Cohomology of group number 39 of order 243

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

General information on the group

  • The group has 3 minimal generators and exponent 9.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t2  +  1)2 · (t3  −  t2  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  t  +  1) · (t2  −  t  +  1)2
  • The a-invariants are -∞,-∞,-4,-3. 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 243

Ring generators

The cohomology ring has 19 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. b_2_2, an element of degree 2
  5. b_2_3, an element of degree 2
  6. b_2_4, an element of degree 2
  7. b_2_5, an element of degree 2
  8. a_3_7, a nilpotent element of degree 3
  9. a_3_8, a nilpotent element of degree 3
  10. a_3_9, a nilpotent element of degree 3
  11. a_5_12, a nilpotent element of degree 5
  12. a_5_15, a nilpotent element of degree 5
  13. b_6_17, an element of degree 6
  14. c_6_18, a Duflot regular element of degree 6
  15. c_6_19, a Duflot regular element of degree 6
  16. a_7_24, a nilpotent element of degree 7
  17. a_7_25, a nilpotent element of degree 7
  18. b_8_33, an element of degree 8
  19. a_9_42, a nilpotent element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

Ring relations

There are 11 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_72, a_3_82, a_3_92, a_5_122, a_5_152, a_7_242, a_7_252, a_9_422

Apart from that, there are 97 minimal relations of maximal degree 17:

  1. a_1_0·a_1_2
  2. b_2_2·a_1_1
  3. b_2_3·a_1_0 − b_2_2·a_1_2
  4. b_2_4·a_1_0 − b_2_3·a_1_2 + b_2_2·a_1_2
  5. b_2_4·a_1_2 + b_2_3·a_1_2 − b_2_2·a_1_2
  6. b_2_5·a_1_0 − b_2_2·a_1_2
  7.  − b_2_32 + b_2_2·b_2_4 + b_2_2·b_2_3
  8.  − b_2_42 + b_2_3·b_2_5 + b_2_3·b_2_4 − b_2_32
  9.  − b_2_3·b_2_4 − b_2_32 + b_2_2·b_2_5
  10. a_1_1·a_3_7
  11.  − b_2_3·b_2_4 − b_2_32 + b_2_2·b_2_3 + a_1_0·a_3_8
  12.  − b_2_42 + b_2_3·b_2_4 + a_1_2·a_3_8
  13.  − b_2_42 − b_2_32 + b_2_2·b_2_3 + a_1_0·a_3_9
  14.  − b_2_4·b_2_5 + b_2_42 + a_1_2·a_3_9
  15. b_2_2·a_3_7
  16. b_2_5·a_3_8 − b_2_3·a_3_8
  17.  − b_2_4·a_3_8 + b_2_3·a_3_9 + b_2_3·a_3_8 + b_2_2·b_2_3·a_1_2 − b_2_22·a_1_2
  18.  − b_2_3·a_3_8 + b_2_2·a_3_9 − b_2_2·a_3_8 − b_2_2·b_2_3·a_1_2 + b_2_22·a_1_2
  19. b_2_4·a_3_9 + b_2_2·b_2_3·a_1_2 − b_2_22·a_1_2
  20. a_3_8·a_3_9 + b_2_2·a_1_2·a_3_9 + b_2_2·a_1_0·a_3_9 − b_2_2·a_1_0·a_3_8
  21. a_1_1·a_5_12 + b_2_5·a_1_2·a_3_7 − b_2_3·a_1_2·a_3_7
  22. a_1_0·a_5_12 + b_2_3·a_1_2·a_3_7
  23. a_1_2·a_5_12 + b_2_5·a_1_2·a_3_7 − b_2_3·a_1_2·a_3_7 + b_2_3·a_1_1·a_3_9
  24. a_1_1·a_5_15
  25. b_2_3·a_5_12 − a_1_0·a_3_7·a_3_9
  26. b_2_2·a_5_12
  27. b_2_4·a_5_12 − a_1_2·a_3_7·a_3_9 + a_1_0·a_3_7·a_3_9
  28. b_6_17·a_1_1 − b_2_52·a_3_7 + a_1_0·a_3_7·a_3_9
  29. b_6_17·a_1_0
  30. b_6_17·a_1_2 − b_2_5·a_5_12 − b_2_52·a_3_7 + a_1_2·a_3_7·a_3_9 + a_1_0·a_3_7·a_3_9
       − b_2_5·a_1_1·a_1_2·a_3_9
  31. a_3_8·a_5_12 − b_2_3·a_3_7·a_3_9
  32. a_3_7·a_5_15
  33.  − a_3_7·a_5_12 + b_2_5·a_1_2·a_5_15 − b_2_3·a_1_2·a_5_15
  34. b_2_3·b_6_17 − b_2_3·a_1_2·a_5_15
  35. b_2_2·b_6_17 − b_2_2·a_1_2·a_5_15
  36. b_2_4·b_6_17 + a_3_9·a_5_12 − b_2_5·a_3_7·a_3_9 + b_2_3·a_1_2·a_5_15
       − b_2_2·a_1_2·a_5_15
  37. a_3_7·a_5_12 + a_1_1·a_7_24
  38. a_1_0·a_7_24 + b_2_2·a_1_2·a_5_15
  39. a_1_2·a_7_24 + b_2_52·a_1_2·a_3_7 + b_2_3·a_3_7·a_3_9 + b_2_3·a_1_2·a_5_15
  40. a_3_7·a_5_12 + a_1_1·a_7_25 − b_2_5·a_3_7·a_3_9 + b_2_3·a_3_7·a_3_9
  41. a_1_0·a_7_25 + b_2_3·a_1_2·a_5_15 − b_2_2·a_1_2·a_5_15 + c_6_18·a_1_0·a_1_1
  42. a_3_9·a_5_12 − a_3_7·a_5_12 + a_1_2·a_7_25 − b_2_5·a_3_7·a_3_9 + b_2_52·a_1_2·a_3_7
       + b_2_3·a_3_7·a_3_9 − b_2_3·a_1_2·a_5_15 + b_2_2·a_1_2·a_5_15 − c_6_18·a_1_1·a_1_2
  43. b_6_17·a_3_8 + a_1_0·a_3_9·a_5_15 − a_1_0·a_3_8·a_5_15
  44. b_6_17·a_3_7 + b_2_52·a_5_15 − b_2_2·b_2_4·a_5_15 − b_2_2·b_2_3·a_5_15
       + a_1_0·a_3_9·a_5_15 − a_1_0·a_3_8·a_5_15
  45. b_2_3·a_7_24 + b_2_2·b_2_4·a_5_15 + b_2_2·b_2_3·a_5_15
  46. b_2_2·a_7_24 + b_2_2·b_2_3·a_5_15
  47. b_2_4·a_7_24 − b_2_2·b_2_4·a_5_15 + a_1_0·a_3_8·a_5_15 − b_2_5·a_1_2·a_3_7·a_3_9
  48. b_2_3·a_7_25 − b_2_2·b_2_4·a_5_15 − a_1_0·a_3_9·a_5_15 − a_1_0·a_3_8·a_5_15
       + b_2_3·c_6_18·a_1_1
  49. b_2_2·a_7_25 + b_2_2·b_2_4·a_5_15 − a_1_0·a_3_8·a_5_15
  50.  − b_6_17·a_3_9 + b_2_5·a_7_25 − b_2_5·a_7_24 − b_2_52·a_5_15 − b_2_2·b_2_4·a_5_15
       + a_1_0·a_3_9·a_5_15 − b_2_52·a_1_1·a_1_2·a_3_9 + b_2_5·c_6_18·a_1_1
  51. b_2_4·a_7_25 − b_2_2·b_2_4·a_5_15 − a_1_2·a_3_9·a_5_15 + a_1_0·a_3_8·a_5_15
       − b_2_5·a_1_2·a_3_7·a_3_9 + b_2_4·c_6_18·a_1_1
  52. b_8_33·a_1_1 − b_2_52·a_5_15 + b_2_2·b_2_4·a_5_15 + b_2_2·b_2_3·a_5_15
       − a_1_0·a_3_9·a_5_15 + a_1_0·a_3_8·a_5_15 + b_2_5·a_1_2·a_3_7·a_3_9
       − b_2_5·c_6_19·a_1_1 − b_2_4·c_6_19·a_1_1 − b_2_3·c_6_19·a_1_1
  53. b_8_33·a_1_0 + a_1_0·a_3_9·a_5_15 − b_2_3·c_6_19·a_1_2 − b_2_2·c_6_19·a_1_2
  54. b_8_33·a_1_2 − b_2_5·a_7_24 − b_2_53·a_3_7 − b_2_2·b_2_4·a_5_15 − b_2_2·b_2_3·a_5_15
       + a_1_2·a_3_9·a_5_15 − a_1_0·a_3_9·a_5_15 + a_1_0·a_3_8·a_5_15
       − b_2_52·a_1_1·a_1_2·a_3_9 − b_2_5·c_6_19·a_1_2 − b_2_2·c_6_19·a_1_2
  55.  − a_5_12·a_5_15 + b_2_5·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_19·a_1_1·a_1_2
  56. a_3_8·a_7_24 + b_2_2·a_3_9·a_5_15 − b_2_2·a_3_8·a_5_15 − b_2_2·b_2_3·a_1_2·a_5_15
       + b_2_22·a_1_2·a_5_15
  57. a_5_12·a_5_15 + a_3_7·a_7_24
  58. a_3_9·a_7_25 − a_3_9·a_7_24 − b_2_5·a_3_9·a_5_15 − b_2_2·b_2_3·a_1_2·a_5_15
       + b_2_22·a_1_2·a_5_15 − c_6_18·a_1_1·a_3_9 + b_2_3·c_6_18·a_1_1·a_1_2
  59. a_3_8·a_7_25 + b_2_3·a_3_9·a_5_15 + b_2_2·a_3_9·a_5_15 − b_2_2·a_3_8·a_5_15
       − c_6_18·a_1_1·a_3_8
  60. a_5_12·a_5_15 + a_3_7·a_7_25 − b_2_5·a_3_9·a_5_15 + b_2_3·a_3_9·a_5_15
  61. b_2_3·b_8_33 + b_2_3·a_3_9·a_5_15 − b_2_2·b_2_3·a_1_2·a_5_15 − b_2_22·a_1_2·a_5_15
       − b_2_2·b_2_4·c_6_19 + b_2_2·b_2_3·c_6_19 − c_6_19·a_1_0·a_3_9
       − b_2_3·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_18·a_1_1·a_1_2
  62. b_2_2·b_8_33 + b_2_2·a_3_9·a_5_15 − b_2_2·b_2_3·a_1_2·a_5_15 − b_2_22·a_1_2·a_5_15
       − b_2_2·b_2_4·c_6_19 + b_2_2·b_2_3·c_6_19 − c_6_19·a_1_0·a_3_8
  63. b_2_4·b_8_33 + a_3_9·a_7_24 − b_2_52·a_3_7·a_3_9 + b_2_3·a_3_9·a_5_15
       − b_2_2·b_2_3·a_1_2·a_5_15 + b_2_22·a_1_2·a_5_15 − c_6_19·a_1_2·a_3_9
       + c_6_19·a_1_0·a_3_9 − c_6_19·a_1_0·a_3_8 + b_2_3·c_6_19·a_1_1·a_1_2
  64.  − a_5_12·a_5_15 + a_1_1·a_9_42 + b_2_5·a_3_9·a_5_15 − b_2_52·a_1_2·a_5_15
       − b_2_3·a_3_9·a_5_15 + b_2_22·a_1_2·a_5_15 + c_6_19·a_1_1·a_3_9 + c_6_19·a_1_1·a_3_8
       − b_2_3·c_6_19·a_1_1·a_1_2
  65. a_1_0·a_9_42 − b_2_2·b_2_3·a_1_2·a_5_15 + c_6_19·a_1_0·a_3_9 + c_6_19·a_1_0·a_3_8
       + c_6_18·a_1_0·a_3_7 − b_2_3·c_6_19·a_1_1·a_1_2
  66. a_5_12·a_5_15 + a_3_9·a_7_24 + a_1_2·a_9_42 − b_2_52·a_3_7·a_3_9 + b_2_53·a_1_2·a_3_7
       + b_2_3·a_3_9·a_5_15 − b_2_22·a_1_2·a_5_15 + c_6_19·a_1_2·a_3_9 + c_6_19·a_1_0·a_3_9
       − c_6_19·a_1_0·a_3_8 + c_6_18·a_1_2·a_3_7 − b_2_3·c_6_18·a_1_1·a_1_2
  67. b_6_17·a_5_15 + b_2_52·c_6_19·a_1_1 − c_6_19·a_1_0·a_1_1·a_3_9
  68. b_6_17·a_5_12 + b_2_52·a_7_24 − b_2_53·a_5_15 + b_2_54·a_3_7 − b_2_22·b_2_3·a_5_15
       − b_2_5·a_1_2·a_3_9·a_5_15 + b_2_52·a_1_2·a_3_7·a_3_9 + b_2_2·a_1_0·a_3_9·a_5_15
       + b_2_2·a_1_0·a_3_8·a_5_15
  69. b_8_33·a_3_8 − b_2_2·a_1_0·a_3_9·a_5_15 + b_2_2·a_1_0·a_3_8·a_5_15
       − b_2_3·c_6_19·a_3_9 − b_2_2·b_2_3·c_6_19·a_1_2 + b_2_22·c_6_19·a_1_2
       + c_6_19·a_1_0·a_1_1·a_3_9 + c_6_18·a_1_0·a_1_1·a_3_9
  70. b_8_33·a_3_7 + b_2_5·a_1_2·a_3_9·a_5_15 − b_2_2·a_1_2·a_3_9·a_5_15
       − b_2_5·c_6_19·a_3_7 + b_2_52·c_6_19·a_1_1 − b_2_4·c_6_19·a_3_7 − b_2_3·c_6_19·a_3_7
       − c_6_19·a_1_0·a_1_1·a_3_9
  71. b_2_3·a_9_42 − b_2_22·b_2_3·a_5_15 − b_2_2·a_1_2·a_3_9·a_5_15
       − b_2_2·a_1_0·a_3_8·a_5_15 + b_2_3·c_6_19·a_3_9 + b_2_3·c_6_18·a_3_7
       + b_2_2·c_6_19·a_3_9 − b_2_2·c_6_19·a_3_8 + c_6_19·a_1_0·a_1_1·a_3_9
  72. b_2_2·a_9_42 − b_2_22·b_2_4·a_5_15 − b_2_22·b_2_3·a_5_15 + b_2_2·a_1_2·a_3_9·a_5_15
       + b_2_2·a_1_0·a_3_9·a_5_15 + b_2_2·c_6_19·a_3_9 + b_2_2·c_6_19·a_3_8
       − b_2_2·b_2_3·c_6_19·a_1_2 + b_2_22·c_6_19·a_1_2
  73.  − b_8_33·a_3_9 − b_6_17·a_5_12 + b_2_5·a_9_42 + b_2_53·a_5_15 + b_2_54·a_3_7
       + b_2_22·b_2_3·a_5_15 + b_2_52·a_1_2·a_3_7·a_3_9 − b_2_53·a_1_1·a_1_2·a_3_9
       − b_2_2·a_1_2·a_3_9·a_5_15 + b_2_2·a_1_0·a_3_9·a_5_15 − b_2_5·c_6_19·a_3_9
       + b_2_5·c_6_18·a_3_7 − b_2_52·c_6_19·a_1_2 − b_2_52·c_6_19·a_1_1 + b_2_3·c_6_19·a_3_9
       + b_2_2·c_6_19·a_3_9 − b_2_2·c_6_19·a_3_8 − b_2_2·b_2_3·c_6_19·a_1_2
       − b_2_22·c_6_19·a_1_2 − c_6_19·a_1_1·a_1_2·a_3_9 + c_6_18·a_1_1·a_1_2·a_3_9
       + c_6_18·a_1_0·a_1_1·a_3_9
  74. b_2_4·a_9_42 − b_2_22·b_2_4·a_5_15 − b_2_52·a_1_2·a_3_7·a_3_9
       − b_2_2·a_1_0·a_3_9·a_5_15 − b_2_2·a_1_0·a_3_8·a_5_15 + b_2_4·c_6_18·a_3_7
       + b_2_3·c_6_19·a_3_9 + b_2_2·c_6_19·a_3_9 − b_2_2·c_6_19·a_3_8
       − c_6_19·a_1_1·a_1_2·a_3_9
  75. a_5_12·a_7_24 − b_2_53·a_1_2·a_5_15 + b_2_22·b_2_3·a_1_2·a_5_15
       − b_2_52·c_6_19·a_1_1·a_1_2
  76. a_5_15·a_7_24 + b_2_5·c_6_19·a_1_2·a_3_7 − b_2_3·c_6_19·a_1_2·a_3_7
  77. a_5_15·a_7_25 + b_2_5·c_6_19·a_1_2·a_3_7 + b_2_5·c_6_19·a_1_1·a_3_9
       − b_2_3·c_6_19·a_1_2·a_3_7 − b_2_3·c_6_19·a_1_1·a_3_9
  78. b_6_172 + b_2_52·b_8_33 + b_2_52·a_3_9·a_5_15 − b_2_52·a_1_2·a_7_25
       − b_2_53·a_3_7·a_3_9 + b_2_53·a_1_2·a_5_15 − b_2_54·a_1_2·a_3_7
       + b_2_54·a_1_1·a_3_9 − b_2_22·b_2_3·a_1_2·a_5_15 + b_2_23·a_1_2·a_5_15
       − b_2_53·c_6_19 − b_2_22·b_2_4·c_6_19 − b_2_22·b_2_3·c_6_19
       − b_2_5·c_6_19·a_1_2·a_3_9 + b_2_2·c_6_19·a_1_2·a_3_9
  79. a_5_12·a_7_25 + a_3_9·a_9_42 + b_2_52·a_3_9·a_5_15 − b_2_53·a_3_7·a_3_9
       − b_2_53·a_1_2·a_5_15 + b_2_2·b_2_3·a_3_9·a_5_15 + b_2_23·a_1_2·a_5_15
       − c_6_18·a_3_7·a_3_9 + b_2_5·c_6_19·a_1_2·a_3_9 + b_2_5·c_6_19·a_1_1·a_3_9
       + b_2_5·c_6_18·a_1_2·a_3_7 + b_2_52·c_6_19·a_1_1·a_1_2 + b_2_3·c_6_19·a_1_1·a_3_9
       + b_2_3·c_6_18·a_1_2·a_3_7 − b_2_3·c_6_18·a_1_1·a_3_9 + b_2_2·c_6_19·a_1_0·a_3_9
       − b_2_2·c_6_19·a_1_0·a_3_8
  80. a_3_8·a_9_42 − b_2_2·b_2_3·a_3_9·a_5_15 + b_2_22·a_3_9·a_5_15 − b_2_22·a_3_8·a_5_15
       + b_2_22·b_2_3·a_1_2·a_5_15 − b_2_23·a_1_2·a_5_15 − c_6_18·a_3_7·a_3_8
       − b_2_3·c_6_19·a_1_1·a_3_9
  81. a_3_7·a_9_42 + c_6_19·a_3_7·a_3_9 + c_6_19·a_3_7·a_3_8 + b_2_5·c_6_19·a_1_2·a_3_7
       + b_2_5·c_6_19·a_1_1·a_3_9 − b_2_52·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_19·a_1_1·a_3_9
  82. a_5_12·a_7_25 + b_2_5·a_1_2·a_9_42 + b_2_52·a_3_9·a_5_15 − b_2_53·a_1_2·a_5_15
       + b_2_54·a_1_2·a_3_7 − b_2_2·b_2_3·a_3_9·a_5_15 + b_2_22·b_2_3·a_1_2·a_5_15
       − b_2_23·a_1_2·a_5_15 + b_2_5·c_6_19·a_1_2·a_3_9 − b_2_5·c_6_18·a_1_2·a_3_7
       − b_2_52·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_18·a_1_2·a_3_7 + b_2_2·c_6_19·a_1_2·a_3_9
       − b_2_2·c_6_19·a_1_0·a_3_9 + b_2_2·c_6_19·a_1_0·a_3_8
  83. b_6_17·a_7_24 − b_2_54·a_5_15 + b_2_23·b_2_4·a_5_15 + b_2_23·b_2_3·a_5_15
       − b_2_52·a_1_2·a_3_9·a_5_15 + b_2_22·a_1_2·a_3_9·a_5_15
       − b_2_22·a_1_0·a_3_9·a_5_15 + b_2_22·a_1_0·a_3_8·a_5_15 + b_2_53·c_6_19·a_1_2
       − b_2_22·b_2_3·c_6_19·a_1_2
  84. b_8_33·a_5_12 − b_2_52·a_1_2·a_3_9·a_5_15 + b_2_53·a_1_2·a_3_7·a_3_9
       + b_2_22·a_1_2·a_3_9·a_5_15 − b_2_5·c_6_19·a_5_12 + b_2_53·c_6_19·a_1_2
       − b_2_53·c_6_19·a_1_1 − b_2_22·b_2_3·c_6_19·a_1_2 − c_6_19·a_1_2·a_3_7·a_3_9
       + b_2_5·c_6_19·a_1_1·a_1_2·a_3_9
  85. b_8_33·a_5_15 − b_2_5·c_6_19·a_5_15 − b_2_52·c_6_19·a_3_7 − b_2_4·c_6_19·a_5_15
       − b_2_3·c_6_19·a_5_15 + c_6_19·a_1_0·a_3_7·a_3_9 + b_2_5·c_6_19·a_1_1·a_1_2·a_3_9
  86. b_6_17·a_7_25 + b_2_52·a_9_42 + b_2_53·a_7_24 − b_2_54·a_5_15 − b_2_55·a_3_7
       + b_2_23·b_2_4·a_5_15 + b_2_23·b_2_3·a_5_15 + b_2_52·a_1_2·a_3_9·a_5_15
       − b_2_54·a_1_1·a_1_2·a_3_9 + b_2_22·a_1_2·a_3_9·a_5_15 + b_2_52·c_6_19·a_3_9
       − b_2_52·c_6_18·a_3_7 + b_2_2·b_2_3·c_6_19·a_3_9 − b_2_22·c_6_19·a_3_9
       + b_2_22·c_6_19·a_3_8 + b_2_22·b_2_3·c_6_19·a_1_2 − b_2_23·c_6_19·a_1_2
       − c_6_18·a_1_0·a_3_7·a_3_9 − b_2_5·c_6_19·a_1_1·a_1_2·a_3_9
  87. a_7_24·a_7_25 + b_2_53·a_3_9·a_5_15 − b_2_22·b_2_3·a_3_9·a_5_15
       + b_2_5·c_6_18·a_1_2·a_5_15 + b_2_52·c_6_19·a_1_2·a_3_9 − b_2_52·c_6_19·a_1_2·a_3_7
       − b_2_3·c_6_18·a_1_2·a_5_15 − b_2_22·c_6_19·a_1_2·a_3_9
  88. b_6_17·b_8_33 + b_2_52·a_1_2·a_9_42 − b_2_53·a_3_9·a_5_15 + b_2_54·a_3_7·a_3_9
       + b_2_55·a_1_2·a_3_7 + b_2_22·b_2_3·a_3_9·a_5_15 − b_2_24·a_1_2·a_5_15
       − b_2_5·b_6_17·c_6_19 + b_2_54·c_6_19 − b_2_23·b_2_4·c_6_19 − b_2_23·b_2_3·c_6_19
       − c_6_19·a_1_2·a_7_25 + b_2_5·c_6_19·a_1_2·a_5_15 + b_2_52·c_6_19·a_1_2·a_3_9
       − b_2_52·c_6_19·a_1_2·a_3_7 + b_2_52·c_6_19·a_1_1·a_3_9
       − b_2_52·c_6_18·a_1_2·a_3_7 + b_2_53·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_19·a_3_7·a_3_9
       + b_2_2·c_6_19·a_1_2·a_5_15 − b_2_22·c_6_19·a_1_0·a_3_9 + b_2_22·c_6_19·a_1_0·a_3_8
       + c_6_18·c_6_19·a_1_1·a_1_2
  89. a_5_12·a_9_42 − b_2_54·a_1_2·a_5_15 + b_2_24·a_1_2·a_5_15 + c_6_19·a_1_2·a_7_25
       − b_2_5·c_6_19·a_3_7·a_3_9 − b_2_5·c_6_19·a_1_2·a_5_15 − b_2_5·c_6_18·a_1_2·a_5_15
       + b_2_52·c_6_19·a_1_2·a_3_9 − b_2_52·c_6_19·a_1_2·a_3_7
       − b_2_52·c_6_19·a_1_1·a_3_9 + b_2_53·c_6_19·a_1_1·a_1_2 + b_2_3·c_6_18·a_1_2·a_5_15
       + b_2_2·c_6_19·a_1_2·a_5_15 − b_2_22·c_6_19·a_1_2·a_3_9 − c_6_18·c_6_19·a_1_1·a_1_2
  90. a_5_15·a_9_42 − c_6_19·a_3_9·a_5_15 − c_6_19·a_3_8·a_5_15 − b_2_5·c_6_19·a_3_7·a_3_9
       + b_2_5·c_6_19·a_1_2·a_5_15 − b_2_52·c_6_19·a_1_2·a_3_7 + b_2_3·c_6_19·a_3_7·a_3_9
       − b_2_2·c_6_19·a_1_2·a_5_15
  91. b_8_33·a_7_24 + b_2_53·a_1_2·a_3_9·a_5_15 − b_2_23·a_1_2·a_3_9·a_5_15
       − b_2_5·c_6_19·a_7_24 − b_2_52·c_6_19·a_5_12 − b_2_53·c_6_19·a_3_7
       − b_2_54·c_6_19·a_1_1 + b_2_2·b_2_3·c_6_19·a_5_15 + c_6_19·a_1_0·a_3_8·a_5_15
       − b_2_5·c_6_19·a_1_2·a_3_7·a_3_9
  92. b_8_33·a_7_25 − b_2_5·c_6_19·a_7_25 − b_2_52·c_6_19·a_5_12 + b_2_52·c_6_18·a_5_15
       + b_2_53·c_6_19·a_3_9 + b_2_53·c_6_19·a_3_7 − b_2_54·c_6_19·a_1_1
       + b_2_2·b_2_4·c_6_19·a_5_15 − b_2_2·b_2_4·c_6_18·a_5_15 − b_2_2·b_2_3·c_6_18·a_5_15
       − b_2_22·b_2_3·c_6_19·a_3_9 − c_6_19·a_1_2·a_3_9·a_5_15 − c_6_19·a_1_0·a_3_9·a_5_15
       + c_6_18·a_1_0·a_3_9·a_5_15 − c_6_18·a_1_0·a_3_8·a_5_15
       − b_2_5·c_6_19·a_1_2·a_3_7·a_3_9 + b_2_52·c_6_19·a_1_1·a_1_2·a_3_9
       + b_2_4·c_6_18·c_6_19·a_1_1 + b_2_3·c_6_18·c_6_19·a_1_1
  93. b_6_17·a_9_42 − b_2_55·a_5_15 + b_2_24·b_2_3·a_5_15 + b_2_54·a_1_2·a_3_7·a_3_9
       + b_2_23·a_1_0·a_3_8·a_5_15 + b_2_5·c_6_19·a_7_25 − b_2_5·c_6_19·a_7_24
       − b_2_52·c_6_19·a_5_15 − b_2_52·c_6_19·a_5_12 − b_2_52·c_6_18·a_5_15
       + b_2_53·c_6_19·a_3_9 + b_2_53·c_6_19·a_3_7 − b_2_54·c_6_19·a_1_2
       − b_2_2·b_2_4·c_6_19·a_5_15 + b_2_2·b_2_4·c_6_18·a_5_15 + b_2_2·b_2_3·c_6_18·a_5_15
       − b_2_22·b_2_3·c_6_19·a_3_9 + b_2_24·c_6_19·a_1_2 + c_6_19·a_1_0·a_3_8·a_5_15
       − c_6_18·a_1_0·a_3_9·a_5_15 + c_6_18·a_1_0·a_3_8·a_5_15
       − b_2_5·c_6_19·a_1_2·a_3_7·a_3_9 − b_2_52·c_6_19·a_1_1·a_1_2·a_3_9
       + b_2_5·c_6_18·c_6_19·a_1_1
  94.  − b_8_332 + a_7_24·a_9_42 − b_2_54·a_3_9·a_5_15 + b_2_23·b_2_3·a_3_9·a_5_15
       − b_2_24·b_2_3·a_1_2·a_5_15 + b_2_25·a_1_2·a_5_15 − b_2_5·c_6_19·b_8_33
       + b_2_52·b_6_17·c_6_19 − b_2_5·c_6_19·a_1_2·a_7_25 + b_2_52·c_6_19·a_3_7·a_3_9
       + b_2_52·c_6_18·a_1_2·a_5_15 + b_2_53·c_6_19·a_1_2·a_3_9
       + b_2_53·c_6_19·a_1_2·a_3_7 − b_2_54·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_19·a_3_9·a_5_15
       + b_2_2·c_6_19·a_3_9·a_5_15 − b_2_2·c_6_19·a_3_8·a_5_15 − b_2_22·c_6_18·a_1_2·a_5_15
       − b_2_23·c_6_19·a_1_2·a_3_9 − b_2_52·c_6_192 + b_2_2·b_2_4·c_6_192
       + b_2_2·b_2_3·c_6_192 + c_6_192·a_1_0·a_3_9 − c_6_192·a_1_0·a_3_8
       − b_2_5·c_6_18·c_6_19·a_1_1·a_1_2 + b_2_3·c_6_18·c_6_19·a_1_1·a_1_2
  95.  − b_8_332 + a_7_25·a_9_42 + b_2_54·a_3_9·a_5_15 − b_2_23·b_2_3·a_3_9·a_5_15
       + b_2_24·b_2_3·a_1_2·a_5_15 − b_2_25·a_1_2·a_5_15 − b_2_5·c_6_19·b_8_33
       + b_2_52·b_6_17·c_6_19 − b_2_5·c_6_19·a_3_9·a_5_15 + b_2_5·c_6_18·a_3_9·a_5_15
       + b_2_52·c_6_19·a_3_7·a_3_9 − b_2_52·c_6_18·a_1_2·a_5_15
       − b_2_53·c_6_19·a_1_2·a_3_9 + b_2_53·c_6_19·a_1_2·a_3_7
       − b_2_54·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_18·a_3_9·a_5_15 + b_2_2·c_6_19·a_3_9·a_5_15
       − b_2_2·c_6_19·a_3_8·a_5_15 + b_2_2·b_2_3·c_6_19·a_1_2·a_5_15
       − b_2_22·c_6_19·a_1_2·a_5_15 + b_2_22·c_6_18·a_1_2·a_5_15
       + b_2_23·c_6_19·a_1_2·a_3_9 − b_2_52·c_6_192 + b_2_2·b_2_4·c_6_192
       + b_2_2·b_2_3·c_6_192 + c_6_192·a_1_0·a_3_9 − c_6_192·a_1_0·a_3_8
       − c_6_18·c_6_19·a_1_1·a_3_9 − c_6_18·c_6_19·a_1_1·a_3_8
       − b_2_5·c_6_18·c_6_19·a_1_1·a_1_2 − b_2_3·c_6_18·c_6_19·a_1_1·a_1_2
  96. b_8_332 + b_2_54·a_3_9·a_5_15 − b_2_23·b_2_3·a_3_9·a_5_15
       + b_2_24·b_2_3·a_1_2·a_5_15 − b_2_25·a_1_2·a_5_15 + b_2_5·c_6_19·b_8_33
       − b_2_52·b_6_17·c_6_19 + c_6_19·a_1_2·a_9_42 + b_2_52·c_6_19·a_3_7·a_3_9
       − b_2_52·c_6_18·a_1_2·a_5_15 − b_2_53·c_6_19·a_1_2·a_3_9
       + b_2_53·c_6_19·a_1_2·a_3_7 − b_2_53·c_6_19·a_1_1·a_3_9 − b_2_3·c_6_19·a_3_9·a_5_15
       − b_2_2·b_2_3·c_6_19·a_1_2·a_5_15 + b_2_22·c_6_18·a_1_2·a_5_15
       + b_2_23·c_6_19·a_1_2·a_3_9 + b_2_52·c_6_192 − b_2_2·b_2_4·c_6_192
       − b_2_2·b_2_3·c_6_192 + c_6_192·a_1_2·a_3_9 + c_6_18·c_6_19·a_1_2·a_3_7
       + b_2_5·c_6_192·a_1_1·a_1_2 − b_2_3·c_6_192·a_1_1·a_1_2
  97. b_8_33·a_9_42 − b_2_54·a_1_2·a_3_9·a_5_15 + b_2_24·a_1_2·a_3_9·a_5_15
       − b_2_52·c_6_19·a_7_25 + b_2_52·c_6_19·a_7_24 + b_2_53·c_6_19·a_5_12
       − b_2_54·c_6_19·a_3_7 − b_2_55·c_6_19·a_1_1 + b_2_22·b_2_4·c_6_19·a_5_15
       − b_2_22·b_2_3·c_6_19·a_5_15 − b_2_5·c_6_19·a_1_2·a_3_9·a_5_15
       + b_2_5·c_6_18·a_1_2·a_3_9·a_5_15 + b_2_53·c_6_19·a_1_1·a_1_2·a_3_9
       + b_2_2·c_6_19·a_1_2·a_3_9·a_5_15 − b_2_2·c_6_19·a_1_0·a_3_9·a_5_15
       − b_2_2·c_6_19·a_1_0·a_3_8·a_5_15 − b_2_2·c_6_18·a_1_2·a_3_9·a_5_15
       + b_2_5·c_6_192·a_3_9 + b_2_5·c_6_18·c_6_19·a_3_7 − b_2_52·c_6_192·a_1_2
       − b_2_52·c_6_192·a_1_1 + b_2_52·c_6_18·c_6_19·a_1_1 + b_2_4·c_6_18·c_6_19·a_3_7
       − b_2_3·c_6_192·a_3_9 + b_2_3·c_6_18·c_6_19·a_3_7 + b_2_22·c_6_192·a_1_2
       + c_6_192·a_1_1·a_1_2·a_3_9 + c_6_18·c_6_19·a_1_1·a_1_2·a_3_9
       + c_6_18·c_6_19·a_1_0·a_1_1·a_3_9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 17.
  • 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_6_18, a Duflot regular element of degree 6
    2. c_6_19, a Duflot regular element of degree 6
    3. b_2_52 − b_2_2·b_2_4 − b_2_2·b_2_3 + b_2_22, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 13].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. b_2_5 → 0, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. a_3_9 → 0, an element of degree 3
  11. a_5_12 → 0, an element of degree 5
  12. a_5_15 → 0, an element of degree 5
  13. b_6_17 → 0, an element of degree 6
  14. c_6_18 →  − c_2_23, an element of degree 6
  15. c_6_19 →  − c_2_13, an element of degree 6
  16. a_7_24 → 0, an element of degree 7
  17. a_7_25 → 0, an element of degree 7
  18. b_8_33 → 0, an element of degree 8
  19. a_9_42 → 0, an element of degree 9

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

  1. a_1_0 → a_1_2, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_2_2 → c_2_5, an element of degree 2
  5. b_2_3 →  − a_1_1·a_1_2, an element of degree 2
  6. b_2_4 → a_1_1·a_1_2, an element of degree 2
  7. b_2_5 → 0, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  10. a_3_9 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  11. a_5_12 → 0, an element of degree 5
  12. a_5_15 → c_2_52·a_1_0 − c_2_3·c_2_5·a_1_2, an element of degree 5
  13. b_6_17 → 0, an element of degree 6
  14. c_6_18 →  − c_2_52·a_1_1·a_1_2 + c_2_4·c_2_52 − c_2_43, an element of degree 6
  15. c_6_19 → c_2_52·a_1_0·a_1_2 + c_2_3·c_2_52 − c_2_33, an element of degree 6
  16. a_7_24 → c_2_52·a_1_0·a_1_1·a_1_2, an element of degree 7
  17. a_7_25 → 0, an element of degree 7
  18. b_8_33 →  − c_2_53·a_1_0·a_1_1 + c_2_4·c_2_52·a_1_0·a_1_2 − c_2_3·c_2_52·a_1_1·a_1_2, an element of degree 8
  19. a_9_42 →  − c_2_3·c_2_53·a_1_1 + c_2_3·c_2_4·c_2_52·a_1_2 + c_2_33·c_2_5·a_1_1
       − c_2_33·c_2_4·a_1_2, an element of degree 9

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → a_1_2, an element of degree 1
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_2_4 → a_1_1·a_1_2, an element of degree 2
  7. b_2_5 → c_2_5, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. a_3_9 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  11. a_5_12 →  − c_2_3·c_2_5·a_1_2, an element of degree 5
  12. a_5_15 → 0, an element of degree 5
  13. b_6_17 →  − c_2_3·c_2_52, an element of degree 6
  14. c_6_18 → c_2_4·c_2_52 − c_2_43, an element of degree 6
  15. c_6_19 →  − c_2_33, an element of degree 6
  16. a_7_24 →  − c_2_32·c_2_5·a_1_2, an element of degree 7
  17. a_7_25 → c_2_3·c_2_52·a_1_1 − c_2_3·c_2_4·c_2_5·a_1_2 − c_2_32·c_2_5·a_1_2, an element of degree 7
  18. b_8_33 →  − c_2_3·c_2_52·a_1_1·a_1_2 − c_2_33·a_1_1·a_1_2 − c_2_32·c_2_52 − c_2_33·c_2_5, an element of degree 8
  19. a_9_42 → c_2_32·c_2_52·a_1_2 + c_2_32·c_2_52·a_1_1 − c_2_32·c_2_4·c_2_5·a_1_2
       − c_2_33·c_2_5·a_1_2 − c_2_33·c_2_5·a_1_1 + c_2_33·c_2_4·a_1_2, an element of degree 9

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

  1. a_1_0 → a_1_2, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → a_1_2, an element of degree 1
  4. b_2_2 → c_2_5, an element of degree 2
  5. b_2_3 →  − a_1_1·a_1_2 + c_2_5, an element of degree 2
  6. b_2_4 →  − a_1_1·a_1_2, an element of degree 2
  7. b_2_5 → c_2_5, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  10. a_3_9 → c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
  11. a_5_12 → 0, an element of degree 5
  12. a_5_15 → c_2_52·a_1_0 − c_2_3·c_2_5·a_1_2, an element of degree 5
  13. b_6_17 →  − c_2_52·a_1_0·a_1_2, an element of degree 6
  14. c_6_18 → c_2_4·c_2_52 − c_2_43, an element of degree 6
  15. c_6_19 → c_2_52·a_1_0·a_1_2 + c_2_3·c_2_52 − c_2_33, an element of degree 6
  16. a_7_24 → c_2_52·a_1_0·a_1_1·a_1_2 − c_2_53·a_1_0 + c_2_3·c_2_52·a_1_2, an element of degree 7
  17. a_7_25 →  − c_2_52·a_1_0·a_1_1·a_1_2, an element of degree 7
  18. b_8_33 → c_2_53·a_1_0·a_1_1 − c_2_4·c_2_52·a_1_0·a_1_2 − c_2_3·c_2_52·a_1_1·a_1_2
       − c_2_33·a_1_1·a_1_2 − c_2_3·c_2_53 + c_2_33·c_2_5, an element of degree 8
  19. a_9_42 → c_2_54·a_1_0 − c_2_3·c_2_53·a_1_2, an element of degree 9

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

  1. a_1_0 →  − a_1_2, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → a_1_2, an element of degree 1
  4. b_2_2 →  − c_2_5, an element of degree 2
  5. b_2_3 → a_1_1·a_1_2 + c_2_5, an element of degree 2
  6. b_2_4 → c_2_5, an element of degree 2
  7. b_2_5 → c_2_5, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → c_2_5·a_1_2 + c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
  10. a_3_9 →  − c_2_5·a_1_2, an element of degree 3
  11. a_5_12 → 0, an element of degree 5
  12. a_5_15 → c_2_52·a_1_0 − c_2_3·c_2_5·a_1_2, an element of degree 5
  13. b_6_17 →  − c_2_52·a_1_0·a_1_2, an element of degree 6
  14. c_6_18 → c_2_52·a_1_1·a_1_2 + c_2_53 + c_2_4·c_2_52 − c_2_43, an element of degree 6
  15. c_6_19 →  − c_2_52·a_1_0·a_1_2 + c_2_3·c_2_52 − c_2_33, an element of degree 6
  16. a_7_24 →  − c_2_52·a_1_0·a_1_1·a_1_2 − c_2_53·a_1_0 + c_2_3·c_2_52·a_1_2, an element of degree 7
  17. a_7_25 →  − c_2_52·a_1_0·a_1_1·a_1_2 − c_2_53·a_1_0 + c_2_3·c_2_52·a_1_2, an element of degree 7
  18. b_8_33 →  − c_2_53·a_1_0·a_1_2 + c_2_3·c_2_52·a_1_1·a_1_2 − c_2_33·a_1_1·a_1_2, an element of degree 8
  19. a_9_42 → c_2_53·a_1_0·a_1_1·a_1_2 + c_2_54·a_1_0 + c_2_3·c_2_53·a_1_2 − c_2_3·c_2_53·a_1_1
       + c_2_3·c_2_4·c_2_52·a_1_2 + c_2_33·c_2_5·a_1_2 + c_2_33·c_2_5·a_1_1
       − c_2_33·c_2_4·a_1_2, an element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243


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