Cohomology of group number 1332 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_0·b_1_1 + a_1_3·b_1_0 + a_1_32
  2. b_1_0·b_1_2
  3. b_1_1·b_1_2 + a_1_32
  4. a_1_33
  5. a_1_32·b_1_1
  6. b_1_2·b_3_10 + c_2_7·a_1_3·b_1_2 + c_2_7·a_1_32
  7. 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 + c_2_7·b_1_12 + c_2_7·a_1_32
  9. b_1_2·b_3_12
  10. b_1_2·b_3_13 + b_1_1·b_3_12 + b_1_1·b_3_10 + a_1_3·b_3_12 + a_1_3·b_3_11 + a_1_3·b_3_10
       + c_2_7·b_1_12 + c_2_7·a_1_32
  11. b_1_1·b_3_12 + b_1_1·b_3_10 + b_1_0·b_3_13 + b_1_0·b_3_12 + b_1_0·b_3_10 + a_1_3·b_3_10
       + a_1_3·b_1_03 + c_2_7·b_1_12
  12. b_1_0·b_3_14 + b_1_0·b_3_10 + c_2_7·a_1_32
  13. b_1_1·b_3_14 + b_1_1·b_3_12 + a_1_3·b_3_12 + a_1_3·b_3_10 + c_2_7·a_1_3·b_1_1
       + c_2_7·a_1_32
  14. b_1_2·b_3_15 + b_1_1·b_3_12 + b_1_1·b_3_10 + a_1_3·b_3_14 + a_1_3·b_3_12 + a_1_3·b_3_11
       + c_2_7·b_1_12 + c_2_7·a_1_3·b_1_1
  15. b_1_1·b_3_12 + b_1_1·b_3_10 + b_1_0·b_3_15 + b_1_0·b_3_10 + a_1_3·b_3_12 + c_2_7·b_1_12
       + c_2_7·a_1_3·b_1_1 + c_2_7·a_1_32
  16. b_1_1·b_3_12 + b_1_1·b_3_10 + a_1_3·b_3_15 + a_1_3·b_3_13 + a_1_3·b_3_10 + c_2_7·b_1_12
       + c_2_7·a_1_32
  17. b_1_1·b_3_15 + b_1_1·b_3_13 + b_1_1·b_3_12 + b_1_1·b_3_10 + a_1_3·b_3_13 + a_1_3·b_3_12
       + c_2_7·b_1_12 + c_2_7·a_1_3·b_1_1
  18. a_1_32·b_3_15
  19. b_3_10·b_3_11 + c_2_7·b_1_1·b_3_10 + c_2_7·b_1_14 + 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_13 + c_2_72·b_1_12 + c_2_72·a_1_32
  20. b_3_11·b_3_12 + c_2_7·b_1_14
  21. b_3_12·b_3_14 + b_3_10·b_3_14 + b_3_10·b_3_12 + b_3_102 + c_2_7·a_1_3·b_3_14
       + c_2_7·a_1_3·b_3_10 + c_2_72·b_1_12 + c_2_72·a_1_3·b_1_1
  22. b_3_13·b_3_14 + b_3_12·b_3_15 + b_3_11·b_3_15 + b_3_11·b_3_13 + b_3_102
       + a_1_3·b_1_02·b_3_10 + c_2_7·b_1_14 + c_2_72·b_1_12 + c_2_72·a_1_32
  23. b_3_152 + b_3_132 + b_3_12·b_3_15 + b_3_122 + b_3_10·b_3_13 + b_3_102
       + a_1_3·b_1_02·b_3_10 + c_2_7·b_1_1·b_3_13 + c_2_7·a_1_3·b_3_13 + c_2_72·b_1_12
       + c_2_72·a_1_32
  24. b_3_12·b_3_14 + b_3_10·b_3_12 + c_2_7·b_1_1·b_3_10 + c_2_7·a_1_3·b_3_15
       + 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_10 + c_2_7·a_1_3·b_1_13
       + c_2_72·b_1_12 + c_2_72·a_1_32
  25. b_3_142 + b_3_102 + b_1_23·b_3_14 + a_1_3·b_1_22·b_3_14 + a_1_3·b_1_22·b_3_11
       + c_4_26·b_1_22 + c_2_72·b_1_12 + c_2_72·a_1_32
  26. b_3_14·b_3_15 + b_3_13·b_3_14 + b_3_10·b_3_15 + b_3_10·b_3_13 + a_1_3·b_1_22·b_3_14
       + c_4_26·a_1_3·b_1_2 + 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_10
       + c_2_72·a_1_3·b_1_1 + c_2_72·a_1_32
  27. b_3_122 + b_3_102 + b_1_03·b_3_10 + a_1_3·b_1_02·b_3_12 + c_4_26·b_1_02
       + c_2_72·b_1_12 + c_2_72·a_1_32
  28. b_3_12·b_3_15 + b_3_12·b_3_14 + b_3_12·b_3_13 + b_3_122 + b_3_10·b_3_15 + b_3_10·b_3_13
       + a_1_3·b_1_02·b_3_12 + 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_10 + c_2_72·a_1_3·b_1_1 + c_2_72·a_1_32
  29. b_3_12·b_3_15 + b_3_10·b_3_13 + b_3_102 + a_1_3·b_1_02·b_3_10 + c_2_7·b_1_1·b_3_13
       + c_2_7·b_1_14 + c_2_7·a_1_3·b_3_13 + c_4_26·a_1_32 + c_2_72·b_1_12
       + c_2_72·a_1_32
  30. b_3_142 + b_3_112 + b_3_102 + c_4_27·b_1_22 + c_2_7·b_1_14 + c_2_72·b_1_12
       + c_2_72·a_1_32
  31. b_3_14·b_3_15 + b_3_13·b_3_15 + b_3_13·b_3_14 + b_3_132 + b_3_12·b_3_15 + b_3_12·b_3_14
       + b_3_122 + b_3_11·b_3_13 + a_1_3·b_1_12·b_3_10 + c_4_27·a_1_3·b_1_2
       + c_4_26·a_1_3·b_1_1
  32. b_3_122 + b_1_03·b_3_12 + b_1_03·b_3_10 + a_1_3·b_1_02·b_3_12
       + a_1_3·b_1_02·b_3_10 + c_4_27·b_1_02 + c_2_7·b_1_14
  33. b_3_13·b_3_15 + b_3_132 + b_3_12·b_3_13 + b_3_10·b_3_15 + b_3_102
       + a_1_3·b_1_12·b_3_10 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_14 + c_4_27·a_1_3·b_1_0
       + c_4_26·a_1_3·b_1_1 + c_2_7·a_1_3·b_3_12 + c_2_7·a_1_3·b_3_10 + c_2_72·b_1_12
       + c_2_72·a_1_3·b_1_1
  34. b_3_132 + b_3_12·b_3_15 + b_3_122 + b_3_10·b_3_13 + b_1_13·b_3_10
       + a_1_3·b_1_12·b_3_13 + a_1_3·b_1_02·b_3_10 + c_4_26·b_1_12 + c_2_7·b_1_1·b_3_13
       + c_2_7·b_1_14 + c_2_7·a_1_3·b_3_13 + c_2_7·a_1_3·b_1_13 + c_4_27·a_1_32
  35. b_3_12·b_3_14 + b_3_112 + b_3_10·b_3_12 + b_3_10·b_3_11 + b_1_2·b_5_43
       + a_1_3·b_1_22·b_3_14 + a_1_3·b_1_22·b_3_11 + c_2_7·a_1_3·b_1_23
  36. b_3_122 + b_3_10·b_3_12 + b_1_0·b_5_43 + b_1_03·b_3_10 + a_1_3·b_1_02·b_3_10
       + a_1_3·b_1_05 + c_2_7·b_1_1·b_3_10 + c_2_7·b_1_0·b_3_12 + c_2_7·b_1_0·b_3_10
       + 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_13 + c_2_7·a_1_3·b_1_03
       + c_2_72·b_1_12
  37. b_3_13·b_3_15 + b_3_132 + b_3_12·b_3_14 + b_3_12·b_3_13 + b_3_11·b_3_13 + b_3_10·b_3_15
       + b_3_10·b_3_13 + a_1_3·b_5_43 + a_1_3·b_1_12·b_3_13 + a_1_3·b_1_12·b_3_10
       + a_1_3·b_1_02·b_3_12 + a_1_3·b_1_02·b_3_10 + c_2_7·b_1_14 + c_4_26·a_1_3·b_1_1
       + 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_13 + c_2_72·a_1_3·b_1_1
       + c_2_72·a_1_32
  38. b_3_13·b_3_15 + b_3_12·b_3_15 + b_3_12·b_3_14 + b_3_12·b_3_13 + b_3_122 + b_3_10·b_3_15
       + b_1_1·b_5_43 + b_1_13·b_3_13 + b_1_13·b_3_10 + a_1_3·b_1_12·b_3_10
       + a_1_3·b_1_02·b_3_12 + c_4_26·b_1_12 + c_2_7·b_1_14 + c_4_26·a_1_3·b_1_1
       + 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_13 + c_2_72·a_1_3·b_1_1
       + c_2_72·a_1_32
  39. b_3_14·b_3_15 + b_3_13·b_3_15 + b_3_13·b_3_14 + b_3_132 + b_3_12·b_3_14 + b_3_122
       + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_112 + b_3_10·b_3_13 + b_3_102 + b_1_2·b_5_44
       + a_1_3·b_1_22·b_3_14 + a_1_3·b_1_12·b_3_10 + a_1_3·b_1_02·b_3_10
       + c_2_7·b_1_2·b_3_14 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_14 + c_4_26·a_1_3·b_1_1
       + c_2_7·a_1_3·b_3_13 + c_2_7·a_1_3·b_1_23 + c_2_72·b_1_12 + c_2_72·a_1_32
  40. b_3_13·b_3_15 + b_3_132 + b_1_0·b_5_44 + b_1_03·b_3_12 + b_1_03·b_3_10
       + a_1_3·b_1_12·b_3_10 + a_1_3·b_1_02·b_3_12 + a_1_3·b_1_05 + c_2_7·b_1_0·b_3_12
       + c_4_26·a_1_3·b_1_1 + c_2_7·a_1_3·b_1_03
  41. b_3_13·b_3_15 + b_3_13·b_3_14 + b_3_12·b_3_13 + b_3_122 + b_3_11·b_3_13 + b_3_10·b_3_15
       + b_3_10·b_3_14 + b_3_10·b_3_12 + b_1_13·b_3_10 + a_1_3·b_5_44 + a_1_3·b_1_12·b_3_13
       + a_1_3·b_1_02·b_3_10 + c_4_26·b_1_12 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_14
       + c_2_7·a_1_3·b_3_10 + c_2_72·a_1_3·b_1_1 + c_2_72·a_1_32
  42. b_3_132 + b_3_12·b_3_13 + b_3_10·b_3_13 + b_1_1·b_5_44 + c_2_7·b_1_1·b_3_13
       + c_2_7·b_1_1·b_3_10 + c_2_7·b_1_14 + c_4_27·a_1_3·b_1_1 + 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_10 + c_2_72·b_1_12 + c_2_72·a_1_32
  43. b_3_15·b_5_43 + b_3_13·b_5_43 + b_1_03·b_5_43 + b_1_05·b_3_12 + b_1_05·b_3_10
       + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_14·b_3_10 + a_1_3·b_1_04·b_3_12
       + a_1_3·b_1_04·b_3_10 + a_1_3·b_1_07 + c_4_27·b_1_0·b_3_12 + c_4_27·b_1_0·b_3_10
       + c_4_27·b_1_04 + c_4_26·b_1_04 + c_2_7·b_1_0·b_5_43 + c_2_7·b_1_03·b_3_12
       + c_4_27·a_1_3·b_3_14 + 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_12
       + c_4_26·a_1_3·b_3_10 + c_4_26·a_1_3·b_1_23 + c_4_26·a_1_3·b_1_13
       + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_15 + c_2_72·b_1_0·b_3_12
       + c_2_72·b_1_0·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_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_1_13 + c_2_72·a_1_3·b_1_03 + c_2_73·a_1_32
  44. b_3_15·b_5_43 + b_3_13·b_5_43 + b_3_10·b_5_43 + b_1_13·b_5_43 + b_1_15·b_3_13
       + b_1_05·b_3_12 + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_12·b_5_43 + a_1_3·b_1_14·b_3_10
       + a_1_3·b_1_04·b_3_12 + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_04 + c_4_26·b_1_0·b_3_12
       + c_4_26·b_1_04 + c_2_7·b_1_13·b_3_13 + c_2_7·b_1_13·b_3_10 + c_2_7·b_1_16
       + c_2_7·b_1_03·b_3_10 + c_4_27·a_1_3·b_3_14 + 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_12 + c_4_26·a_1_3·b_3_10
       + c_4_26·a_1_3·b_1_23 + c_4_26·a_1_3·b_1_13 + c_2_7·a_1_3·b_5_43
       + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_02·b_3_12 + c_2_7·c_4_26·b_1_02
       + c_2_72·b_1_1·b_3_13 + c_2_7·c_4_27·a_1_3·b_1_1 + c_2_7·c_4_26·a_1_3·b_1_0
  45. b_3_12·b_5_43 + b_3_10·b_5_43 + b_1_05·b_3_12 + a_1_3·b_1_12·b_5_43
       + a_1_3·b_1_14·b_3_13 + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_04
       + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04 + c_2_7·b_1_1·b_5_43 + c_2_7·b_1_03·b_3_10
       + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_22·b_3_14
       + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_12·b_3_10 + c_2_7·a_1_3·b_1_15
       + 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·c_4_26·b_1_02
       + c_2_7·c_4_27·a_1_3·b_1_2 + c_2_7·c_4_26·a_1_3·b_1_2 + c_2_72·a_1_3·b_3_13
       + c_2_7·c_4_26·a_1_32
  46. b_3_15·b_5_43 + b_3_13·b_5_43 + b_3_12·b_5_43 + b_3_11·b_5_43 + b_1_23·b_5_44
       + b_1_25·b_3_14 + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_14·b_3_10
       + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_12 + c_4_27·b_1_2·b_3_11 + c_4_27·b_1_24
       + c_4_26·b_1_2·b_3_11 + c_4_26·b_1_24 + c_2_7·b_1_23·b_3_14 + c_4_27·a_1_3·b_3_14
       + c_4_27·a_1_3·b_3_10 + c_4_27·a_1_3·b_1_23 + c_4_27·a_1_3·b_1_03
       + c_4_26·a_1_3·b_3_14 + 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_4_26·a_1_3·b_1_13 + c_4_26·a_1_3·b_1_03
       + c_2_7·a_1_3·b_1_22·b_3_14 + c_2_7·a_1_3·b_1_22·b_3_11 + c_2_7·a_1_3·b_1_25
       + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_15 + c_2_7·a_1_3·b_1_02·b_3_10
       + 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_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_1_13 + c_2_7·c_4_27·a_1_32
       + c_2_7·c_4_26·a_1_32
  47. b_3_13·b_5_43 + b_1_15·b_3_10 + b_1_05·b_3_12 + a_1_3·b_1_22·b_5_44
       + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_12·b_5_43 + a_1_3·b_1_14·b_3_10
       + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_12 + c_4_27·b_1_04 + c_4_26·b_1_1·b_3_10
       + c_4_26·b_1_14 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04 + c_2_7·b_1_13·b_3_13
       + c_2_7·b_1_16 + c_2_7·b_1_03·b_3_10 + c_4_27·a_1_3·b_3_15 + c_4_27·a_1_3·b_3_13
       + c_4_27·a_1_3·b_3_11 + 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_11 + c_4_26·a_1_3·b_3_10 + c_4_26·a_1_3·b_1_23
       + c_4_26·a_1_3·b_1_13 + c_2_7·a_1_3·b_1_22·b_3_14 + c_2_7·a_1_3·b_1_02·b_3_10
       + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_14 + c_2_7·c_4_27·a_1_3·b_1_1
  48. b_3_12·b_5_44 + b_1_03·b_5_43 + a_1_3·b_1_12·b_5_43 + a_1_3·b_1_14·b_3_13
       + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_12 + a_1_3·b_1_07 + c_4_27·b_1_0·b_3_12
       + c_4_27·b_1_0·b_3_10 + c_4_26·b_1_1·b_3_10 + c_4_26·b_1_04 + c_2_7·b_1_16
       + c_4_27·a_1_3·b_3_12 + 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_10 + c_4_26·a_1_3·b_1_03
       + c_2_7·a_1_3·b_1_05 + c_2_7·c_4_27·b_1_02 + c_2_7·c_4_26·b_1_12
       + c_2_72·b_1_14 + c_2_72·a_1_3·b_1_13 + c_2_7·c_4_27·a_1_32
       + c_2_7·c_4_26·a_1_32
  49. b_3_15·b_5_43 + b_3_14·b_5_43 + b_3_13·b_5_44 + b_3_10·b_5_43 + b_1_25·b_3_14
       + b_1_13·b_5_43 + b_1_15·b_3_13 + b_1_15·b_3_10 + b_1_05·b_3_12
       + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_14·b_3_10
       + a_1_3·b_1_04·b_3_12 + a_1_3·b_1_04·b_3_10 + c_4_27·b_1_2·b_3_14
       + c_4_27·b_1_0·b_3_12 + c_4_27·b_1_0·b_3_10 + c_4_27·b_1_04 + c_4_26·b_1_2·b_3_14
       + c_4_26·b_1_24 + c_4_26·b_1_1·b_3_13 + c_4_26·b_1_1·b_3_10 + c_4_26·b_1_14
       + c_4_26·b_1_04 + c_2_7·b_1_13·b_3_13 + c_2_7·b_1_03·b_3_10 + c_4_27·a_1_3·b_3_13
       + 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_13 + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_15
       + 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·c_4_26·b_1_02
       + c_2_72·b_1_1·b_3_13 + c_2_7·c_4_27·a_1_3·b_1_2 + 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_7·c_4_27·a_1_32
  50. b_3_14·b_5_43 + b_3_12·b_5_43 + b_3_10·b_5_44 + b_3_10·b_5_43 + b_1_25·b_3_14
       + b_1_13·b_5_43 + b_1_15·b_3_13 + b_1_03·b_5_43 + a_1_3·b_1_24·b_3_14
       + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_14·b_3_13 + a_1_3·b_1_02·b_5_43
       + a_1_3·b_1_04·b_3_12 + a_1_3·b_1_04·b_3_10 + a_1_3·b_1_07 + c_4_27·b_1_2·b_3_14
       + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_24 + c_4_26·b_1_1·b_3_10 + c_4_26·b_1_0·b_3_12
       + c_4_26·b_1_04 + c_2_7·b_1_13·b_3_13 + c_4_27·a_1_3·b_3_10 + c_4_27·a_1_3·b_1_03
       + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_22·b_3_14 + c_2_7·a_1_3·b_1_12·b_3_10
       + 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_1·b_3_13 + c_2_72·b_1_1·b_3_10 + c_2_7·c_4_27·a_1_3·b_1_0
       + c_2_72·a_1_3·b_3_14 + c_2_72·a_1_3·b_3_13 + c_2_73·b_1_12 + c_2_73·a_1_3·b_1_1
  51. b_3_14·b_5_44 + b_3_12·b_5_43 + b_3_11·b_5_43 + b_1_25·b_3_14 + b_1_03·b_5_43
       + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_12·b_5_43 + a_1_3·b_1_14·b_3_13
       + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_12 + a_1_3·b_1_04·b_3_10 + a_1_3·b_1_07
       + c_4_27·b_1_2·b_3_14 + c_4_27·b_1_2·b_3_11 + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_24
       + c_4_26·b_1_1·b_3_10 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04 + c_2_7·b_1_23·b_3_14
       + c_2_7·b_1_16 + c_4_27·a_1_3·b_3_14 + c_4_27·a_1_3·b_1_23 + 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_1_23 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_22·b_3_14
       + 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_7·c_4_26·b_1_22 + c_2_7·c_4_26·b_1_12 + c_2_72·b_1_14
       + c_2_7·c_4_27·a_1_3·b_1_2 + 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_72·a_1_3·b_1_13 + c_2_7·c_4_26·a_1_32
  52. b_3_15·b_5_44 + b_3_14·b_5_43 + b_1_25·b_3_14 + b_1_03·b_5_43 + b_1_05·b_3_12
       + a_1_3·b_1_24·b_3_14 + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_12·b_5_43
       + a_1_3·b_1_14·b_3_13 + a_1_3·b_1_14·b_3_10 + a_1_3·b_1_04·b_3_10 + a_1_3·b_1_07
       + c_4_27·b_1_2·b_3_14 + c_4_27·b_1_04 + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_24
       + c_4_26·b_1_1·b_3_13 + c_2_7·b_1_13·b_3_13 + c_2_7·b_1_13·b_3_10
       + c_2_7·b_1_03·b_3_10 + c_4_27·a_1_3·b_3_13 + c_4_27·a_1_3·b_3_11
       + c_4_27·a_1_3·b_3_10 + 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_10 + c_4_26·a_1_3·b_1_13 + 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_05 + c_2_7·c_4_27·b_1_02
       + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_14 + 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_7·c_4_26·a_1_3·b_1_1 + c_2_72·a_1_3·b_1_13
       + c_2_7·c_4_27·a_1_32
  53. b_3_15·b_5_43 + b_3_12·b_5_43 + b_3_11·b_5_44 + b_3_11·b_5_43 + b_1_25·b_3_14
       + b_1_15·b_3_10 + b_1_05·b_3_12 + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_14·b_3_13
       + c_4_27·b_1_2·b_3_14 + c_4_27·b_1_04 + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_24
       + c_4_26·b_1_14 + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04 + c_2_7·b_1_2·b_5_44
       + c_2_7·b_1_23·b_3_14 + c_2_7·b_1_13·b_3_13 + c_2_7·b_1_03·b_3_10
       + c_4_27·a_1_3·b_3_15 + c_4_27·a_1_3·b_3_14 + c_4_27·a_1_3·b_3_13
       + c_4_27·a_1_3·b_1_23 + c_4_27·a_1_3·b_1_03 + c_4_26·a_1_3·b_3_14
       + 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_4_26·a_1_3·b_1_23 + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_1_22·b_3_14
       + c_2_7·a_1_3·b_1_22·b_3_11 + c_2_7·a_1_3·b_1_12·b_3_13 + c_2_7·a_1_3·b_1_15
       + c_2_7·c_4_27·b_1_22 + c_2_7·c_4_26·b_1_22 + c_2_7·c_4_26·b_1_12
       + c_2_7·c_4_26·b_1_02 + c_2_72·b_1_2·b_3_14 + 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_1_23 + c_2_7·c_4_27·a_1_32
       + c_2_7·c_4_26·a_1_32
  54. b_3_14·b_5_43 + b_3_12·b_5_43 + b_1_25·b_3_14 + b_1_05·b_3_12 + a_1_3·b_1_24·b_3_14
       + a_1_3·b_1_24·b_3_11 + a_1_3·b_1_02·b_5_43 + a_1_3·b_1_04·b_3_10
       + c_4_27·b_1_2·b_3_14 + c_4_27·b_1_04 + c_4_26·b_1_2·b_3_14 + c_4_26·b_1_24
       + c_4_26·b_1_0·b_3_12 + c_4_26·b_1_04 + c_2_7·b_1_03·b_3_10 + c_4_27·a_1_3·b_1_03
       + c_4_26·a_1_3·b_1_03 + c_2_7·a_1_3·b_5_44 + c_2_7·a_1_3·b_1_12·b_3_10
       + c_2_7·a_1_3·b_1_02·b_3_10 + c_2_7·c_4_26·b_1_02 + c_2_7·c_4_27·a_1_3·b_1_2
       + 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_14
       + 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_13
       + c_2_7·c_4_27·a_1_32 + c_2_73·a_1_3·b_1_1 + c_2_73·a_1_32
  55. b_5_432 + b_1_27·b_3_14 + b_1_17·b_3_10 + b_1_05·b_5_43 + b_1_07·b_3_12
       + a_1_3·b_1_26·b_3_14 + a_1_3·b_1_26·b_3_11 + a_1_3·b_1_16·b_3_13
       + a_1_3·b_1_06·b_3_12 + a_1_3·b_1_06·b_3_10 + a_1_3·b_1_09 + c_4_27·b_1_03·b_3_10
       + c_4_27·b_1_06 + c_4_26·b_1_26 + c_4_26·b_1_16 + c_4_26·b_1_03·b_3_12
       + c_4_26·b_1_03·b_3_10 + c_2_7·b_1_15·b_3_10 + c_2_7·b_1_05·b_3_12
       + c_2_7·b_1_05·b_3_10 + c_4_27·a_1_3·b_1_02·b_3_12 + c_4_26·a_1_3·b_1_02·b_3_12
       + c_4_26·a_1_3·b_1_02·b_3_10 + c_4_26·a_1_3·b_1_05 + c_2_7·a_1_3·b_1_14·b_3_13
       + c_2_7·a_1_3·b_1_17 + c_2_7·a_1_3·b_1_04·b_3_12 + c_2_7·a_1_3·b_1_07
       + c_4_272·b_1_22 + c_4_26·c_4_27·b_1_02 + c_4_262·b_1_22
       + c_2_7·c_4_26·b_1_14 + c_2_72·b_1_13·b_3_10 + c_2_72·b_1_16
       + c_2_72·b_1_03·b_3_10 + c_2_72·a_1_3·b_1_12·b_3_13 + c_2_72·a_1_3·b_1_15
       + c_2_72·a_1_3·b_1_02·b_3_12 + c_2_72·c_4_26·b_1_12 + c_2_72·c_4_26·b_1_02
       + c_2_73·a_1_3·b_1_13 + c_2_72·c_4_27·a_1_32 + c_2_72·c_4_26·a_1_32
  56. b_5_442 + b_1_05·b_5_43 + b_1_07·b_3_10 + a_1_3·b_1_09 + c_4_27·b_1_23·b_3_14
       + c_4_27·b_1_03·b_3_10 + c_4_26·b_1_03·b_3_12 + c_4_26·b_1_03·b_3_10
       + c_2_7·b_1_18 + c_2_7·b_1_05·b_3_12 + c_2_7·b_1_05·b_3_10
       + c_4_27·a_1_3·b_1_22·b_3_14 + c_4_27·a_1_3·b_1_22·b_3_11
       + c_4_27·a_1_3·b_1_02·b_3_12 + c_4_26·a_1_3·b_1_02·b_3_12
       + c_4_26·a_1_3·b_1_02·b_3_10 + c_4_26·a_1_3·b_1_05 + c_2_7·a_1_3·b_1_04·b_3_12
       + c_2_7·a_1_3·b_1_07 + c_4_272·b_1_22 + c_4_26·c_4_27·b_1_22
       + c_4_26·c_4_27·b_1_02 + c_4_262·b_1_12 + c_2_72·b_1_23·b_3_14
       + c_2_72·b_1_03·b_3_12 + c_2_72·b_1_03·b_3_10 + c_2_72·a_1_3·b_1_22·b_3_14
       + c_2_72·a_1_3·b_1_22·b_3_11 + 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_2_72·c_4_27·b_1_02
       + c_2_72·c_4_26·b_1_22 + c_2_73·b_1_14
  57. b_5_43·b_5_44 + b_1_25·b_5_44 + b_1_15·b_5_43 + b_1_17·b_3_13 + b_1_07·b_3_10
       + a_1_3·b_1_14·b_5_43 + a_1_3·b_1_16·b_3_10 + a_1_3·b_1_04·b_5_43
       + a_1_3·b_1_06·b_3_12 + c_4_27·b_1_2·b_5_44 + c_4_27·b_1_23·b_3_14 + c_4_27·b_1_26
       + c_4_27·b_1_03·b_3_12 + c_4_26·b_1_2·b_5_44 + c_4_26·b_1_23·b_3_14
       + c_4_26·b_1_23·b_3_11 + c_4_26·b_1_1·b_5_43 + c_4_26·b_1_03·b_3_12
       + c_4_26·b_1_03·b_3_10 + c_4_26·b_1_06 + c_2_7·b_1_15·b_3_10 + c_2_7·b_1_18
       + c_4_27·a_1_3·b_5_43 + c_4_27·a_1_3·b_1_02·b_3_12 + c_4_27·a_1_3·b_1_05
       + c_4_26·a_1_3·b_1_25 + c_4_26·a_1_3·b_1_15 + c_4_26·a_1_3·b_1_02·b_3_12
       + c_4_26·a_1_3·b_1_02·b_3_10 + c_2_7·a_1_3·b_1_24·b_3_11 + c_2_7·a_1_3·b_1_27
       + c_2_7·a_1_3·b_1_17 + c_2_7·a_1_3·b_1_02·b_5_43 + c_4_26·c_4_27·b_1_02
       + c_2_7·c_4_27·b_1_0·b_3_12 + c_2_7·c_4_27·b_1_0·b_3_10 + c_2_7·c_4_26·b_1_24
       + c_2_7·c_4_26·b_1_0·b_3_12 + c_2_72·b_1_1·b_5_43 + c_2_72·b_1_13·b_3_10
       + c_2_72·b_1_16 + c_2_72·b_1_0·b_5_43 + c_2_72·b_1_03·b_3_10
       + c_4_272·a_1_3·b_1_2 + c_4_26·c_4_27·a_1_3·b_1_2 + c_2_7·c_4_27·a_1_3·b_3_15
       + c_2_7·c_4_27·a_1_3·b_3_14 + c_2_7·c_4_27·a_1_3·b_3_13 + c_2_7·c_4_27·a_1_3·b_3_11
       + c_2_7·c_4_27·a_1_3·b_1_23 + c_2_7·c_4_26·a_1_3·b_3_14 + c_2_7·c_4_26·a_1_3·b_3_11
       + c_2_7·c_4_26·a_1_3·b_3_10 + 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_1_12·b_3_13 + c_2_72·a_1_3·b_1_02·b_3_12
       + c_2_72·a_1_3·b_1_05 + c_4_26·c_4_27·a_1_32 + c_4_262·a_1_32
       + c_2_73·b_1_1·b_3_13 + c_2_73·b_1_0·b_3_12 + c_2_73·b_1_0·b_3_10
       + c_2_72·c_4_27·a_1_3·b_1_2 + c_2_72·c_4_27·a_1_3·b_1_0
       + c_2_72·c_4_26·a_1_3·b_1_2 + c_2_73·a_1_3·b_3_15 + c_2_73·a_1_3·b_3_14
       + c_2_73·a_1_3·b_3_13 + c_2_73·a_1_3·b_3_12 + c_2_73·a_1_3·b_3_10
       + c_2_73·a_1_3·b_1_13 + c_2_73·a_1_3·b_1_03 + c_2_72·c_4_27·a_1_32
       + c_2_72·c_4_26·a_1_32 + 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_12 + 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_12, 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_24 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_24, 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_1·c_1_3 + c_1_12, an element of degree 2
  6. b_3_10 → c_1_2·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → c_1_22·c_1_3, an element of degree 3
  9. b_3_13 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  10. b_3_14 → c_1_2·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  11. b_3_15 → c_1_2·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  12. c_4_26 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_24 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_24 + c_1_0·c_1_33 + c_1_02·c_1_32, 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_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_0·c_1_34 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_1·c_1_33
       + c_1_0·c_1_12·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_22·c_1_3
       + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·c_1_3, an element of degree 5
  15. b_5_44 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_24·c_1_3
       + c_1_1·c_1_22·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_34
       + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_22·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 → c_1_3, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. c_2_7 → c_1_12, an element of degree 2
  6. b_3_10 → c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  7. b_3_11 → c_1_1·c_1_32, an element of degree 3
  8. b_3_12 → c_1_1·c_1_32, an element of degree 3
  9. b_3_13 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  10. b_3_14 → c_1_1·c_1_32, an element of degree 3
  11. b_3_15 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  12. c_4_26 → c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_2·c_1_33 + c_1_24 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
  14. b_5_43 → c_1_2·c_1_34 + c_1_22·c_1_33 + 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_13·c_1_32 + c_1_0·c_1_34 + c_1_0·c_1_1·c_1_33 + c_1_0·c_1_12·c_1_32
       + c_1_02·c_1_33 + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·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_13·c_1_32 + c_1_02·c_1_33 + 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_1·c_1_3 + c_1_12, an element of degree 2
  6. b_3_10 → 0, an element of degree 3
  7. b_3_11 → c_1_02·c_1_3, 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 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. c_4_26 → c_1_2·c_1_33 + c_1_24 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  13. c_4_27 → c_1_22·c_1_32 + c_1_24 + c_1_02·c_1_32, an element of degree 4
  14. b_5_43 → c_1_04·c_1_3, an element of degree 5
  15. b_5_44 → 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_0·c_1_1·c_1_33 + c_1_0·c_1_12·c_1_32
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_1·c_1_32
       + c_1_02·c_1_12·c_1_3 + c_1_03·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