Cohomology of group number 929 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 1.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 3 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t9  −  t8  +  t6  −  t5  −  t4  +  t3  −  t2  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-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 14 minimal generators of maximal degree 8:

  1. a_1_1, a nilpotent element of degree 1
  2. b_1_0, 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_8, an element of degree 3
  7. b_4_9, an element of degree 4
  8. b_4_12, an element of degree 4
  9. b_5_17, an element of degree 5
  10. b_5_18, an element of degree 5
  11. b_5_19, an element of degree 5
  12. b_6_27, an element of degree 6
  13. b_8_43, an element of degree 8
  14. c_8_44, a Duflot regular element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 53 minimal relations of maximal degree 16:

  1. a_1_1·b_1_0
  2. b_1_0·b_1_2
  3. a_1_12·b_1_2 + a_1_13
  4. b_2_4·b_1_2 + a_1_13
  5. b_2_5·a_1_1 + a_1_13
  6. a_1_14
  7. b_2_4·b_2_5
  8. b_1_0·b_3_8 + b_2_4·a_1_12
  9. a_1_12·b_3_8
  10. b_4_9·a_1_1
  11. b_4_9·b_1_0
  12. b_4_12·b_1_0
  13. b_2_4·b_4_9 + b_2_4·a_1_1·b_3_8
  14. b_3_82 + b_1_2·b_5_18 + b_1_2·b_5_17 + b_2_5·b_4_12 + b_2_4·b_4_12 + a_1_1·b_5_17
       + b_4_12·a_1_1·b_1_2 + b_4_12·a_1_12
  15. b_1_2·b_5_17 + b_4_12·b_1_22 + b_2_5·b_1_2·b_3_8 + a_1_1·b_5_18 + a_1_1·b_5_17
  16. b_1_0·b_5_18
  17. b_1_2·b_5_19 + b_1_23·b_3_8 + b_4_12·b_1_22 + b_4_9·b_1_22 + a_1_1·b_1_22·b_3_8
  18. a_1_1·b_5_19 + a_1_1·b_1_22·b_3_8 + b_4_12·a_1_1·b_1_2 + b_2_4·a_1_1·b_3_8
  19. b_1_0·b_5_19 + b_1_0·b_5_17 + b_2_42·a_1_12
  20. a_1_12·b_5_18
  21. b_2_5·b_5_19 + b_2_5·b_5_17 + b_2_5·b_1_22·b_3_8 + b_2_5·b_4_9·b_1_2 + b_2_52·b_3_8
       + a_1_12·b_5_17
  22. b_2_4·b_5_19 + b_2_4·b_5_18 + b_2_4·b_5_17 + b_2_42·b_3_8 + a_1_12·b_5_17
  23. b_1_22·b_5_18 + b_6_27·b_1_2 + b_4_9·b_3_8 + b_4_9·b_1_23 + b_2_5·b_5_18
       + b_2_5·b_4_12·b_1_2 + b_2_5·b_4_9·b_1_2 + b_2_4·b_4_12·a_1_1 + a_1_12·b_5_17
  24. a_1_1·b_1_2·b_5_18 + b_6_27·a_1_1 + b_2_4·b_4_12·a_1_1 + a_1_12·b_5_17
       + b_4_12·a_1_13
  25. b_1_02·b_5_17 + b_6_27·b_1_0 + b_2_5·b_5_17 + b_2_5·b_4_12·b_1_2 + b_2_52·b_3_8
       + a_1_12·b_5_17
  26. b_4_92 + b_2_52·b_1_2·b_3_8 + b_2_52·b_4_12
  27. a_1_13·b_5_17
  28. b_3_8·b_5_19 + b_6_27·b_1_22 + b_4_12·b_1_2·b_3_8 + b_4_12·b_1_24 + b_4_9·b_1_24
       + b_2_5·b_1_2·b_5_18 + b_2_5·b_1_23·b_3_8 + b_2_5·b_4_9·b_1_22 + b_2_42·b_4_12
  29. b_2_4·b_1_0·b_5_17 + b_2_4·b_6_27 + b_2_42·b_4_12 + b_2_4·a_1_1·b_5_17
       + b_2_42·a_1_1·b_3_8 + b_2_43·a_1_12
  30. b_4_9·b_5_17 + b_4_9·b_4_12·b_1_2 + b_2_5·b_4_9·b_3_8 + a_1_1·b_3_8·b_5_17
       + b_4_12·a_1_1·b_1_2·b_3_8
  31. b_4_12·b_5_19 + b_4_12·b_1_22·b_3_8 + b_4_122·b_1_2 + b_4_9·b_4_12·b_1_2
       + b_2_4·b_4_12·b_3_8 + a_1_1·b_3_8·b_5_17
  32. b_4_9·b_5_19 + b_4_9·b_1_22·b_3_8 + b_4_9·b_4_12·b_1_2 + b_2_52·b_1_22·b_3_8
       + b_2_52·b_4_12·b_1_2 + b_2_42·b_4_12·a_1_1
  33. b_1_26·b_3_8 + b_8_43·b_1_2 + b_4_122·b_1_2 + b_4_9·b_5_18 + b_4_9·b_1_25
       + b_2_5·b_4_12·b_3_8 + b_2_52·b_4_9·b_1_2 + a_1_1·b_3_8·b_5_18 + a_1_1·b_1_25·b_3_8
       + b_4_12·a_1_1·b_1_2·b_3_8 + b_4_12·a_1_1·b_1_24
  34. a_1_1·b_1_25·b_3_8 + b_8_43·a_1_1 + b_4_122·a_1_1 + b_2_44·a_1_1
  35. b_8_43·b_1_0 + b_6_27·b_1_03 + b_2_5·b_6_27·b_1_0 + b_2_52·b_5_17
       + b_2_52·b_4_12·b_1_2 + b_2_53·b_3_8 + b_2_42·b_1_05 + b_2_43·b_1_03
       + b_2_44·b_1_0
  36. b_5_192 + b_5_18·b_5_19 + b_5_17·b_5_19 + b_6_27·b_1_2·b_3_8 + b_6_27·b_1_24
       + b_4_12·b_1_2·b_5_18 + b_4_12·b_1_23·b_3_8 + b_4_12·b_1_26 + b_4_9·b_1_26
       + b_2_5·b_3_8·b_5_18 + b_2_5·b_1_25·b_3_8 + b_2_5·b_4_12·b_1_24
       + b_2_5·b_4_9·b_1_2·b_3_8 + b_2_5·b_4_9·b_1_24 + b_2_5·b_4_9·b_4_12 + b_2_43·b_4_12
       + b_6_27·a_1_1·b_1_23 + b_4_12·a_1_1·b_5_18 + b_4_12·a_1_1·b_5_17
       + b_4_12·a_1_1·b_1_25
  37. b_5_192 + b_5_17·b_5_19 + b_6_27·b_1_24 + b_4_12·b_1_23·b_3_8 + b_4_12·b_1_26
       + b_4_9·b_1_23·b_3_8 + b_4_9·b_1_26 + b_4_9·b_4_12·b_1_22 + b_2_5·b_1_25·b_3_8
       + b_2_5·b_4_12·b_1_2·b_3_8 + b_2_5·b_4_12·b_1_24 + b_2_5·b_4_9·b_1_2·b_3_8
       + b_2_5·b_4_9·b_1_24 + b_2_4·b_3_8·b_5_17 + b_2_43·b_4_12 + b_6_27·a_1_1·b_3_8
       + b_6_27·a_1_1·b_1_23 + b_4_12·a_1_1·b_5_18 + b_4_12·a_1_1·b_5_17
       + b_4_12·a_1_1·b_1_25 + b_2_42·a_1_1·b_5_17
  38. b_5_192 + b_5_172 + b_6_27·b_1_24 + b_4_12·b_1_26 + b_4_9·b_1_23·b_3_8
       + b_4_9·b_1_26 + b_2_5·b_1_25·b_3_8 + b_2_5·b_6_27·b_1_22
       + b_2_5·b_4_9·b_1_2·b_3_8 + b_2_52·b_1_23·b_3_8 + b_2_52·b_4_12·b_1_22
       + b_2_52·b_4_9·b_1_22 + b_2_53·b_1_2·b_3_8 + b_2_53·b_4_12 + b_2_4·b_4_122
       + b_2_43·b_4_12 + b_6_27·a_1_1·b_1_23 + b_4_12·a_1_1·b_5_17 + b_4_12·a_1_1·b_1_25
       + b_4_122·a_1_1·b_1_2 + b_2_42·a_1_1·b_5_17 + b_4_122·a_1_12 + b_2_44·a_1_12
       + c_8_44·a_1_12
  39. b_5_192 + b_6_27·b_1_24 + b_4_12·b_1_26 + b_4_122·b_1_22 + b_4_9·b_1_23·b_3_8
       + b_4_9·b_1_26 + b_2_5·b_1_25·b_3_8 + b_2_5·b_6_27·b_1_22
       + b_2_5·b_4_9·b_1_2·b_3_8 + b_2_52·b_1_2·b_5_18 + b_2_52·b_1_23·b_3_8
       + b_2_52·b_4_9·b_1_22 + b_2_4·b_6_27·b_1_02 + b_2_43·b_4_12 + b_2_44·b_1_02
       + b_6_27·a_1_1·b_1_23 + b_4_12·a_1_1·b_1_25 + b_2_42·a_1_1·b_5_17
       + c_8_44·b_1_02
  40. b_4_9·b_1_2·b_5_18 + b_4_9·b_6_27 + b_2_5·b_1_25·b_3_8 + b_2_5·b_8_43
       + b_2_5·b_6_27·b_1_02 + b_2_5·b_4_122 + b_2_5·b_4_9·b_1_24 + b_2_5·b_4_9·b_4_12
       + b_2_52·b_1_2·b_5_18 + b_2_52·b_1_23·b_3_8 + b_2_52·b_1_0·b_5_17 + b_2_53·b_4_9
       + b_2_4·b_4_12·a_1_1·b_3_8
  41. b_5_192 + b_5_18·b_5_19 + b_5_182 + b_5_172 + b_8_43·b_1_22 + b_4_9·b_1_2·b_5_18
       + b_2_5·b_6_27·b_1_22 + b_2_5·b_4_12·b_1_24 + b_2_5·b_4_122
       + b_2_5·b_4_9·b_1_2·b_3_8 + b_2_52·b_4_12·b_1_22 + b_2_53·b_1_2·b_3_8
       + b_2_53·b_4_12 + b_2_4·b_3_8·b_5_17 + b_2_43·b_4_12 + b_4_12·a_1_1·b_5_18
       + b_4_12·a_1_1·b_5_17 + b_4_12·a_1_1·b_1_22·b_3_8 + b_4_12·a_1_1·b_1_25
       + b_2_4·b_4_12·a_1_1·b_3_8 + c_8_44·b_1_22
  42. b_5_17·b_5_19 + b_5_17·b_5_18 + b_5_172 + b_4_12·b_1_2·b_5_18 + b_4_12·b_1_23·b_3_8
       + b_4_9·b_4_12·b_1_22 + b_2_5·b_3_8·b_5_18 + b_2_5·b_6_27·b_1_22
       + b_2_5·b_4_12·b_1_2·b_3_8 + b_2_5·b_4_12·b_1_24 + b_2_5·b_4_9·b_1_24
       + b_2_52·b_1_23·b_3_8 + b_2_52·b_4_12·b_1_22 + b_2_52·b_4_9·b_1_22
       + b_2_53·b_1_2·b_3_8 + b_2_53·b_4_12 + b_2_4·b_3_8·b_5_17 + b_8_43·a_1_1·b_1_2
       + b_6_27·a_1_1·b_1_23 + b_4_12·a_1_1·b_5_18 + b_4_12·a_1_1·b_5_17
       + b_4_12·a_1_1·b_1_25 + b_4_122·a_1_12 + c_8_44·a_1_1·b_1_2
  43. b_2_4·b_8_43 + b_2_4·b_6_27·b_1_02 + b_2_4·b_4_122 + b_2_43·b_1_04
       + b_2_44·b_1_02 + b_2_45 + b_2_42·a_1_1·b_5_17 + b_2_44·a_1_12
  44. b_6_27·b_5_19 + b_6_27·b_1_22·b_3_8 + b_4_12·b_6_27·b_1_2 + b_4_9·b_6_27·b_1_2
       + b_2_4·b_6_27·b_1_03 + b_2_42·b_4_12·b_3_8 + b_2_44·b_1_03
       + b_6_27·a_1_1·b_1_2·b_3_8 + b_2_43·b_4_12·a_1_1 + c_8_44·b_1_03
       + b_2_5·c_8_44·b_1_0
  45. b_6_27·b_5_19 + b_6_27·b_5_17 + b_6_27·b_1_22·b_3_8 + b_4_9·b_6_27·b_1_2
       + b_2_5·b_6_27·b_3_8 + b_2_4·b_4_12·b_5_17 + b_2_42·b_4_12·b_3_8
       + b_8_43·a_1_1·b_1_22 + b_6_27·a_1_1·b_1_24 + b_4_12·a_1_1·b_1_26
       + b_2_4·a_1_1·b_3_8·b_5_17 + b_2_4·b_4_122·a_1_1 + b_2_43·b_4_12·a_1_1
       + c_8_44·a_1_1·b_1_22
  46. b_8_43·b_3_8 + b_8_43·b_1_23 + b_6_27·b_5_19 + b_6_27·b_5_18 + b_6_27·b_5_17
       + b_4_12·b_6_27·b_1_2 + b_4_122·b_3_8 + b_4_9·b_6_27·b_1_2 + b_4_9·b_4_12·b_3_8
       + b_2_5·b_8_43·b_1_2 + b_2_5·b_6_27·b_3_8 + b_2_5·b_4_12·b_1_22·b_3_8
       + b_2_5·b_4_122·b_1_2 + b_2_5·b_4_9·b_5_18 + b_2_5·b_4_9·b_1_25
       + b_2_52·b_6_27·b_1_2 + b_2_52·b_4_12·b_3_8 + b_2_52·b_4_9·b_1_23 + b_2_53·b_5_18
       + b_2_53·b_1_22·b_3_8 + b_2_53·b_4_12·b_1_2 + b_2_42·b_4_12·b_3_8 + b_2_44·b_3_8
       + b_6_27·a_1_1·b_1_24 + b_2_4·b_4_122·a_1_1 + b_2_43·b_4_12·a_1_1
       + b_4_12·a_1_12·b_5_17 + b_4_122·a_1_13 + c_8_44·b_1_23 + b_2_5·c_8_44·b_1_2
       + c_8_44·a_1_13
  47. b_6_27·b_1_2·b_5_18 + b_6_272 + b_4_9·b_3_8·b_5_18 + b_4_9·b_6_27·b_1_22
       + b_2_5·b_8_43·b_1_22 + b_2_5·b_6_27·b_1_2·b_3_8 + b_2_5·b_6_27·b_1_24
       + b_2_5·b_4_12·b_1_26 + b_2_5·b_4_9·b_1_26 + b_2_5·b_4_9·b_6_27
       + b_2_52·b_3_8·b_5_18 + b_2_52·b_4_12·b_1_2·b_3_8 + b_2_52·b_4_12·b_1_24
       + b_2_52·b_4_9·b_1_24 + b_2_53·b_1_2·b_5_18 + b_2_53·b_1_23·b_3_8
       + b_2_53·b_4_12·b_1_22 + b_2_54·b_1_2·b_3_8 + b_2_4·b_6_27·b_1_04
       + b_2_42·b_4_122 + b_2_44·b_1_04 + b_2_4·b_4_12·a_1_1·b_5_17 + c_8_44·b_1_04
       + b_2_5·c_8_44·b_1_22 + b_2_52·c_8_44
  48. b_4_9·b_1_25·b_3_8 + b_4_9·b_8_43 + b_4_9·b_4_122 + b_2_5·b_4_12·b_1_2·b_5_18
       + b_2_5·b_4_12·b_6_27 + b_2_5·b_4_9·b_4_12·b_1_22 + b_2_52·b_3_8·b_5_18
       + b_2_52·b_1_25·b_3_8 + b_2_52·b_4_12·b_1_24 + b_2_52·b_4_122
       + b_2_52·b_4_9·b_4_12 + b_2_54·b_1_2·b_3_8 + b_2_54·b_4_12 + b_2_44·a_1_1·b_3_8
  49. b_8_43·b_5_17 + b_4_12·b_8_43·b_1_2 + b_4_122·b_5_17 + b_4_123·b_1_2
       + b_2_5·b_8_43·b_1_23 + b_2_5·b_6_27·b_5_18 + b_2_5·b_6_27·b_1_22·b_3_8
       + b_2_5·b_4_12·b_6_27·b_1_2 + b_2_5·b_4_9·b_4_12·b_3_8 + b_2_52·b_8_43·b_1_2
       + b_2_52·b_4_12·b_1_22·b_3_8 + b_2_52·b_4_122·b_1_2 + b_2_52·b_4_9·b_5_18
       + b_2_52·b_4_9·b_1_25 + b_2_53·b_6_27·b_1_2 + b_2_53·b_4_12·b_3_8
       + b_2_53·b_4_9·b_1_23 + b_2_54·b_5_18 + b_2_54·b_1_22·b_3_8
       + b_2_54·b_4_12·b_1_2 + b_2_4·b_6_27·b_1_05 + b_2_42·b_6_27·b_1_03
       + b_2_43·b_6_27·b_1_0 + b_2_44·b_5_17 + b_2_44·b_1_05 + b_6_27·a_1_1·b_1_23·b_3_8
       + b_4_12·b_8_43·a_1_1 + b_4_123·a_1_1 + b_2_42·b_4_122·a_1_1 + b_2_44·b_4_12·a_1_1
       + c_8_44·b_1_05 + b_2_5·c_8_44·b_1_23 + b_2_52·c_8_44·b_1_2
  50. b_8_43·b_5_18 + b_8_43·b_5_17 + b_6_27·b_1_24·b_3_8 + b_4_12·b_1_2·b_3_8·b_5_18
       + b_4_12·b_8_43·b_1_2 + b_4_12·b_6_27·b_3_8 + b_4_122·b_5_18 + b_4_122·b_5_17
       + b_4_123·b_1_2 + b_4_9·b_6_27·b_3_8 + b_4_9·b_6_27·b_1_23
       + b_4_9·b_4_12·b_1_22·b_3_8 + b_2_5·b_6_27·b_1_22·b_3_8 + b_2_5·b_6_27·b_1_25
       + b_2_5·b_4_12·b_1_24·b_3_8 + b_2_5·b_4_12·b_1_27 + b_2_5·b_4_122·b_3_8
       + b_2_5·b_4_9·b_1_27 + b_2_5·b_4_9·b_6_27·b_1_2 + b_2_5·b_4_9·b_4_12·b_1_23
       + b_2_52·b_1_2·b_3_8·b_5_18 + b_2_52·b_8_43·b_1_2 + b_2_52·b_6_27·b_1_23
       + b_2_52·b_4_12·b_5_18 + b_2_52·b_4_122·b_1_2 + b_2_52·b_4_9·b_5_18
       + b_2_52·b_4_9·b_1_25 + b_2_53·b_1_24·b_3_8 + b_2_53·b_4_12·b_1_23
       + b_2_53·b_4_9·b_3_8 + b_2_53·b_4_9·b_1_23 + b_2_54·b_4_9·b_1_2
       + b_2_4·b_6_27·b_1_05 + b_2_4·b_4_122·b_3_8 + b_2_42·b_6_27·b_1_03
       + b_2_43·b_6_27·b_1_0 + b_2_44·b_5_18 + b_2_44·b_5_17 + b_2_44·b_1_05
       + b_6_27·a_1_1·b_1_23·b_3_8 + b_6_27·a_1_1·b_1_26 + b_6_272·a_1_1
       + b_4_12·a_1_1·b_3_8·b_5_17 + b_4_12·a_1_1·b_1_28 + b_2_42·b_4_122·a_1_1
       + c_8_44·b_1_05 + b_4_9·c_8_44·b_1_2 + c_8_44·a_1_1·b_1_2·b_3_8
  51. b_8_43·b_5_19 + b_8_43·b_5_17 + b_6_27·b_1_27 + b_4_12·b_1_29 + b_4_122·b_5_17
       + b_4_122·b_1_22·b_3_8 + b_4_123·b_1_2 + b_4_9·b_1_29 + b_4_9·b_8_43·b_1_2
       + b_4_9·b_6_27·b_3_8 + b_2_5·b_8_43·b_1_23 + b_2_5·b_4_12·b_1_27
       + b_2_5·b_4_12·b_6_27·b_1_2 + b_2_5·b_4_9·b_1_24·b_3_8 + b_2_5·b_4_9·b_6_27·b_1_2
       + b_2_5·b_4_9·b_4_12·b_1_23 + b_2_52·b_1_2·b_3_8·b_5_18 + b_2_52·b_8_43·b_1_2
       + b_2_52·b_6_27·b_1_23 + b_2_52·b_4_12·b_5_18 + b_2_52·b_4_122·b_1_2
       + b_2_53·b_4_9·b_3_8 + b_2_54·b_4_9·b_1_2 + b_2_4·b_4_122·b_3_8 + b_2_44·b_5_18
       + b_2_45·b_3_8 + b_6_27·a_1_1·b_1_23·b_3_8 + b_6_27·a_1_1·b_1_26 + b_6_272·a_1_1
       + b_4_12·a_1_1·b_3_8·b_5_17 + b_4_12·a_1_1·b_1_28 + b_4_122·a_1_1·b_1_24
       + b_2_42·a_1_1·b_3_8·b_5_17 + b_2_42·b_4_122·a_1_1
  52. b_6_27·b_1_25·b_3_8 + b_6_27·b_8_43 + b_4_122·b_6_27 + b_4_9·b_8_43·b_1_22
       + b_4_9·b_6_27·b_1_2·b_3_8 + b_4_9·b_4_12·b_1_26 + b_4_9·b_4_12·b_6_27
       + b_2_5·b_4_12·b_3_8·b_5_18 + b_2_5·b_4_9·b_6_27·b_1_22
       + b_2_52·b_6_27·b_1_2·b_3_8 + b_2_52·b_6_27·b_1_24 + b_2_52·b_4_12·b_6_27
       + b_2_52·b_4_9·b_6_27 + b_2_52·b_4_9·b_4_12·b_1_22 + b_2_53·b_1_25·b_3_8
       + b_2_53·b_6_27·b_1_22 + b_2_53·b_4_122 + b_2_53·b_4_9·b_1_24
       + b_2_54·b_1_2·b_5_18 + b_2_54·b_1_23·b_3_8 + b_2_54·b_4_12·b_1_22
       + b_2_54·b_4_9·b_1_22 + b_2_4·b_6_27·b_1_06 + b_2_42·b_6_27·b_1_04
       + b_2_43·b_6_27·b_1_02 + b_2_44·b_1_06 + b_2_44·b_6_27 + b_6_27·a_1_1·b_1_27
       + b_6_272·a_1_1·b_1_2 + b_4_12·a_1_1·b_1_29 + b_4_12·b_8_43·a_1_1·b_1_2
       + b_4_122·a_1_1·b_1_22·b_3_8 + b_4_123·a_1_1·b_1_2 + b_2_4·b_4_122·a_1_1·b_3_8
       + b_2_42·b_4_12·a_1_1·b_5_17 + c_8_44·b_1_06 + b_4_9·c_8_44·b_1_22
       + b_2_5·c_8_44·b_1_04 + b_2_5·b_4_9·c_8_44 + b_2_52·c_8_44·b_1_22
       + c_8_44·a_1_1·b_1_22·b_3_8
  53. b_8_432 + b_6_27·b_8_43·b_1_22 + b_6_272·b_1_2·b_3_8 + b_6_272·b_1_24
       + b_4_12·b_6_272 + b_4_122·b_1_25·b_3_8 + b_4_122·b_1_28 + b_4_123·b_1_24
       + b_4_124 + b_4_9·b_6_27·b_1_23·b_3_8 + b_4_9·b_6_272
       + b_4_9·b_4_12·b_6_27·b_1_22 + b_4_9·b_4_122·b_1_2·b_3_8 + b_4_9·b_4_122·b_1_24
       + b_2_5·b_6_27·b_1_28 + b_2_5·b_6_27·b_8_43 + b_2_5·b_4_12·b_1_210
       + b_2_5·b_4_12·b_6_27·b_1_2·b_3_8 + b_2_5·b_4_12·b_6_27·b_1_24
       + b_2_5·b_4_122·b_1_2·b_5_18 + b_2_5·b_4_122·b_1_26 + b_2_5·b_4_122·b_6_27
       + b_2_5·b_4_9·b_1_210 + b_2_5·b_4_9·b_8_43·b_1_22
       + b_2_5·b_4_9·b_6_27·b_1_2·b_3_8 + b_2_5·b_4_9·b_6_27·b_1_24
       + b_2_5·b_4_9·b_4_12·b_1_26 + b_2_5·b_4_9·b_4_122·b_1_22
       + b_2_52·b_6_27·b_1_26 + b_2_52·b_6_272 + b_2_52·b_4_12·b_1_25·b_3_8
       + b_2_52·b_4_12·b_1_28 + b_2_52·b_4_12·b_8_43 + b_2_52·b_4_122·b_1_24
       + b_2_52·b_4_9·b_6_27·b_1_22 + b_2_52·b_4_9·b_4_122 + b_2_53·b_8_43·b_1_22
       + b_2_53·b_4_12·b_1_2·b_5_18 + b_2_53·b_4_12·b_1_23·b_3_8
       + b_2_53·b_4_122·b_1_22 + b_2_53·b_4_9·b_1_26 + b_2_53·b_4_9·b_4_12·b_1_22
       + b_2_54·b_4_12·b_1_24 + b_2_54·b_4_122 + b_2_55·b_1_2·b_5_18 + b_2_56·b_4_12
       + b_2_4·b_6_27·b_1_08 + b_2_42·b_4_123 + b_2_46·b_1_04 + b_2_48
       + b_6_27·b_8_43·a_1_1·b_1_2 + b_4_12·b_6_27·a_1_1·b_5_18
       + b_4_12·b_6_27·a_1_1·b_1_25 + b_4_122·b_6_27·a_1_1·b_1_2
       + b_2_4·b_4_122·a_1_1·b_5_17 + b_2_42·b_4_122·a_1_1·b_3_8 + c_8_44·b_1_25·b_3_8
       + c_8_44·b_1_08 + b_4_12·c_8_44·b_1_24 + b_2_5·c_8_44·b_1_06
       + b_2_53·c_8_44·b_1_22 + b_2_54·c_8_44 + c_8_44·a_1_1·b_1_24·b_3_8
       + b_4_12·c_8_44·a_1_1·b_1_23

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 17.
  • However, the last relation was already found in degree 16 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_44, a Duflot regular element of degree 8
    2. b_1_24 + b_1_04 + b_4_12 + b_2_52 + b_2_42, an element of degree 4
    3. b_4_12·b_1_22 + b_2_5·b_4_12 + b_2_52·b_1_22 + b_2_52·b_1_02 + b_2_4·b_4_12
         + b_2_42·b_1_02, an element of degree 6
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 14, 16].
  • 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 1

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → 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_8 → 0, an element of degree 3
  7. b_4_9 → 0, an element of degree 4
  8. b_4_12 → 0, an element of degree 4
  9. b_5_17 → 0, an element of degree 5
  10. b_5_18 → 0, an element of degree 5
  11. b_5_19 → 0, an element of degree 5
  12. b_6_27 → 0, an element of degree 6
  13. b_8_43 → 0, an element of degree 8
  14. c_8_44 → c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → c_1_1, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_4_9 → 0, an element of degree 4
  8. b_4_12 → 0, an element of degree 4
  9. b_5_17 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_18 → 0, an element of degree 5
  11. b_5_19 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  12. b_6_27 → c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_12·c_1_22
       + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  13. b_8_43 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_16·c_1_22 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16
       + c_1_04·c_1_14, an element of degree 8
  14. c_8_44 → c_1_28 + c_1_14·c_1_24 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → c_1_1, 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 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_4_9 → 0, an element of degree 4
  8. b_4_12 → 0, an element of degree 4
  9. b_5_17 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_18 → 0, an element of degree 5
  11. b_5_19 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  12. b_6_27 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_24
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2
       + c_1_04·c_1_12, an element of degree 6
  13. b_8_43 → c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_14, an element of degree 8
  14. c_8_44 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_4 → c_1_12, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_4_9 → 0, an element of degree 4
  8. b_4_12 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  9. b_5_17 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. b_5_18 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  11. b_5_19 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  12. b_6_27 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
  13. b_8_43 → c_1_28 + c_1_14·c_1_24 + c_1_18, an element of degree 8
  14. c_8_44 → c_1_28 + c_1_16·c_1_22 + c_1_18 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_2 → c_1_1, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1, an element of degree 3
  7. b_4_9 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2, an element of degree 4
  8. b_4_12 → c_1_34 + c_1_22·c_1_32 + 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, an element of degree 4
  9. b_5_17 → c_1_23·c_1_32 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_12·c_1_2·c_1_32
       + c_1_13·c_1_32 + c_1_13·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2, an element of degree 5
  10. b_5_18 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_1·c_1_34 + c_1_1·c_1_22·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_13·c_1_32 + c_1_13·c_1_2·c_1_3
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
  11. b_5_19 → c_1_1·c_1_34 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_2·c_1_32 + c_1_13·c_1_32
       + c_1_13·c_1_2·c_1_3 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_13, an element of degree 5
  12. b_6_27 → c_1_24·c_1_32 + c_1_25·c_1_3 + c_1_1·c_1_2·c_1_34 + 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_22·c_1_32
       + c_1_13·c_1_2·c_1_32 + c_1_14·c_1_32 + 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_22·c_1_32 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_22·c_1_3 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_22·c_1_32
       + c_1_02·c_1_23·c_1_3 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_2·c_1_32
       + c_1_02·c_1_12·c_1_2·c_1_3 + c_1_02·c_1_13·c_1_2 + c_1_03·c_1_1·c_1_22
       + c_1_03·c_1_12·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  13. b_8_43 → c_1_38 + c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_24·c_1_34 + c_1_25·c_1_33
       + c_1_27·c_1_3 + c_1_1·c_1_2·c_1_36 + c_1_1·c_1_23·c_1_34
       + c_1_1·c_1_25·c_1_32 + c_1_12·c_1_2·c_1_35 + c_1_12·c_1_22·c_1_34
       + c_1_13·c_1_2·c_1_34 + c_1_13·c_1_23·c_1_32 + c_1_14·c_1_34
       + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_22·c_1_32 + c_1_15·c_1_22·c_1_3
       + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_25·c_1_32 + c_1_0·c_1_1·c_1_26
       + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_23·c_1_32
       + c_1_0·c_1_12·c_1_24·c_1_3 + c_1_0·c_1_12·c_1_25
       + c_1_0·c_1_13·c_1_22·c_1_32 + c_1_0·c_1_13·c_1_24
       + c_1_0·c_1_14·c_1_2·c_1_32 + c_1_0·c_1_14·c_1_22·c_1_3 + c_1_0·c_1_14·c_1_23
       + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_22·c_1_34
       + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_1·c_1_2·c_1_34
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_12·c_1_23·c_1_3
       + c_1_02·c_1_13·c_1_2·c_1_32 + c_1_02·c_1_16 + c_1_03·c_1_1·c_1_24
       + c_1_03·c_1_13·c_1_22 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_23·c_1_3
       + c_1_04·c_1_1·c_1_2·c_1_32 + c_1_04·c_1_12·c_1_2·c_1_3 + c_1_05·c_1_1·c_1_22
       + c_1_05·c_1_12·c_1_2, an element of degree 8
  14. c_8_44 → c_1_38 + c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_25·c_1_33
       + c_1_1·c_1_22·c_1_35 + c_1_1·c_1_24·c_1_33 + c_1_1·c_1_25·c_1_32
       + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_23·c_1_33 + c_1_13·c_1_22·c_1_33
       + c_1_13·c_1_23·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32
       + c_1_14·c_1_23·c_1_3 + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_25·c_1_32
       + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_23·c_1_32
       + c_1_0·c_1_12·c_1_24·c_1_3 + c_1_0·c_1_13·c_1_22·c_1_32
       + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_14·c_1_2·c_1_32 + c_1_0·c_1_14·c_1_22·c_1_3
       + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_25·c_1_3
       + c_1_02·c_1_1·c_1_2·c_1_34 + c_1_02·c_1_12·c_1_34
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_12·c_1_23·c_1_3
       + c_1_02·c_1_13·c_1_22·c_1_3 + c_1_02·c_1_14·c_1_32
       + c_1_02·c_1_14·c_1_2·c_1_3 + c_1_02·c_1_16 + c_1_03·c_1_1·c_1_24
       + c_1_03·c_1_13·c_1_22 + c_1_04·c_1_34 + c_1_04·c_1_23·c_1_3
       + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_2·c_1_32 + c_1_04·c_1_12·c_1_32
       + c_1_04·c_1_12·c_1_2·c_1_3 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14
       + c_1_05·c_1_1·c_1_22 + c_1_05·c_1_12·c_1_2 + c_1_06·c_1_12 + c_1_08, an element of degree 8

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