42·b_1_22 + c_4_17·b_1_22 + c_4_17·b_1_12
       + c_4_16·b_1_22
  29. b_3_82 + b_3_72 + b_4_14·b_1_22 + b_2_5·b_1_2·b_3_9 + b_2_52·b_1_22
       + b_2_4·b_1_0·b_3_7 + b_2_4·b_4_15 + b_2_4·b_2_5·b_1_22 + b_2_42·b_1_22
       + b_2_42·b_2_5
  30. b_1_2·b_5_27 + b_4_14·b_1_1·b_1_2 + b_2_5·b_1_2·b_3_9 + b_2_52·b_1_22
  31. b_3_7·b_3_8 + b_1_0·b_5_27 + b_2_5·b_1_0·b_3_7 + b_2_4·b_1_0·b_3_7 + b_2_42·b_1_02
       + b_2_42·b_2_5 + b_2_43 + c_4_17·b_1_02
  32. b_3_92 + b_1_1·b_5_27 + b_4_14·b_1_1·b_1_2 + b_2_5·b_1_2·b_3_9 + b_2_4·b_2_5·b_1_22
       + b_2_42·b_1_22 + c_4_17·b_1_22 + c_4_17·b_1_12 + c_4_16·b_1_22
  33. b_4_13·b_3_7 + b_4_13·b_1_03 + b_2_5·b_4_13·b_1_0 + b_2_42·b_3_9 + b_2_42·b_3_8
       + b_2_42·b_3_7 + b_2_42·b_1_03 + b_2_42·b_2_5·b_1_2 + b_2_43·b_1_2
       + c_4_17·b_1_03 + c_4_16·b_1_03 + b_2_4·c_4_17·b_1_0 + b_2_4·c_4_16·b_1_0
  34. b_4_13·b_3_9 + b_2_4·b_2_5·b_3_9 + b_2_42·b_3_9 + b_2_42·b_2_5·b_1_2 + b_2_43·b_1_2
       + b_2_4·c_4_17·b_1_2 + b_2_4·c_4_16·b_1_2
  35. b_4_14·b_3_7 + b_2_5·b_4_13·b_1_0 + b_2_52·b_3_7 + b_2_4·b_4_14·b_1_2
       + b_2_4·b_2_5·b_3_8 + b_2_4·b_2_5·b_3_7 + b_2_4·b_2_52·b_1_0 + b_2_42·b_3_7
       + b_2_43·b_1_2 + b_2_5·c_4_17·b_1_0 + b_2_5·c_4_16·b_1_0
  36. b_4_15·b_3_7 + b_4_13·b_3_8 + b_2_4·b_4_14·b_1_2 + b_2_42·b_3_9 + b_2_42·b_3_7
       + b_2_42·b_1_03 + b_2_42·b_2_5·b_1_2 + b_2_42·b_2_5·b_1_0 + b_2_43·b_1_2
       + b_2_4·c_4_17·b_1_0 + b_2_4·c_4_16·b_1_0
  37. b_4_15·b_3_8 + b_4_13·b_3_8 + b_2_5·b_4_14·b_1_2 + b_2_52·b_3_9 + b_2_4·b_4_14·b_1_2
       + b_2_42·b_3_9 + b_2_42·b_1_03 + b_2_43·b_1_0 + b_2_4·c_4_17·b_1_0
  38. b_4_15·b_3_9 + b_4_14·b_3_9 + b_4_14·b_1_12·b_1_2 + b_2_52·b_3_9 + b_2_4·b_2_52·b_1_2
       + b_2_42·b_3_9 + b_2_42·b_2_5·b_1_2 + c_4_17·b_1_12·b_1_2 + b_2_5·c_4_17·b_1_2
       + b_2_5·c_4_16·b_1_2
  39. b_4_14·b_3_8 + b_2_5·b_5_27 + b_2_5·b_4_14·b_1_2 + b_2_52·b_3_9 + b_2_52·b_3_8
       + b_2_4·b_4_14·b_1_2 + b_2_4·b_2_5·b_3_8 + b_2_42·b_3_8 + b_2_42·b_2_5·b_1_0
       + b_2_43·b_1_2 + b_2_5·c_4_17·b_1_0
  40. b_4_13·b_3_8 + b_4_13·b_1_03 + b_2_5·b_4_13·b_1_0 + b_2_4·b_5_27 + b_2_4·b_4_13·b_1_0
       + b_2_4·b_2_5·b_3_8 + b_2_4·b_2_5·b_3_7 + b_2_42·b_3_9 + b_2_42·b_1_03
       + b_2_42·b_2_5·b_1_2 + b_2_42·b_2_5·b_1_0 + b_2_43·b_1_2 + c_4_17·b_1_03
       + c_4_16·b_1_03 + b_2_4·c_4_17·b_1_0
  41. b_6_39·b_1_2 + b_4_14·b_3_9 + b_4_14·b_1_13 + b_2_5·b_4_14·b_1_2 + b_2_4·b_2_52·b_1_2
       + c_4_17·b_1_12·b_1_2 + c_4_17·b_1_13 + b_2_5·c_4_16·b_1_2 + b_2_4·c_4_17·b_1_2
  42. b_6_39·b_1_0 + b_4_13·b_3_8 + b_2_4·b_2_5·b_3_9 + b_2_4·b_2_5·b_3_8
       + b_2_4·b_2_52·b_1_2 + b_2_4·b_2_52·b_1_0 + b_2_42·b_3_9 + b_2_42·b_3_8
       + b_2_42·b_1_03 + b_2_42·b_2_5·b_1_2 + b_2_43·b_1_0 + c_4_16·b_1_03
       + b_2_5·c_4_17·b_1_0 + b_2_4·c_4_16·b_1_0
  43. b_6_39·b_1_1 + b_4_14·b_1_12·b_1_2 + b_4_14·b_1_13 + c_4_17·b_1_12·b_1_2
       + c_4_17·b_1_13 + c_4_16·b_1_12·b_1_2
  44. b_4_13·b_1_04 + b_4_132 + b_2_5·b_4_13·b_1_02 + b_2_4·b_2_5·b_4_13
       + b_2_42·b_1_2·b_3_9 + b_2_42·b_1_04 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22
       + b_2_42·b_2_5·b_1_02 + b_2_43·b_2_5 + b_2_44 + c_4_17·b_1_04 + c_4_16·b_1_04
       + b_2_4·c_4_16·b_1_02 + b_2_42·c_4_17 + b_2_42·c_4_16
  45. b_4_14·b_1_2·b_3_9 + b_4_14·b_1_14 + b_4_142 + b_4_13·b_4_14 + b_2_5·b_4_13·b_1_02
       + b_2_52·b_1_2·b_3_9 + b_2_52·b_4_15 + b_2_52·b_4_14 + b_2_52·b_4_13
       + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_5·b_4_14 + b_2_4·b_2_52·b_1_22
       + b_2_42·b_1_2·b_3_9 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_2_43·b_2_5 + b_2_44 + c_4_17·b_1_13·b_1_2 + c_4_17·b_1_14
       + b_2_5·c_4_17·b_1_22 + b_2_5·c_4_17·b_1_02 + b_2_5·c_4_16·b_1_22
       + b_2_52·c_4_17 + b_2_52·c_4_16 + b_2_4·c_4_17·b_1_22 + b_2_4·b_2_5·c_4_17
       + b_2_4·b_2_5·c_4_16
  46. b_4_14·b_1_2·b_3_9 + b_4_14·b_1_13·b_1_2 + b_4_13·b_4_14 + b_2_5·b_4_13·b_1_02
       + b_2_52·b_1_2·b_3_9 + b_2_52·b_4_13 + b_2_4·b_2_5·b_1_2·b_3_9
       + b_2_4·b_2_52·b_1_02 + b_2_42·b_1_2·b_3_9 + b_2_42·b_1_0·b_3_7 + b_2_42·b_4_15
       + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_02 + b_2_42·b_2_52 + b_2_43·b_1_02
       + c_4_17·b_1_13·b_1_2 + b_2_5·c_4_17·b_1_22 + b_2_5·c_4_17·b_1_02
       + b_2_5·c_4_16·b_1_22 + b_2_4·c_4_16·b_1_02 + b_2_4·b_2_5·c_4_17
       + b_2_4·b_2_5·c_4_16
  47. b_4_13·b_1_04 + b_4_13·b_4_15 + b_4_13·b_4_14 + b_2_52·b_4_13
       + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_52·b_1_22 + b_2_4·b_2_52·b_1_02
       + b_2_42·b_1_2·b_3_9 + b_2_42·b_1_04 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_2_44 + c_4_17·b_1_04 + c_4_16·b_1_0·b_3_7 + c_4_16·b_1_04
       + b_2_5·c_4_17·b_1_02 + b_2_4·c_4_17·b_1_02 + b_2_4·b_2_5·c_4_17
       + b_2_4·b_2_5·c_4_16
  48. b_4_152 + b_4_14·b_1_2·b_3_9 + b_4_14·b_1_14 + b_4_13·b_1_04 + b_4_13·b_4_14
       + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_5·b_4_15
       + b_2_4·b_2_52·b_1_22 + b_2_4·b_2_52·b_1_02 + b_2_42·b_1_2·b_3_9
       + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_02 + b_2_42·b_2_52 + b_2_43·b_1_22
       + b_2_43·b_1_02 + b_2_43·b_2_5 + b_2_44 + c_4_17·b_1_13·b_1_2 + c_4_17·b_1_14
       + c_4_17·b_1_04 + c_4_16·b_1_04 + b_2_5·c_4_17·b_1_22 + b_2_5·c_4_17·b_1_02
       + b_2_5·c_4_16·b_1_22 + b_2_4·c_4_17·b_1_22 + b_2_4·c_4_16·b_1_22
       + b_2_4·b_2_5·c_4_17 + b_2_4·b_2_5·c_4_16 + b_2_42·c_4_16
  49. b_3_7·b_5_27 + b_4_13·b_1_04 + b_2_5·b_4_13·b_1_02 + b_2_4·b_4_13·b_1_02
       + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_52·b_1_22 + b_2_42·b_1_2·b_3_9
       + b_2_42·b_1_0·b_3_7 + b_2_42·b_1_04 + b_2_42·b_4_13 + b_2_43·b_1_22
       + b_2_43·b_1_02 + c_4_17·b_1_04 + c_4_16·b_1_0·b_3_7 + c_4_16·b_1_04
       + b_2_4·c_4_17·b_1_22 + b_2_4·c_4_16·b_1_22 + b_2_4·c_4_16·b_1_02
       + b_2_42·c_4_17 + b_2_42·c_4_16
  50. b_3_8·b_5_27 + b_4_13·b_1_04 + b_2_5·b_4_13·b_1_02 + b_2_52·b_1_2·b_3_9
       + b_2_53·b_1_22 + b_2_4·b_4_13·b_1_02 + b_2_4·b_2_5·b_4_13
       + b_2_4·b_2_52·b_1_02 + b_2_42·b_1_2·b_3_9 + b_2_42·b_1_0·b_3_7 + b_2_42·b_1_04
       + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52 + b_2_43·b_1_22
       + c_4_17·b_1_04 + c_4_16·b_1_04 + b_2_5·c_4_16·b_1_02 + b_2_4·c_4_17·b_1_22
       + b_2_4·c_4_16·b_1_02 + b_2_42·c_4_17
  51. b_3_9·b_5_27 + b_4_14·b_1_13·b_1_2 + b_4_14·b_1_14 + b_2_4·b_2_52·b_1_22
       + b_2_42·b_2_5·b_1_22 + c_4_17·b_1_14 + b_2_5·c_4_17·b_1_22
       + b_2_5·c_4_16·b_1_22
  52. b_4_14·b_1_13·b_1_2 + b_4_14·b_4_15 + b_4_142 + b_2_5·b_6_39 + b_2_52·b_4_14
       + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_5·b_4_15
       + b_2_4·b_2_5·b_4_14 + b_2_4·b_2_5·b_4_13 + b_2_42·b_1_0·b_3_7 + b_2_42·b_4_14
       + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52 + b_2_43·b_1_02 + b_2_43·b_2_5
       + c_4_17·b_1_13·b_1_2 + b_2_5·c_4_16·b_1_02 + b_2_52·c_4_16 + b_2_4·c_4_16·b_1_02
       + b_2_4·b_2_5·c_4_17
  53. b_4_14·b_1_2·b_3_9 + b_4_14·b_1_13·b_1_2 + b_2_52·b_1_2·b_3_9 + b_2_4·b_6_39
       + b_2_4·b_4_13·b_1_02 + b_2_4·b_2_5·b_1_2·b_3_9 + b_2_4·b_2_5·b_4_15
       + b_2_4·b_2_5·b_4_13 + b_2_4·b_2_52·b_1_22 + b_2_42·b_1_0·b_3_7
       + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_5·b_1_02 + b_2_43·b_1_02 + b_2_43·b_2_5
       + c_4_17·b_1_13·b_1_2 + c_4_16·b_1_0·b_3_7 + b_2_5·c_4_17·b_1_22
       + b_2_5·c_4_16·b_1_22 + b_2_4·c_4_17·b_1_02 + b_2_4·c_4_16·b_1_02
       + b_2_4·b_2_5·c_4_17 + b_2_42·c_4_17
  54. b_4_14·b_5_27 + b_4_142·b_1_1 + b_2_5·b_4_14·b_3_9 + b_2_5·b_4_13·b_1_03
       + b_2_52·b_4_14·b_1_2 + b_2_52·b_4_13·b_1_0 + b_2_4·b_2_5·b_5_27
       + b_2_4·b_2_52·b_3_9 + b_2_4·b_2_52·b_3_8 + b_2_4·b_2_52·b_3_7 + b_2_42·b_5_27
       + b_2_42·b_2_5·b_3_9 + b_2_42·b_2_5·b_3_8 + b_2_42·b_2_52·b_1_0 + b_2_43·b_3_8
       + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_5·c_4_17·b_3_8 + b_2_5·c_4_17·b_3_7
       + b_2_5·c_4_17·b_1_03 + b_2_5·c_4_16·b_3_8 + b_2_5·c_4_16·b_1_03
       + b_2_52·c_4_17·b_1_2 + b_2_52·c_4_16·b_1_2 + b_2_4·c_4_16·b_1_03
       + b_2_4·b_2_5·c_4_16·b_1_2
  55. b_4_13·b_5_27 + b_4_132·b_1_0 + b_2_4·b_4_13·b_1_03 + b_2_42·b_2_5·b_3_9
       + b_2_42·b_2_5·b_3_8 + b_2_42·b_2_5·b_3_7 + b_2_42·b_2_52·b_1_2 + b_2_43·b_3_8
       + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_43·b_2_5·b_1_0 + b_4_13·c_4_16·b_1_0
       + b_2_4·c_4_17·b_3_8 + b_2_4·c_4_17·b_3_7 + b_2_4·c_4_17·b_1_03 + b_2_4·c_4_16·b_3_8
       + b_2_4·c_4_16·b_3_7 + b_2_4·c_4_16·b_1_03 + b_2_42·c_4_16·b_1_0
  56. b_4_15·b_5_27 + b_4_142·b_1_2 + b_4_142·b_1_1 + b_4_132·b_1_0 + b_2_5·b_4_14·b_3_9
       + b_2_52·b_4_14·b_1_2 + b_2_53·b_3_9 + b_2_4·b_4_13·b_1_03
       + b_2_4·b_2_5·b_4_14·b_1_2 + b_2_4·b_2_53·b_1_2 + b_2_42·b_4_14·b_1_2
       + b_2_42·b_4_13·b_1_0 + b_2_42·b_2_5·b_3_8 + b_2_42·b_2_5·b_3_7
       + b_2_42·b_2_52·b_1_2 + b_2_43·b_3_8 + b_2_43·b_1_03 + b_2_43·b_2_5·b_1_2
       + b_2_44·b_1_2 + b_2_44·b_1_0 + b_4_13·c_4_16·b_1_0 + b_2_4·c_4_16·b_3_8
       + b_2_4·c_4_16·b_3_7 + b_2_4·b_2_5·c_4_16·b_1_2 + b_2_42·c_4_17·b_1_2
       + b_2_42·c_4_16·b_1_0
  57. b_6_39·b_3_7 + b_4_132·b_1_0 + b_2_5·b_4_13·b_1_03 + b_2_52·b_4_13·b_1_0
       + b_2_4·b_4_14·b_3_9 + b_2_4·b_2_5·b_4_14·b_1_2 + b_2_42·b_4_13·b_1_0
       + b_2_42·b_2_52·b_1_2 + b_2_42·b_2_52·b_1_0 + b_2_43·b_3_8 + b_2_43·b_2_5·b_1_2
       + b_2_44·b_1_2 + b_4_13·c_4_17·b_1_0 + b_2_5·c_4_17·b_3_7 + b_2_5·c_4_17·b_1_03
       + b_2_5·c_4_16·b_1_03 + b_2_4·c_4_17·b_3_8 + b_2_4·c_4_17·b_3_7 + b_2_4·c_4_16·b_3_8
       + b_2_4·c_4_16·b_3_7 + b_2_42·c_4_17·b_1_2
  58. b_6_39·b_3_8 + b_4_14·b_5_27 + b_4_142·b_1_1 + b_4_132·b_1_0 + b_2_4·b_4_14·b_3_9
       + b_2_4·b_2_5·b_4_14·b_1_2 + b_2_4·b_2_53·b_1_2 + b_2_42·b_2_52·b_1_2
       + b_2_43·b_3_8 + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_43·b_2_5·b_1_2
       + b_2_43·b_2_5·b_1_0 + b_2_44·b_1_0 + b_4_13·c_4_17·b_1_0 + b_4_13·c_4_16·b_1_0
       + b_2_5·c_4_17·b_3_7 + b_2_5·c_4_16·b_3_8 + b_2_4·c_4_16·b_3_8 + b_2_4·c_4_16·b_3_7
       + b_2_4·c_4_16·b_1_03 + b_2_4·b_2_5·c_4_16·b_1_2 + b_2_42·c_4_17·b_1_2
       + b_2_42·c_4_17·b_1_0 + b_2_42·c_4_16·b_1_0
  59. b_6_39·b_3_9 + b_4_142·b_1_2 + b_4_142·b_1_1 + b_2_53·b_3_9 + b_2_4·b_2_52·b_3_9
       + b_2_42·b_2_5·b_3_9 + b_4_14·c_4_17·b_1_2 + b_4_14·c_4_17·b_1_1 + b_4_14·c_4_16·b_1_2
       + b_2_5·c_4_16·b_3_9 + b_2_52·c_4_17·b_1_2 + b_2_52·c_4_16·b_1_2 + b_2_4·c_4_17·b_3_9
       + b_2_42·c_4_17·b_1_2
  60. b_5_272 + b_4_142·b_1_12 + b_4_132·b_1_02 + b_2_53·b_1_2·b_3_9
       + b_2_54·b_1_22 + b_2_4·b_2_52·b_1_2·b_3_9 + b_2_4·b_2_52·b_4_13
       + b_2_4·b_2_53·b_1_02 + b_2_42·b_4_13·b_1_02 + b_2_42·b_2_5·b_1_2·b_3_9
       + b_2_42·b_2_5·b_4_13 + b_2_42·b_2_52·b_1_22 + b_2_42·b_2_52·b_1_02
       + b_2_42·b_2_53 + b_2_43·b_1_2·b_3_9 + b_2_43·b_4_13 + b_2_43·b_2_5·b_1_02
       + b_2_43·b_2_52 + b_2_44·b_1_22 + b_2_44·b_1_02 + b_2_44·b_2_5
       + b_2_5·b_4_13·c_4_16 + b_2_52·c_4_17·b_1_22 + b_2_52·c_4_16·b_1_22
       + b_2_52·c_4_16·b_1_02 + b_2_4·b_4_15·c_4_16 + b_2_4·b_4_14·c_4_16
       + b_2_4·b_2_5·c_4_17·b_1_22 + b_2_4·b_2_5·c_4_17·b_1_02
       + b_2_4·b_2_5·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16·b_1_02 + b_2_42·c_4_16·b_1_22
       + b_2_42·b_2_5·c_4_17 + c_4_16·c_4_17·b_1_02 + c_4_162·b_1_02
  61. b_4_14·b_6_39 + b_4_142·b_1_1·b_1_2 + b_4_142·b_1_12 + b_2_5·b_4_132
       + b_2_52·b_4_13·b_1_02 + b_2_53·b_4_13 + b_2_54·b_1_22 + b_2_4·b_2_5·b_6_39
       + b_2_4·b_2_52·b_4_15 + b_2_4·b_2_52·b_4_13 + b_2_42·b_6_39
       + b_2_42·b_4_13·b_1_02 + b_2_42·b_2_52·b_1_22 + b_2_42·b_2_53
       + b_2_43·b_1_2·b_3_9 + b_2_43·b_4_13 + b_2_43·b_2_5·b_1_02 + b_2_44·b_1_22
       + b_2_44·b_2_5 + b_4_14·c_4_17·b_1_1·b_1_2 + b_4_14·c_4_17·b_1_12
       + b_4_14·c_4_16·b_1_1·b_1_2 + b_4_13·c_4_16·b_1_02 + b_2_5·b_4_15·c_4_17
       + b_2_5·b_4_15·c_4_16 + b_2_5·b_4_14·c_4_17 + b_2_52·c_4_17·b_1_02
       + b_2_52·c_4_16·b_1_02 + b_2_4·c_4_17·b_1_2·b_3_9 + b_2_4·c_4_16·b_1_0·b_3_7
       + b_2_4·b_4_14·c_4_17 + b_2_4·b_2_52·c_4_16 + b_2_42·c_4_16·b_1_02
       + b_2_42·b_2_5·c_4_17 + b_2_43·c_4_17
  62. b_4_14·b_6_39 + b_4_142·b_1_1·b_1_2 + b_4_142·b_1_12 + b_4_13·b_6_39
       + b_4_132·b_1_02 + b_2_52·b_4_13·b_1_02 + b_2_53·b_4_13 + b_2_54·b_1_22
       + b_2_4·b_2_5·b_6_39 + b_2_4·b_2_52·b_1_2·b_3_9 + b_2_4·b_2_52·b_4_15
       + b_2_4·b_2_52·b_4_13 + b_2_42·b_4_13·b_1_02 + b_2_42·b_2_5·b_1_2·b_3_9
       + b_2_42·b_2_5·b_4_15 + b_2_42·b_2_53 + b_2_43·b_4_15 + b_2_43·b_2_5·b_1_22
       + b_2_44·b_1_22 + b_2_44·b_2_5 + b_4_14·c_4_17·b_1_1·b_1_2 + b_4_14·c_4_17·b_1_12
       + b_4_14·c_4_16·b_1_1·b_1_2 + b_4_13·c_4_17·b_1_02 + b_2_5·b_4_15·c_4_17
       + b_2_5·b_4_15·c_4_16 + b_2_5·b_4_14·c_4_17 + b_2_5·b_4_13·c_4_17
       + b_2_52·c_4_17·b_1_02 + b_2_52·c_4_16·b_1_02 + b_2_4·c_4_17·b_1_2·b_3_9
       + b_2_4·c_4_17·b_1_0·b_3_7 + b_2_4·c_4_16·b_1_0·b_3_7 + b_2_4·b_4_15·c_4_17
       + b_2_4·b_4_15·c_4_16 + b_2_4·b_4_14·c_4_17 + b_2_4·b_4_13·c_4_17
       + b_2_4·b_2_5·c_4_17·b_1_22 + b_2_4·b_2_5·c_4_17·b_1_02
       + b_2_4·b_2_5·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16·b_1_02 + b_2_4·b_2_52·c_4_16
       + b_2_42·c_4_17·b_1_02 + b_2_42·c_4_16·b_1_22 + b_2_42·c_4_16·b_1_02
       + b_2_43·c_4_17
  63. b_4_15·b_6_39 + b_4_142·b_1_12 + b_4_132·b_1_02 + b_2_5·b_4_132
       + b_2_53·b_1_2·b_3_9 + b_2_54·b_1_22 + b_2_4·b_2_5·b_6_39 + b_2_42·b_4_13·b_1_02
       + b_2_42·b_2_5·b_1_2·b_3_9 + b_2_42·b_2_5·b_4_15 + b_2_42·b_2_5·b_4_13
       + b_2_43·b_1_04 + b_2_43·b_2_5·b_1_22 + b_2_43·b_2_52 + b_2_44·b_1_02
       + b_4_14·c_4_17·b_1_12 + b_4_14·c_4_16·b_1_1·b_1_2 + b_4_13·c_4_17·b_1_02
       + b_2_5·b_4_15·c_4_17 + b_2_5·b_4_13·c_4_16 + b_2_52·c_4_17·b_1_22
       + b_2_52·c_4_16·b_1_22 + b_2_4·c_4_17·b_1_2·b_3_9 + b_2_4·c_4_17·b_1_0·b_3_7
       + b_2_4·c_4_16·b_1_2·b_3_9 + b_2_4·c_4_16·b_1_04 + b_2_4·b_4_15·c_4_17
       + b_2_4·b_4_15·c_4_16 + b_2_4·b_4_14·c_4_16 + b_2_4·b_4_13·c_4_16
       + b_2_4·b_2_5·c_4_17·b_1_22 + b_2_4·b_2_5·c_4_16·b_1_22 + b_2_4·b_2_52·c_4_17
       + b_2_42·b_2_5·c_4_16 + c_4_162·b_1_02
  64. b_6_39·b_5_27 + b_4_142·b_1_12·b_1_2 + b_4_142·b_1_13 + b_4_132·b_1_03
       + b_2_5·b_4_132·b_1_0 + b_2_52·b_4_14·b_3_9 + b_2_53·b_4_14·b_1_2
       + b_2_4·b_4_132·b_1_0 + b_2_4·b_2_52·b_4_14·b_1_2 + b_2_4·b_2_53·b_3_9
       + b_2_4·b_2_54·b_1_2 + b_2_42·b_4_13·b_1_03 + b_2_42·b_2_5·b_4_14·b_1_2
       + b_2_42·b_2_52·b_3_8 + b_2_42·b_2_52·b_3_7 + b_2_42·b_2_53·b_1_2
       + b_2_43·b_5_27 + b_2_43·b_2_5·b_3_9 + b_2_43·b_2_5·b_3_7 + b_2_43·b_2_52·b_1_2
       + b_2_44·b_3_8 + b_2_44·b_2_5·b_1_0 + b_2_45·b_1_2 + b_2_45·b_1_0
       + b_4_14·c_4_17·b_1_12·b_1_2 + b_4_14·c_4_17·b_1_13 + b_4_14·c_4_16·b_1_12·b_1_2
       + b_4_13·c_4_17·b_1_03 + b_2_5·c_4_17·b_5_27 + b_2_5·b_4_14·c_4_17·b_1_2
       + b_2_5·b_4_14·c_4_16·b_1_2 + b_2_5·b_4_13·c_4_16·b_1_0 + b_2_52·c_4_17·b_3_9
       + b_2_52·c_4_16·b_3_9 + b_2_53·c_4_17·b_1_2 + b_2_53·c_4_16·b_1_2
       + b_2_4·c_4_17·b_5_27 + b_2_4·c_4_16·b_5_27 + b_2_4·b_4_13·c_4_17·b_1_0
       + b_2_4·b_2_5·c_4_17·b_3_8 + b_2_4·b_2_5·c_4_17·b_3_7 + b_2_4·b_2_5·c_4_16·b_3_9
       + b_2_4·b_2_5·c_4_16·b_3_8 + b_2_4·b_2_5·c_4_16·b_3_7 + b_2_4·b_2_52·c_4_17·b_1_2
       + b_2_42·c_4_17·b_3_8 + b_2_42·c_4_17·b_3_7 + b_2_42·c_4_17·b_1_03
       + b_2_42·b_2_5·c_4_17·b_1_2 + b_2_42·b_2_5·c_4_17·b_1_0
       + b_2_42·b_2_5·c_4_16·b_1_0 + b_2_43·c_4_17·b_1_0 + b_2_4·c_4_16·c_4_17·b_1_0
  65. b_6_392 + b_4_142·b_4_15 + b_4_133 + b_2_5·b_4_132·b_1_02 + b_2_53·b_6_39
       + b_2_54·b_4_15 + b_2_54·b_4_13 + b_2_55·b_1_22 + b_2_4·b_2_53·b_1_2·b_3_9
       + b_2_4·b_2_53·b_4_15 + b_2_4·b_2_54·b_1_02 + b_2_42·b_4_132
       + b_2_42·b_2_5·b_6_39 + b_2_42·b_2_52·b_4_15 + b_2_42·b_2_52·b_4_14
       + b_2_42·b_2_52·b_4_13 + b_2_42·b_2_53·b_1_22 + b_2_42·b_2_53·b_1_02
       + b_2_42·b_2_54 + b_2_43·b_2_5·b_1_2·b_3_9 + b_2_43·b_2_5·b_4_14
       + b_2_43·b_2_52·b_1_22 + b_2_43·b_2_53 + b_2_44·b_1_2·b_3_9 + b_2_44·b_1_04
       + b_2_44·b_4_15 + b_2_44·b_4_13 + b_2_45·b_1_02 + b_2_45·b_2_5 + b_4_142·c_4_17
       + b_4_132·c_4_17 + b_2_5·b_4_13·c_4_17·b_1_02 + b_2_5·b_4_13·c_4_16·b_1_02
       + b_2_52·c_4_17·b_1_2·b_3_9 + b_2_52·c_4_16·b_1_2·b_3_9 + b_2_52·b_4_15·c_4_16
       + b_2_52·b_4_14·c_4_17 + b_2_53·c_4_16·b_1_02 + b_2_54·c_4_17
       + b_2_4·b_4_13·c_4_16·b_1_02 + b_2_4·b_2_5·c_4_17·b_1_2·b_3_9
       + b_2_4·b_2_5·c_4_16·b_1_2·b_3_9 + b_2_4·b_2_5·b_4_15·c_4_17
       + b_2_4·b_2_5·b_4_15·c_4_16 + b_2_4·b_2_5·b_4_14·c_4_17 + b_2_4·b_2_53·c_4_17
       + b_2_42·c_4_17·b_1_2·b_3_9 + b_2_42·c_4_16·b_1_2·b_3_9 + b_2_42·b_4_14·c_4_17
       + b_2_42·b_4_14·c_4_16 + b_2_42·b_2_5·c_4_16·b_1_22 + b_2_42·b_2_52·c_4_16
       + b_2_43·b_2_5·c_4_16 + b_2_44·c_4_17 + c_4_162·b_1_04 + b_2_52·c_4_16·c_4_17
       + b_2_4·c_4_16·c_4_17·b_1_02 + b_2_4·c_4_162·b_1_22 + b_2_4·c_4_162·b_1_02
       + b_2_42·c_4_162

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_16, a Duflot regular element of degree 4
    2. c_4_17, a Duflot regular element of degree 4
    3. b_1_22 + b_1_12 + b_1_02 + b_2_5, an element of degree 2
    4. b_1_22 + b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
  • 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. b_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. b_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. b_4_15 → 0, an element of degree 4
  12. c_4_16 → c_1_14, an element of degree 4
  13. c_4_17 → c_1_14 + c_1_04, an element of degree 4
  14. b_5_27 → 0, an element of degree 5
  15. b_6_39 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 3

  1. b_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. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_7 → 0, an element of degree 3
  7. b_3_8 → 0, an element of degree 3
  8. b_3_9 → 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_4_13 → 0, an element of degree 4
  10. b_4_14 → 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
  11. b_4_15 → 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
  12. c_4_16 → 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. c_4_17 → c_1_1·c_1_23 + c_1_14 + c_1_0·c_1_23 + c_1_04, an element of degree 4
  14. b_5_27 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  15. b_6_39 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → c_1_2, 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 → c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. b_3_7 → c_1_1·c_1_22 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. b_3_8 → 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
  8. b_3_9 → 0, an element of degree 3
  9. b_4_13 → c_1_0·c_1_23 + c_1_0·c_1_1·c_1_22 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2, an element of degree 4
  10. b_4_14 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·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_32 + c_1_02·c_1_2·c_1_3, an element of degree 4
  11. b_4_15 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_13·c_1_2
       + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  12. c_4_16 → 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_12·c_1_22 + c_1_14, an element of degree 4
  13. c_4_17 → 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_13·c_1_2 + c_1_14 + 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_0·c_1_1·c_1_22 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_2
       + c_1_04, an element of degree 4
  14. b_5_27 → c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_24
       + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_23 + c_1_02·c_1_12·c_1_2, an element of degree 5
  15. b_6_39 → c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3 + c_1_12·c_1_34
       + c_1_12·c_1_23·c_1_3 + c_1_13·c_1_2·c_1_32 + c_1_13·c_1_22·c_1_3
       + c_1_13·c_1_23 + c_1_14·c_1_32 + c_1_14·c_1_2·c_1_3 + c_1_15·c_1_2
       + c_1_0·c_1_2·c_1_34 + c_1_0·c_1_24·c_1_3 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_23 + c_1_0·c_1_13·c_1_22 + 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_12·c_1_22
       + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_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_32, an element of degree 2
  5. b_2_5 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_7 → c_1_33, an element of degree 3
  7. b_3_8 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. b_3_9 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  9. b_4_13 → c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
  10. b_4_14 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  11. b_4_15 → c_1_34 + c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32, 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_33 + c_1_1·c_1_2·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_33 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32, an element of degree 4
  13. c_4_17 → c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·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_2·c_1_3 + c_1_12·c_1_22 + c_1_14
       + c_1_0·c_1_33 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. b_5_27 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32
       + 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_39 → c_1_2·c_1_35 + c_1_22·c_1_34 + c_1_1·c_1_2·c_1_34 + c_1_1·c_1_24·c_1_3
       + c_1_12·c_1_34 + 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_14·c_1_22 + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3
       + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_12·c_1_33 + c_1_02·c_1_23·c_1_3
       + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_33 + c_1_02·c_1_12·c_1_32
       + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3 + c_1_04·c_1_2·c_1_3
       + c_1_04·c_1_22, 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