Cohomology of group number 1189 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 4.
  • Its center has rank 3.
  • It has 3 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 coincides with the Duflot bound.
  • The Poincaré series is
    (t3  +  2·t  +  1) · (t3  +  t2  +  1)
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-5,-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_3, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_1, an element of degree 1
  4. b_1_2, an element of degree 1
  5. c_2_7, a Duflot regular element of degree 2
  6. b_3_10, an element of degree 3
  7. b_3_11, an element of degree 3
  8. b_3_12, an element of degree 3
  9. b_3_13, an element of degree 3
  10. b_3_14, an element of degree 3
  11. b_3_15, an element of degree 3
  12. c_4_26, a Duflot regular element of degree 4
  13. c_4_27, a Duflot regular element of degree 4
  14. b_5_43, an element of degree 5
  15. b_5_44, 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 57 minimal relations of maximal degree 10:

  1. b_1_12 + b_1_0·b_1_1 + a_1_3·b_1_0
  2. b_1_0·b_1_2 + a_1_32
  3. b_1_1·b_1_2
  4. a_1_32·b_1_0
  5. a_1_32·b_1_2
  6. b_1_2·b_3_10
  7. b_1_1·b_3_10 + b_1_0·b_3_11
  8. b_1_1·b_3_11 + b_1_1·b_3_10 + a_1_3·b_3_10
  9. b_1_2·b_3_13 + c_2_7·a_1_32
  10. b_1_1·b_3_12 + b_1_0·b_3_13 + c_2_7·b_1_0·b_1_1 + c_2_7·b_1_02
  11. b_1_2·b_3_11 + b_1_1·b_3_13 + b_1_1·b_3_12 + a_1_3·b_3_12 + c_2_7·a_1_3·b_1_0
  12. b_1_2·b_3_11 + b_1_0·b_3_14 + a_1_3·b_3_12 + c_2_7·b_1_0·b_1_1 + c_2_7·b_1_02
  13. b_1_1·b_3_14 + a_1_3·b_3_13 + c_2_7·a_1_3·b_1_1
  14. b_1_2·b_3_15 + b_1_2·b_3_12 + a_1_3·b_3_14 + c_2_7·a_1_3·b_1_1 + c_2_7·a_1_3·b_1_0
  15. b_1_0·b_3_15 + a_1_3·b_3_11 + a_1_3·b_3_10
  16. b_1_2·b_3_11 + a_1_3·b_3_15
  17. b_1_1·b_3_15
  18. a_1_32·b_3_12
  19. b_3_11·b_3_13 + b_3_10·b_3_14 + b_3_10·b_3_13 + c_2_7·a_1_3·b_3_10
  20. b_3_13·b_3_15 + c_2_7·a_1_3·b_3_11 + c_2_7·a_1_3·b_3_10
  21. b_3_10·b_3_15 + c_2_7·a_1_3·b_1_02·b_1_1 + c_2_7·a_1_3·b_1_03
  22. b_3_11·b_3_15 + b_3_11·b_3_12 + b_3_10·b_3_13 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
  23. b_3_14·b_3_15 + b_3_132 + b_3_12·b_3_14 + b_3_12·b_3_13 + a_1_3·b_1_22·b_3_12
       + c_2_7·a_1_3·b_3_11 + c_2_7·a_1_3·b_3_10 + c_2_72·b_1_0·b_1_1 + c_2_72·b_1_02
       + c_2_72·a_1_3·b_1_0
  24. b_3_152 + b_3_12·b_3_15 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_10·b_3_13
       + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
  25. b_3_12·b_3_15 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_10·b_3_13 + c_4_26·b_1_22
       + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
  26. b_3_11·b_3_12 + b_3_10·b_3_13 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
       + c_4_26·a_1_3·b_1_2
  27. b_3_112 + b_3_102 + a_1_3·b_1_02·b_3_11 + a_1_3·b_1_02·b_3_10
       + c_2_7·b_1_03·b_1_1 + c_2_7·b_1_04 + c_2_7·a_1_3·b_1_03 + c_4_26·a_1_32
  28. b_3_102 + b_1_03·b_3_13 + b_1_03·b_3_11 + a_1_3·b_1_02·b_3_13
       + a_1_3·b_1_02·b_3_12 + c_4_26·b_1_0·b_1_1 + c_2_7·b_1_03·b_1_1 + c_4_26·a_1_3·b_1_0
       + c_2_7·a_1_3·b_1_02·b_1_1 + c_2_7·a_1_3·b_1_03
  29. b_3_10·b_3_11 + b_3_102 + a_1_3·b_1_02·b_3_13 + a_1_3·b_1_02·b_3_10
       + c_2_7·b_1_03·b_1_1 + c_2_7·b_1_04 + c_4_26·a_1_3·b_1_1 + c_2_7·a_1_3·b_1_02·b_1_1
       + c_2_7·a_1_3·b_1_03
  30. b_3_12·b_3_15 + b_3_122 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_10·b_3_13
       + c_4_27·b_1_02 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
  31. b_3_132 + b_3_12·b_3_13 + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12 + c_4_27·a_1_3·b_1_0
       + c_2_72·a_1_3·b_1_0
  32. b_3_142 + b_1_23·b_3_12 + c_4_27·a_1_32 + c_2_72·b_1_0·b_1_1 + c_2_72·b_1_02
       + c_2_72·a_1_3·b_1_0
  33. b_3_12·b_3_13 + c_4_27·b_1_0·b_1_1 + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12
       + c_2_72·b_1_0·b_1_1 + c_2_72·b_1_02
  34. b_3_13·b_3_14 + c_4_27·a_1_3·b_1_1 + c_2_7·a_1_3·b_3_13 + c_2_72·b_1_0·b_1_1
       + c_2_72·b_1_02 + c_2_72·a_1_3·b_1_1
  35. b_3_142 + b_3_11·b_3_12 + b_3_112 + b_3_10·b_3_13 + b_3_102 + b_1_2·b_5_43
       + b_1_23·b_3_12 + a_1_3·b_1_02·b_3_11 + a_1_3·b_1_02·b_3_10 + c_2_7·b_1_2·b_3_12
       + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10 + c_2_7·b_1_03·b_1_1 + c_2_7·b_1_04
       + c_2_7·a_1_3·b_3_15 + c_2_7·a_1_3·b_1_03 + c_2_72·b_1_22 + c_2_72·b_1_0·b_1_1
       + c_2_72·b_1_02 + c_2_72·a_1_3·b_1_2 + c_2_72·a_1_3·b_1_0 + c_2_72·a_1_32
  36. b_3_12·b_3_15 + b_3_122 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_10·b_3_13 + b_3_10·b_3_12
       + b_1_0·b_5_43 + b_1_03·b_3_13 + b_1_03·b_3_11 + b_1_03·b_3_10 + a_1_3·b_1_02·b_3_13
       + a_1_3·b_1_02·b_3_10 + c_4_26·b_1_02 + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12
       + c_2_7·b_1_0·b_3_10 + c_2_7·b_1_03·b_1_1 + c_4_26·a_1_3·b_1_0
       + c_2_7·a_1_3·b_1_02·b_1_1 + c_2_72·b_1_0·b_1_1 + c_2_72·a_1_3·b_1_0
       + c_2_72·a_1_32
  37. b_3_132 + b_3_12·b_3_13 + b_3_11·b_3_13 + b_3_112 + b_3_10·b_3_13 + b_3_102
       + a_1_3·b_5_43 + a_1_3·b_1_02·b_3_13 + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12
       + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10 + c_2_7·b_1_03·b_1_1 + c_2_7·b_1_04
       + c_4_26·a_1_3·b_1_0 + c_2_7·a_1_3·b_3_13 + c_2_7·a_1_3·b_3_12 + c_2_7·a_1_3·b_3_11
       + c_2_7·a_1_3·b_3_10 + c_2_7·a_1_3·b_1_02·b_1_1 + c_2_7·a_1_3·b_1_03
       + c_2_72·a_1_3·b_1_2 + c_2_72·a_1_3·b_1_1 + c_2_72·a_1_3·b_1_0 + c_2_72·a_1_32
  38. b_3_12·b_3_13 + b_3_10·b_3_13 + b_3_10·b_3_11 + b_1_1·b_5_43 + b_1_03·b_3_11
       + a_1_3·b_1_02·b_3_13 + a_1_3·b_1_02·b_3_11 + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12
       + c_2_7·b_1_0·b_3_10 + c_4_26·a_1_3·b_1_0 + c_2_7·a_1_3·b_3_15 + c_2_7·a_1_3·b_3_12
       + c_2_7·a_1_3·b_3_10 + c_2_7·a_1_3·b_1_03 + c_2_72·b_1_02 + c_2_72·a_1_3·b_1_1
  39. b_3_142 + b_3_132 + b_3_12·b_3_14 + b_3_12·b_3_13 + b_3_11·b_3_12 + b_3_10·b_3_13
       + b_1_2·b_5_44 + b_1_23·b_3_12 + a_1_3·b_1_22·b_3_12 + c_4_27·b_1_22
       + c_2_7·b_1_2·b_3_12 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10 + c_2_72·b_1_22
       + c_2_72·a_1_32
  40. b_3_142 + b_3_12·b_3_15 + b_3_12·b_3_13 + b_3_122 + b_3_11·b_3_14 + b_3_11·b_3_13
       + b_3_10·b_3_12 + b_1_23·b_3_12 + b_1_0·b_5_44 + b_1_03·b_3_13 + b_1_03·b_3_11
       + b_1_03·b_3_10 + a_1_3·b_1_02·b_3_13 + a_1_3·b_1_02·b_3_12 + c_2_7·b_1_03·b_1_1
       + c_2_7·b_1_04 + c_4_26·a_1_3·b_1_0 + c_2_7·a_1_3·b_1_02·b_1_1 + c_2_72·a_1_3·b_1_0
       + c_2_72·a_1_32
  41. b_3_13·b_3_14 + b_3_132 + b_3_12·b_3_13 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_112
       + b_3_10·b_3_13 + b_3_102 + a_1_3·b_5_44 + a_1_3·b_1_02·b_3_13 + c_2_7·b_1_0·b_3_13
       + c_2_7·b_1_0·b_3_12 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10 + c_2_7·b_1_03·b_1_1
       + c_2_7·b_1_04 + c_4_27·a_1_3·b_1_2 + c_2_7·a_1_3·b_3_12 + c_2_7·a_1_3·b_1_02·b_1_1
       + c_2_72·b_1_0·b_1_1 + c_2_72·b_1_02 + c_2_72·a_1_3·b_1_2 + c_2_72·a_1_3·b_1_1
       + c_2_72·a_1_3·b_1_0
  42. b_3_132 + b_3_12·b_3_13 + b_3_11·b_3_13 + b_3_10·b_3_13 + b_3_10·b_3_11 + b_3_102
       + b_1_1·b_5_44 + b_1_03·b_3_13 + a_1_3·b_1_02·b_3_13 + a_1_3·b_1_02·b_3_12
       + c_2_7·b_1_0·b_3_13 + c_2_7·b_1_0·b_3_12 + c_2_7·b_1_0·b_3_11 + c_2_7·b_1_0·b_3_10
       + c_2_7·a_1_3·b_3_15 + c_2_7·a_1_3·b_3_12 + c_2_7·a_1_3·b_3_10 + c_2_7·a_1_3·b_1_03
  43. b_3_13·b_5_43 + b_3_11·b_5_43 + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_13
       + a_1_3·b_1_04·b_3_11 + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_0·b_3_13
       + c_4_26·b_1_0·b_3_11 + c_2_7·b_1_0·b_5_43 + c_4_27·a_1_3·b_1_02·b_1_1
       + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_3_15 + c_4_26·a_1_3·b_3_12
       + c_4_26·a_1_3·b_3_11 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_02·b_3_12
       + c_2_7·a_1_3·b_1_05 + c_2_7·c_4_27·b_1_02 + c_2_72·b_1_0·b_3_13
       + c_2_7·c_4_27·a_1_3·b_1_0 + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_7·c_4_26·a_1_3·b_1_1
       + c_2_7·c_4_26·a_1_3·b_1_0 + c_2_72·a_1_3·b_3_13 + c_2_72·a_1_3·b_3_12
       + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_3_10 + c_2_72·a_1_3·b_1_02·b_1_1
       + c_2_7·c_4_26·a_1_32 + c_2_73·b_1_02 + c_2_73·a_1_3·b_1_0
  44. b_3_13·b_5_43 + b_3_12·b_5_43 + b_3_10·b_5_43 + b_1_03·b_5_44 + b_1_05·b_3_13
       + b_1_05·b_3_11 + b_1_05·b_3_10 + a_1_3·b_1_04·b_3_13 + a_1_3·b_1_04·b_3_12
       + a_1_3·b_1_04·b_3_11 + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_0·b_3_13
       + c_4_27·b_1_0·b_3_12 + c_4_27·b_1_0·b_3_11 + c_4_27·b_1_04 + c_4_26·b_1_0·b_3_12
       + c_4_26·b_1_0·b_3_10 + c_2_7·b_1_03·b_3_13 + c_2_7·b_1_03·b_3_12
       + c_4_27·a_1_3·b_1_02·b_1_1 + c_4_26·a_1_3·b_3_15 + c_4_26·a_1_3·b_3_13
       + c_4_26·a_1_3·b_3_10 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_5_43
       + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·a_1_3·b_1_02·b_3_10 + c_2_7·a_1_3·b_1_04·b_1_1
       + c_2_7·a_1_3·b_1_05 + c_2_7·c_4_27·b_1_0·b_1_1 + c_2_7·c_4_27·b_1_02
       + c_2_7·c_4_26·b_1_22 + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_2·b_3_12
       + c_2_72·b_1_0·b_3_10 + c_2_72·b_1_04 + c_2_7·c_4_26·a_1_3·b_1_2
       + c_2_7·c_4_26·a_1_3·b_1_1 + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_12
       + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_1_02·b_1_1 + c_2_7·c_4_26·a_1_32
       + c_2_73·b_1_0·b_1_1 + c_2_73·b_1_02 + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
  45. b_3_11·b_5_43 + a_1_3·b_1_02·b_5_44 + a_1_3·b_1_02·b_5_43 + c_4_27·b_1_0·b_3_11
       + c_4_27·b_1_03·b_1_1 + c_4_26·b_1_0·b_3_13 + c_4_26·b_1_0·b_3_11
       + c_4_27·a_1_3·b_1_02·b_1_1 + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_3_15
       + c_4_26·a_1_3·b_3_13 + c_4_26·a_1_3·b_3_12 + c_4_26·a_1_3·b_3_11
       + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_5_43 + c_2_7·a_1_3·b_1_02·b_3_12
       + c_2_7·a_1_3·b_1_04·b_1_1 + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_0·b_3_11
       + c_2_72·b_1_03·b_1_1 + c_2_7·c_4_27·a_1_3·b_1_0 + c_2_7·c_4_26·a_1_3·b_1_2
       + c_2_7·c_4_26·a_1_3·b_1_1 + c_2_7·c_4_26·a_1_3·b_1_0 + c_2_72·a_1_3·b_3_15
       + c_2_72·a_1_3·b_3_13 + c_2_72·a_1_3·b_3_12 + c_2_72·a_1_3·b_1_02·b_1_1
       + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1 + c_2_73·a_1_32
  46. b_3_13·b_5_43 + b_3_12·b_5_44 + b_3_12·b_5_43 + b_3_10·b_5_43 + a_1_3·b_1_04·b_3_11
       + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_2·b_3_12 + c_4_27·b_1_0·b_3_10
       + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_0·b_3_10 + c_2_7·b_1_05·b_1_1
       + c_2_7·b_1_06 + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_3_15 + c_4_26·a_1_3·b_3_13
       + c_4_26·a_1_3·b_3_12 + c_4_26·a_1_3·b_3_10 + c_4_26·a_1_3·b_1_23
       + c_2_7·a_1_3·b_5_43 + c_2_7·a_1_3·b_1_02·b_3_10 + c_2_7·a_1_3·b_1_05
       + c_2_7·c_4_27·b_1_0·b_1_1 + c_2_7·c_4_27·b_1_02 + c_2_7·c_4_26·b_1_02
       + c_2_72·b_1_0·b_3_13 + c_2_72·b_1_0·b_3_12 + c_2_72·b_1_0·b_3_10 + c_2_72·b_1_04
       + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_7·c_4_26·a_1_3·b_1_1 + c_2_72·a_1_3·b_3_15
       + c_2_72·a_1_3·b_3_12 + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_1_02·b_1_1
       + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
  47. b_3_13·b_5_44 + b_3_11·b_5_43 + b_3_10·b_5_43 + a_1_3·b_1_02·b_5_43
       + a_1_3·b_1_04·b_3_13 + c_4_27·b_1_0·b_3_11 + c_4_27·b_1_0·b_3_10
       + c_4_27·b_1_03·b_1_1 + c_4_26·b_1_0·b_3_11 + c_4_26·b_1_0·b_3_10
       + c_2_7·b_1_03·b_3_13 + c_2_7·b_1_03·b_3_12 + c_2_7·b_1_05·b_1_1 + c_2_7·b_1_06
       + c_4_27·a_1_3·b_3_15 + c_4_27·a_1_3·b_3_12 + c_4_27·a_1_3·b_3_10
       + c_4_27·a_1_3·b_1_02·b_1_1 + c_4_26·a_1_3·b_3_11 + c_4_26·a_1_3·b_3_10
       + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·a_1_3·b_1_02·b_3_11
       + c_2_7·c_4_27·b_1_0·b_1_1 + c_2_7·c_4_27·b_1_02 + c_2_72·b_1_0·b_3_13
       + c_2_72·b_1_0·b_3_12 + c_2_72·b_1_0·b_3_11 + c_2_72·b_1_0·b_3_10
       + c_2_72·b_1_03·b_1_1 + c_2_72·b_1_04 + c_2_7·c_4_26·a_1_3·b_1_2
       + c_2_7·c_4_26·a_1_3·b_1_0 + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_11
       + c_2_72·a_1_3·b_3_10 + c_2_72·a_1_3·b_1_02·b_1_1 + c_2_72·a_1_3·b_1_03
       + c_2_7·c_4_27·a_1_32 + c_2_7·c_4_26·a_1_32 + c_2_73·a_1_3·b_1_0
       + c_2_73·a_1_32
  48. b_3_13·b_5_43 + b_3_12·b_5_43 + b_3_10·b_5_44 + b_1_03·b_5_43 + b_1_05·b_3_13
       + b_1_05·b_3_11 + b_1_05·b_3_10 + c_4_27·b_1_0·b_3_13 + c_4_27·b_1_0·b_3_12
       + c_4_27·b_1_04 + c_4_26·b_1_0·b_3_13 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04
       + c_2_7·b_1_03·b_3_13 + c_2_7·b_1_03·b_3_12 + c_2_7·b_1_03·b_3_10 + c_2_7·b_1_06
       + c_4_27·a_1_3·b_1_02·b_1_1 + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_3_12
       + c_4_26·a_1_3·b_3_10 + c_4_26·a_1_3·b_1_02·b_1_1 + c_4_26·a_1_3·b_1_03
       + c_2_7·a_1_3·b_5_43 + c_2_7·a_1_3·b_1_05 + c_2_7·c_4_27·b_1_0·b_1_1
       + c_2_7·c_4_27·b_1_02 + c_2_7·c_4_26·b_1_22 + c_2_72·b_1_2·b_3_12
       + c_2_72·b_1_0·b_3_11 + c_2_72·b_1_0·b_3_10 + c_2_72·b_1_04
       + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_7·c_4_26·a_1_3·b_1_1 + c_2_72·a_1_3·b_3_15
       + c_2_72·a_1_3·b_3_12 + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_3_10
       + c_2_72·a_1_3·b_1_02·b_1_1 + c_2_72·a_1_3·b_1_03 + c_2_7·c_4_26·a_1_32
       + c_2_73·b_1_0·b_1_1 + c_2_73·b_1_02 + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
  49. b_3_13·b_5_43 + b_3_12·b_5_43 + b_3_11·b_5_44 + b_3_11·b_5_43 + b_3_10·b_5_43
       + b_1_03·b_5_43 + b_1_05·b_3_13 + b_1_05·b_3_11 + b_1_05·b_3_10
       + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_13 + a_1_3·b_1_04·b_3_11
       + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_0·b_3_13 + c_4_27·b_1_0·b_3_12 + c_4_27·b_1_04
       + c_4_26·b_1_0·b_3_13 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_0·b_3_11 + c_4_26·b_1_0·b_3_10
       + c_4_26·b_1_04 + c_2_7·b_1_03·b_3_13 + c_2_7·b_1_03·b_3_12 + c_2_7·b_1_03·b_3_11
       + c_2_7·b_1_06 + c_4_27·a_1_3·b_3_15 + c_4_27·a_1_3·b_3_10 + c_4_26·a_1_3·b_3_14
       + c_4_26·a_1_3·b_3_13 + c_4_26·a_1_3·b_3_12 + c_4_26·a_1_3·b_3_10
       + c_4_26·a_1_3·b_1_02·b_1_1 + c_2_7·a_1_3·b_5_43 + c_2_7·c_4_27·b_1_0·b_1_1
       + c_2_7·c_4_27·b_1_02 + c_2_7·c_4_26·b_1_22 + c_2_72·b_1_2·b_3_12
       + c_2_72·b_1_0·b_3_11 + c_2_72·b_1_0·b_3_10 + c_2_72·b_1_04
       + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_12
       + c_2_72·a_1_3·b_3_10 + c_2_72·a_1_3·b_1_02·b_1_1 + c_2_72·a_1_3·b_1_03
       + c_2_73·b_1_0·b_1_1 + c_2_73·b_1_02 + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
  50. b_3_15·b_5_43 + b_3_14·b_5_44 + b_3_14·b_5_43 + a_1_3·b_1_22·b_5_44
       + c_4_27·b_1_2·b_3_14 + c_4_26·b_1_24 + c_4_27·a_1_3·b_3_13 + c_4_27·a_1_3·b_3_10
       + c_4_27·a_1_3·b_1_23 + c_4_26·a_1_3·b_3_12 + c_4_26·a_1_3·b_3_11
       + c_4_26·a_1_3·b_3_10 + c_2_7·a_1_3·b_1_22·b_3_12 + c_2_7·a_1_3·b_1_02·b_3_12
       + c_2_7·a_1_3·b_1_04·b_1_1 + c_2_7·a_1_3·b_1_05 + c_2_7·c_4_26·b_1_22
       + c_2_7·c_4_26·b_1_0·b_1_1 + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_2·b_3_12
       + c_2_72·b_1_03·b_1_1 + c_2_72·b_1_04 + c_2_7·c_4_27·a_1_3·b_1_1
       + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_13
       + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_1_23 + c_2_73·a_1_3·b_1_0
  51. b_3_15·b_5_44 + b_3_15·b_5_43 + c_4_27·b_1_2·b_3_12 + c_4_26·b_1_2·b_3_14
       + c_4_27·a_1_3·b_3_14 + c_4_26·a_1_3·b_3_11 + c_4_26·a_1_3·b_3_10
       + c_2_7·a_1_3·b_1_02·b_3_11 + c_2_7·a_1_3·b_1_02·b_3_10 + c_2_7·c_4_27·a_1_3·b_1_1
       + c_2_7·c_4_27·a_1_3·b_1_0 + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_72·a_1_3·b_3_15
       + c_2_7·c_4_26·a_1_32
  52. b_3_15·b_5_43 + b_3_14·b_5_43 + b_3_10·b_5_43 + a_1_3·b_1_02·b_5_43
       + a_1_3·b_1_04·b_3_13 + c_4_27·b_1_0·b_3_10 + c_4_27·b_1_03·b_1_1
       + c_4_26·b_1_0·b_3_13 + c_4_26·b_1_0·b_3_10 + c_2_7·b_1_2·b_5_44 + c_2_7·b_1_03·b_3_13
       + c_2_7·b_1_03·b_3_12 + c_2_7·b_1_05·b_1_1 + c_2_7·b_1_06 + c_4_27·a_1_3·b_3_15
       + c_4_27·a_1_3·b_3_12 + c_4_27·a_1_3·b_3_11 + c_4_26·a_1_3·b_3_15 + c_4_26·a_1_3·b_3_14
       + c_4_26·a_1_3·b_3_11 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_5_43
       + c_2_7·a_1_3·b_1_22·b_3_12 + c_2_7·a_1_3·b_1_02·b_3_13
       + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·c_4_27·b_1_22 + c_2_7·c_4_27·b_1_0·b_1_1
       + c_2_7·c_4_27·b_1_02 + c_2_7·c_4_26·b_1_22 + c_2_7·c_4_26·b_1_0·b_1_1
       + c_2_72·b_1_2·b_3_14 + c_2_72·b_1_0·b_3_13 + c_2_72·b_1_0·b_3_12
       + c_2_72·b_1_0·b_3_10 + c_2_72·b_1_03·b_1_1 + c_2_7·c_4_27·a_1_3·b_1_1
       + c_2_7·c_4_26·a_1_3·b_1_1 + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_13
       + c_2_72·a_1_3·b_3_12 + c_2_72·a_1_3·b_3_10 + c_2_7·c_4_27·a_1_32
       + c_2_7·c_4_26·a_1_32 + c_2_73·b_1_22 + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_0
  53. b_3_11·b_5_43 + b_3_10·b_5_43 + a_1_3·b_1_04·b_3_11 + a_1_3·b_1_04·b_3_10
       + c_4_27·b_1_0·b_3_11 + c_4_27·b_1_0·b_3_10 + c_4_26·b_1_0·b_3_11 + c_4_26·b_1_0·b_3_10
       + c_2_7·b_1_0·b_5_44 + c_2_7·b_1_03·b_3_12 + c_2_7·b_1_05·b_1_1 + c_2_7·b_1_06
       + c_4_27·a_1_3·b_1_02·b_1_1 + c_4_26·a_1_3·b_3_13 + c_4_26·a_1_3·b_3_11
       + c_4_26·a_1_3·b_3_10 + c_2_7·a_1_3·b_5_43 + c_2_7·a_1_3·b_1_02·b_3_13
       + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·a_1_3·b_1_02·b_3_11 + c_2_7·a_1_3·b_1_04·b_1_1
       + c_2_7·c_4_27·b_1_0·b_1_1 + c_2_7·c_4_27·b_1_02 + c_2_72·b_1_0·b_3_13
       + c_2_72·b_1_0·b_3_12 + c_2_72·b_1_0·b_3_11 + c_2_72·b_1_0·b_3_10
       + c_2_7·c_4_27·a_1_3·b_1_0 + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_7·c_4_26·a_1_3·b_1_0
       + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_13 + c_2_72·a_1_3·b_3_12
       + c_2_72·a_1_3·b_3_10 + c_2_72·a_1_3·b_1_02·b_1_1 + c_2_7·c_4_27·a_1_32
       + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
  54. b_3_15·b_5_43 + c_4_27·a_1_3·b_3_11 + c_4_27·a_1_3·b_3_10 + c_4_26·a_1_3·b_3_15
       + c_4_26·a_1_3·b_3_11 + c_4_26·a_1_3·b_3_10 + c_2_7·a_1_3·b_5_44
       + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·a_1_3·b_1_04·b_1_1 + c_2_7·a_1_3·b_1_05
       + c_2_7·c_4_26·b_1_22 + c_2_72·b_1_2·b_3_12 + c_2_7·c_4_27·a_1_3·b_1_2
       + c_2_7·c_4_27·a_1_3·b_1_1 + c_2_7·c_4_27·a_1_3·b_1_0 + c_2_7·c_4_26·a_1_3·b_1_2
       + c_2_72·a_1_3·b_3_15 + c_2_72·a_1_3·b_3_14 + c_2_72·a_1_3·b_3_13
       + c_2_72·a_1_3·b_3_12 + c_2_72·a_1_3·b_3_11 + c_2_72·a_1_3·b_3_10
       + c_2_7·c_4_26·a_1_32 + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_3·b_1_1
       + c_2_73·a_1_3·b_1_0
  55. b_5_432 + a_1_3·b_1_06·b_3_11 + a_1_3·b_1_06·b_3_10 + c_4_27·b_1_03·b_3_13
       + c_4_27·b_1_03·b_3_11 + c_4_27·b_1_05·b_1_1 + c_2_7·b_1_07·b_1_1 + c_2_7·b_1_08
       + c_4_27·a_1_3·b_1_02·b_3_13 + c_4_27·a_1_3·b_1_02·b_3_12 + c_4_27·a_1_3·b_1_05
       + c_2_7·a_1_3·b_1_07 + c_4_272·b_1_02 + c_4_26·c_4_27·b_1_0·b_1_1
       + c_4_262·b_1_02 + c_2_7·c_4_27·b_1_03·b_1_1 + c_2_72·b_1_03·b_3_13
       + c_2_72·b_1_03·b_3_11 + c_2_72·b_1_06 + c_4_26·c_4_27·a_1_3·b_1_0
       + c_2_7·c_4_27·a_1_3·b_1_02·b_1_1 + c_2_7·c_4_27·a_1_3·b_1_03
       + c_2_72·a_1_3·b_1_02·b_3_13 + c_2_72·a_1_3·b_1_02·b_3_12
       + c_2_72·a_1_3·b_1_02·b_3_11 + c_2_72·a_1_3·b_1_02·b_3_10 + c_4_262·a_1_32
       + c_2_72·c_4_27·b_1_0·b_1_1 + c_2_72·c_4_27·b_1_02 + c_2_72·c_4_26·b_1_22
       + c_2_72·c_4_26·b_1_0·b_1_1 + c_2_73·b_1_04 + c_2_72·c_4_27·a_1_3·b_1_0
       + c_2_72·c_4_26·a_1_3·b_1_0 + c_2_73·a_1_3·b_1_02·b_1_1 + c_2_72·c_4_26·a_1_32
       + c_2_74·b_1_22 + c_2_74·b_1_02 + c_2_74·a_1_32
  56. b_5_442 + a_1_3·b_1_06·b_3_11 + a_1_3·b_1_06·b_3_10 + c_4_27·b_1_05·b_1_1
       + c_4_26·b_1_23·b_3_12 + c_2_7·b_1_07·b_1_1 + c_2_7·b_1_08
       + c_4_27·a_1_3·b_1_02·b_3_11 + c_4_27·a_1_3·b_1_02·b_3_10 + c_4_27·a_1_3·b_1_05
       + c_2_7·a_1_3·b_1_07 + c_4_272·b_1_22 + c_4_272·b_1_0·b_1_1 + c_4_272·b_1_02
       + c_2_7·c_4_27·b_1_03·b_1_1 + c_2_7·c_4_27·b_1_04 + c_4_272·a_1_3·b_1_0
       + c_2_7·c_4_27·a_1_3·b_1_03 + c_4_26·c_4_27·a_1_32 + c_4_262·a_1_32
       + c_2_72·c_4_27·b_1_0·b_1_1 + c_2_72·c_4_27·b_1_02 + c_2_72·c_4_26·b_1_22
       + c_2_72·c_4_27·a_1_3·b_1_0 + c_2_74·b_1_22 + c_2_74·b_1_0·b_1_1
       + c_2_74·b_1_02 + c_2_74·a_1_3·b_1_0
  57. b_5_43·b_5_44 + a_1_3·b_1_04·b_5_44 + a_1_3·b_1_06·b_3_13 + c_4_27·b_1_03·b_3_13
       + c_4_27·b_1_03·b_3_11 + c_4_27·b_1_05·b_1_1 + c_4_26·b_1_0·b_5_44
       + c_2_7·b_1_03·b_5_43 + c_2_7·b_1_07·b_1_1 + c_2_7·b_1_08
       + c_4_27·a_1_3·b_1_02·b_3_10 + c_4_27·a_1_3·b_1_04·b_1_1 + c_4_26·a_1_3·b_5_44
       + c_4_26·a_1_3·b_5_43 + c_4_26·a_1_3·b_1_02·b_3_13 + c_2_7·a_1_3·b_1_02·b_5_44
       + c_2_7·a_1_3·b_1_02·b_5_43 + c_2_7·a_1_3·b_1_04·b_3_13
       + c_2_7·a_1_3·b_1_04·b_3_11 + c_2_7·a_1_3·b_1_04·b_3_10 + c_2_7·a_1_3·b_1_07
       + c_4_272·b_1_0·b_1_1 + c_4_272·b_1_02 + c_2_7·c_4_27·b_1_2·b_3_12
       + c_2_7·c_4_27·b_1_03·b_1_1 + c_2_7·c_4_26·b_1_2·b_3_14 + c_2_7·c_4_26·b_1_04
       + c_2_72·b_1_2·b_5_44 + c_2_72·b_1_03·b_3_13 + c_2_72·b_1_03·b_3_11
       + c_2_72·b_1_06 + c_4_26·c_4_27·a_1_3·b_1_1 + c_4_262·a_1_3·b_1_0
       + c_2_7·c_4_27·a_1_3·b_3_12 + c_2_7·c_4_27·a_1_3·b_1_03 + c_2_7·c_4_26·a_1_3·b_3_14
       + c_2_7·c_4_26·a_1_3·b_3_13 + c_2_7·c_4_26·a_1_3·b_1_23
       + c_2_7·c_4_26·a_1_3·b_1_03 + c_2_72·a_1_3·b_5_44 + c_2_72·a_1_3·b_5_43
       + c_2_72·a_1_3·b_1_02·b_3_12 + c_2_72·a_1_3·b_1_02·b_3_10 + c_4_272·a_1_32
       + c_4_262·a_1_32 + c_2_72·c_4_27·b_1_0·b_1_1 + c_2_72·c_4_27·b_1_02
       + c_2_72·c_4_26·b_1_22 + c_2_73·b_1_2·b_3_12 + c_2_73·b_1_04
       + c_2_72·c_4_26·a_1_3·b_1_0 + c_2_73·a_1_3·b_3_13 + c_2_73·a_1_3·b_3_11
       + c_2_73·a_1_3·b_3_10 + c_2_72·c_4_26·a_1_32 + c_2_74·b_1_0·b_1_1
       + c_2_74·b_1_02 + c_2_74·a_1_3·b_1_2 + c_2_74·a_1_3·b_1_1

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_2_7, a Duflot regular element of degree 2
    2. c_4_26, a Duflot regular element of degree 4
    3. c_4_27, a Duflot regular element of degree 4
    4. b_1_22 + b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 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 3

  1. a_1_3 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. b_3_10 → 0, an element of degree 3
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. c_4_26 → c_1_04, an element of degree 4
  13. c_4_27 → c_1_14 + c_1_04, an element of degree 4
  14. b_5_43 → 0, an element of degree 5
  15. b_5_44 → 0, an element of degree 5

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

  1. a_1_3 → 0, an element of degree 1
  2. b_1_0 → c_1_3, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. b_3_10 → c_1_2·c_1_32, an element of degree 3
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  9. b_3_13 → c_1_22·c_1_3, an element of degree 3
  10. b_3_14 → c_1_22·c_1_3, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. c_4_26 → c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  14. b_5_43 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_1·c_1_34
       + c_1_1·c_1_2·c_1_33 + c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_14·c_1_3 + c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33
       + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + 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_5_44 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_2·c_1_33 + c_1_1·c_1_22·c_1_32
       + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_14·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_33
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, an element of degree 5

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

  1. a_1_3 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_2 → c_1_3, an element of degree 1
  5. c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_10 → 0, an element of degree 3
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → c_1_02·c_1_3, an element of degree 3
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → c_1_0·c_1_32, an element of degree 3
  11. b_3_15 → c_1_02·c_1_3, an element of degree 3
  12. c_4_26 → c_1_04, an element of degree 4
  13. c_4_27 → c_1_12·c_1_32 + c_1_14 + c_1_04, an element of degree 4
  14. b_5_43 → c_1_22·c_1_33 + c_1_24·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_5_44 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_33 + c_1_14·c_1_3
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_03·c_1_32 + c_1_04·c_1_3, an element of degree 5

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

  1. a_1_3 → 0, an element of degree 1
  2. b_1_0 → c_1_3, an element of degree 1
  3. b_1_1 → c_1_3, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_10 → c_1_33 + c_1_2·c_1_32 + c_1_1·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  7. b_3_11 → c_1_33 + c_1_2·c_1_32 + c_1_1·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  8. b_3_12 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  9. b_3_13 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. c_4_26 → c_1_34 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_34 + c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  14. b_5_43 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_24·c_1_3 + c_1_1·c_1_34
       + c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_13·c_1_32 + c_1_14·c_1_3
       + c_1_0·c_1_34 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_12·c_1_32
       + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_12·c_1_3 + c_1_04·c_1_3, an element of degree 5
  15. b_5_44 → c_1_35 + c_1_1·c_1_34 + c_1_12·c_1_33 + c_1_0·c_1_34 + c_1_02·c_1_33, 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