Cohomology of group number 3 of order 343

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General information on the group

  • The group is also known as E343, the Extraspecial 7-group of order 343 and exponent 7.
  • The group has 2 minimal generators and exponent 7.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has 8 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    (t2  +  1) · (t10  +  t6  +  t2  +  1)
    (t  −  1)2 · (t6  −  t5  +  t4  −  t3  +  t2  −  t  +  1) · (t6  +  t5  +  t4  +  t3  +  t2  +  t  +  1)
  • The a-invariants are -∞,-4,-2. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 16 minimal generators of maximal degree 14:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. a_3_4, a nilpotent element of degree 3
  8. a_3_5, a nilpotent element of degree 3
  9. a_7_9, a nilpotent element of degree 7
  10. a_8_5, a nilpotent element of degree 8
  11. a_9_11, a nilpotent element of degree 9
  12. a_10_6, a nilpotent element of degree 10
  13. a_11_13, a nilpotent element of degree 11
  14. b_12_13, an element of degree 12
  15. a_13_15, a nilpotent element of degree 13
  16. c_14_16, a Duflot regular element of degree 14

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Ring relations

There are 8 "obvious" relations:
   a_1_02, a_1_12, a_3_42, a_3_52, a_7_92, a_9_112, a_11_132, a_13_152

Apart from that, there are 96 minimal relations of maximal degree 25:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_0
  3. a_2_1·a_1_1 + a_2_0·a_1_1
  4. a_2_1·a_1_0 − a_2_0·a_1_1
  5. b_2_3·a_1_0 − b_2_2·a_1_1
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9.  − 2·a_2_1·b_2_2 − a_2_0·b_2_3 − 2·a_2_0·b_2_2 + a_1_1·a_3_4
  10. a_2_0·b_2_2 + a_1_0·a_3_4
  11.  − a_2_1·b_2_3 + 3·a_2_1·b_2_2 − 3·a_2_0·b_2_3 + 3·a_2_0·b_2_2 + a_1_1·a_3_5
  12. a_2_1·b_2_2 + 2·a_2_0·b_2_3 + 3·a_2_0·b_2_2 + a_1_0·a_3_5
  13. a_2_0·a_3_4
  14.  − b_2_3·a_3_4 + b_2_2·a_3_5 − 2·b_2_2·a_3_4 + 3·b_2_2·b_2_3·a_1_1 + 3·b_2_22·a_1_1
  15.  − a_2_1·a_3_4 + a_2_0·a_3_5
  16. a_2_1·a_3_5 − a_2_1·a_3_4
  17. a_1_1·a_7_9 − b_2_32·a_1_1·a_3_5 − b_2_2·b_2_3·a_1_1·a_3_5 − 2·b_2_22·a_1_1·a_3_5
       + b_2_22·a_1_0·a_3_5 − 2·b_2_22·a_1_0·a_3_4
  18. a_1_0·a_7_9 − b_2_2·b_2_3·a_1_1·a_3_5 − b_2_22·a_1_1·a_3_5 − 2·b_2_22·a_1_0·a_3_5
       + b_2_22·a_1_0·a_3_4
  19. b_2_2·a_7_9 − b_2_2·b_2_32·a_3_5 − b_2_22·b_2_3·a_3_5 + b_2_22·b_2_32·a_1_1
       − 2·b_2_23·a_3_5 + b_2_23·a_3_4 + 3·b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1
  20. a_2_0·a_7_9
  21. a_2_1·a_7_9
  22. b_2_3·a_7_9 − b_2_33·a_3_5 − b_2_2·b_2_32·a_3_5 + b_2_2·b_2_33·a_1_1
       − 2·b_2_22·b_2_3·a_3_5 + 3·b_2_22·b_2_32·a_1_1 + b_2_23·a_3_5 − 2·b_2_23·a_3_4
       + 2·b_2_23·b_2_3·a_1_1 + 3·b_2_24·a_1_1
  23. a_8_5·a_1_1
  24. a_8_5·a_1_0
  25. a_3_5·a_7_9 − b_2_2·b_2_32·a_1_1·a_3_5 − 3·b_2_22·b_2_3·a_1_1·a_3_5
       − 2·b_2_23·a_1_1·a_3_5 + 3·b_2_23·a_1_0·a_3_5 + b_2_23·a_1_0·a_3_4
  26. a_3_4·a_7_9 − 3·b_2_2·b_2_32·a_1_1·a_3_5 − b_2_22·b_2_3·a_1_1·a_3_5
       − 3·b_2_23·a_1_1·a_3_5 + b_2_23·a_1_0·a_3_5 − 2·b_2_23·a_1_0·a_3_4
  27. b_2_2·a_8_5 + b_2_2·b_2_32·a_1_1·a_3_5 + 2·b_2_22·b_2_3·a_1_1·a_3_5
       + 2·b_2_23·a_1_1·a_3_5 − 2·b_2_23·a_1_0·a_3_5
  28. a_2_0·a_8_5
  29. a_2_1·a_8_5
  30. b_2_3·a_8_5 + b_2_33·a_1_1·a_3_5 + 2·b_2_2·b_2_32·a_1_1·a_3_5
       + 2·b_2_22·b_2_3·a_1_1·a_3_5 − 2·b_2_23·a_1_1·a_3_5
  31. a_1_1·a_9_11 − b_2_33·a_1_1·a_3_5 − 3·b_2_2·b_2_32·a_1_1·a_3_5
       − 2·b_2_22·b_2_3·a_1_1·a_3_5 + 3·b_2_23·a_1_0·a_3_5 + b_2_23·a_1_0·a_3_4
  32. a_1_0·a_9_11 − b_2_2·b_2_32·a_1_1·a_3_5 − 3·b_2_22·b_2_3·a_1_1·a_3_5
       − 2·b_2_23·a_1_1·a_3_5 + 3·b_2_23·a_1_0·a_3_4
  33. a_8_5·a_3_5
  34. a_8_5·a_3_4
  35. b_2_2·a_9_11 − b_2_2·b_2_33·a_3_5 − 3·b_2_22·b_2_32·a_3_5 + b_2_22·b_2_33·a_1_1
       − 2·b_2_23·b_2_3·a_3_5 + 3·b_2_23·b_2_32·a_1_1 + 3·b_2_24·a_3_4 − b_2_25·a_1_1
  36. a_2_0·a_9_11
  37. a_2_1·a_9_11
  38. b_2_3·a_9_11 − b_2_34·a_3_5 − 3·b_2_2·b_2_33·a_3_5 + b_2_2·b_2_34·a_1_1
       − 2·b_2_22·b_2_32·a_3_5 + 3·b_2_22·b_2_33·a_1_1 + 3·b_2_24·a_3_5 + b_2_24·a_3_4
       + b_2_24·b_2_3·a_1_1 + 2·b_2_25·a_1_1
  39. a_10_6·a_1_1
  40. a_10_6·a_1_0
  41. a_3_5·a_9_11 − b_2_2·b_2_33·a_1_1·a_3_5 − 3·b_2_22·b_2_32·a_1_1·a_3_5
       − b_2_24·a_1_1·a_3_5 + 2·b_2_24·a_1_0·a_3_5 + 3·b_2_24·a_1_0·a_3_4
  42. a_3_4·a_9_11 − 3·b_2_2·b_2_33·a_1_1·a_3_5 + 3·b_2_23·b_2_3·a_1_1·a_3_5
       + 2·b_2_24·a_1_1·a_3_5 − 3·b_2_24·a_1_0·a_3_5 − b_2_24·a_1_0·a_3_4
  43. b_2_2·a_10_6 + b_2_2·b_2_33·a_1_1·a_3_5 − 2·b_2_22·b_2_32·a_1_1·a_3_5
       − b_2_23·b_2_3·a_1_1·a_3_5 − 2·b_2_24·a_1_1·a_3_5 + 3·b_2_24·a_1_0·a_3_5
       − b_2_24·a_1_0·a_3_4
  44. a_2_0·a_10_6
  45. a_2_1·a_10_6
  46. b_2_3·a_10_6 + b_2_34·a_1_1·a_3_5 − 2·b_2_2·b_2_33·a_1_1·a_3_5
       − b_2_22·b_2_32·a_1_1·a_3_5 − 2·b_2_23·b_2_3·a_1_1·a_3_5 + 3·b_2_24·a_1_1·a_3_5
       − b_2_24·a_1_0·a_3_5 + 2·b_2_24·a_1_0·a_3_4
  47. a_1_1·a_11_13 − b_2_34·a_1_1·a_3_5 − 2·b_2_2·b_2_33·a_1_1·a_3_5
       − 2·b_2_22·b_2_32·a_1_1·a_3_5 − b_2_23·b_2_3·a_1_1·a_3_5 − 2·b_2_24·a_1_1·a_3_5
  48. a_1_0·a_11_13 − b_2_2·b_2_33·a_1_1·a_3_5 − 2·b_2_22·b_2_32·a_1_1·a_3_5
       − 2·b_2_23·b_2_3·a_1_1·a_3_5 − b_2_24·a_1_1·a_3_5 − 2·b_2_24·a_1_0·a_3_5
  49. a_10_6·a_3_5
  50. a_10_6·a_3_4
  51. b_2_2·a_11_13 − b_2_2·b_2_34·a_3_5 − 3·b_2_2·b_2_35·a_1_1 − 2·b_2_22·b_2_33·a_3_5
       + b_2_22·b_2_34·a_1_1 − 2·b_2_23·b_2_32·a_3_5 + 3·b_2_23·b_2_33·a_1_1
       − b_2_24·b_2_3·a_3_5 − 2·b_2_24·b_2_32·a_1_1 − 2·b_2_25·a_3_5
       − 2·b_2_25·b_2_3·a_1_1 − 3·b_2_26·a_1_1
  52. a_2_0·a_11_13
  53. a_2_1·a_11_13
  54. b_2_3·a_11_13 − b_2_35·a_3_5 − 2·b_2_2·b_2_34·a_3_5 + b_2_2·b_2_35·a_1_1
       − 2·b_2_22·b_2_33·a_3_5 + 3·b_2_22·b_2_34·a_1_1 − b_2_23·b_2_32·a_3_5
       − 2·b_2_23·b_2_33·a_1_1 − 2·b_2_24·b_2_3·a_3_5 − 2·b_2_24·b_2_32·a_1_1
       − 3·b_2_25·b_2_3·a_1_1 − 3·b_2_26·a_1_1
  55. b_12_13·a_1_1 + b_2_2·b_2_35·a_1_1 + 2·b_2_22·b_2_34·a_1_1 + b_2_24·b_2_32·a_1_1
       − b_2_26·a_1_1
  56. b_12_13·a_1_0 − b_2_2·b_2_35·a_1_1 + b_2_22·b_2_34·a_1_1 + 2·b_2_23·b_2_33·a_1_1
       + b_2_25·b_2_3·a_1_1
  57. a_3_5·a_11_13 − 2·b_2_2·b_2_34·a_1_1·a_3_5 − b_2_22·b_2_33·a_1_1·a_3_5
       − 2·b_2_23·b_2_32·a_1_1·a_3_5 + 3·b_2_24·b_2_3·a_1_1·a_3_5 + b_2_25·a_1_1·a_3_5
  58. a_3_4·a_11_13 − 3·b_2_22·b_2_33·a_1_1·a_3_5 − 2·b_2_23·b_2_32·a_1_1·a_3_5
       − 3·b_2_24·b_2_3·a_1_1·a_3_5 − b_2_25·a_1_1·a_3_5 − b_2_25·a_1_0·a_3_5
       + 2·b_2_25·a_1_0·a_3_4
  59. b_2_2·b_12_13 − b_2_2·b_2_36 + b_2_22·b_2_35 + 2·b_2_23·b_2_34
       + b_2_25·b_2_32 + 3·b_2_2·b_2_34·a_1_1·a_3_5 − b_2_22·b_2_33·a_1_1·a_3_5
       − 2·b_2_23·b_2_32·a_1_1·a_3_5 + b_2_24·b_2_3·a_1_1·a_3_5 + 3·b_2_25·a_1_1·a_3_5
       − 3·b_2_25·a_1_0·a_3_5 + 2·b_2_25·a_1_0·a_3_4
  60. a_2_0·b_12_13 + b_2_2·b_2_34·a_1_1·a_3_5 − 3·b_2_22·b_2_33·a_1_1·a_3_5
       − 3·b_2_23·b_2_32·a_1_1·a_3_5 − b_2_24·b_2_3·a_1_1·a_3_5 + b_2_25·a_1_1·a_3_5
       − 2·b_2_25·a_1_0·a_3_5 − 3·b_2_25·a_1_0·a_3_4
  61. a_2_1·b_12_13 + 3·b_2_22·b_2_33·a_1_1·a_3_5 + 3·b_2_23·b_2_32·a_1_1·a_3_5
       + 2·b_2_24·b_2_3·a_1_1·a_3_5 − 3·b_2_25·a_1_1·a_3_5 − 2·b_2_25·a_1_0·a_3_5
       − 3·b_2_25·a_1_0·a_3_4
  62. b_2_3·b_12_13 + b_2_2·b_2_36 + 2·b_2_22·b_2_35 + b_2_24·b_2_33 − b_2_26·b_2_3
       + 3·b_2_35·a_1_1·a_3_5 − b_2_2·b_2_34·a_1_1·a_3_5 − 2·b_2_22·b_2_33·a_1_1·a_3_5
       + b_2_23·b_2_32·a_1_1·a_3_5 + 3·b_2_24·b_2_3·a_1_1·a_3_5 − 3·b_2_25·a_1_1·a_3_5
       + 2·b_2_25·a_1_0·a_3_5 + 3·b_2_25·a_1_0·a_3_4
  63. a_1_1·a_13_15 − b_2_35·a_1_1·a_3_5 + 3·b_2_2·b_2_34·a_1_1·a_3_5
       + 2·b_2_22·b_2_33·a_1_1·a_3_5 − 2·b_2_23·b_2_32·a_1_1·a_3_5
       − b_2_24·b_2_3·a_1_1·a_3_5 + 2·b_2_25·a_1_1·a_3_5 + 2·b_2_25·a_1_0·a_3_5
       + 3·b_2_25·a_1_0·a_3_4
  64. a_1_0·a_13_15 − 2·b_2_2·b_2_34·a_1_1·a_3_5 − 2·b_2_22·b_2_33·a_1_1·a_3_5
       − 2·b_2_23·b_2_32·a_1_1·a_3_5 − b_2_24·b_2_3·a_1_1·a_3_5 − 3·b_2_25·a_1_1·a_3_5
       − b_2_25·a_1_0·a_3_5 + 2·b_2_25·a_1_0·a_3_4
  65. a_8_5·a_7_9
  66. b_12_13·a_3_5 − b_2_2·b_2_35·a_3_5 − b_2_22·b_2_34·a_3_5 − b_2_22·b_2_35·a_1_1
       − b_2_23·b_2_33·a_3_5 + 2·b_2_23·b_2_34·a_1_1 + 3·b_2_24·b_2_32·a_3_5
       + 3·b_2_24·b_2_33·a_1_1 + 3·b_2_25·b_2_3·a_3_5 + b_2_25·b_2_32·a_1_1
       − 2·b_2_26·b_2_3·a_1_1 − 3·b_2_27·a_1_1
  67. b_12_13·a_3_4 − b_2_2·b_2_35·a_3_5 + 3·b_2_22·b_2_34·a_3_5 − b_2_22·b_2_35·a_1_1
       + 3·b_2_23·b_2_33·a_3_5 − 3·b_2_23·b_2_34·a_1_1 + b_2_24·b_2_32·a_3_5
       − 2·b_2_24·b_2_33·a_1_1 − b_2_25·b_2_3·a_3_5 + 2·b_2_26·a_3_5 + 3·b_2_26·a_3_4
       + 3·b_2_26·b_2_3·a_1_1 + 3·b_2_27·a_1_1
  68. b_2_2·a_13_15 − 2·b_2_2·b_2_35·a_3_5 − 2·b_2_22·b_2_34·a_3_5
       + 3·b_2_22·b_2_35·a_1_1 − 2·b_2_23·b_2_33·a_3_5 − b_2_23·b_2_34·a_1_1
       − b_2_24·b_2_32·a_3_5 + b_2_24·b_2_33·a_1_1 − 3·b_2_25·b_2_3·a_3_5
       + 2·b_2_25·b_2_32·a_1_1 − b_2_26·a_3_5 + 2·b_2_26·a_3_4 + 2·b_2_26·b_2_3·a_1_1
       + b_2_27·a_1_1
  69. a_2_0·a_13_15
  70. a_2_1·a_13_15
  71. b_2_3·a_13_15 − b_2_36·a_3_5 + 3·b_2_2·b_2_35·a_3_5 + 2·b_2_22·b_2_34·a_3_5
       − 2·b_2_22·b_2_35·a_1_1 − 2·b_2_23·b_2_33·a_3_5 + 3·b_2_23·b_2_34·a_1_1
       − b_2_24·b_2_32·a_3_5 − 2·b_2_24·b_2_33·a_1_1 + 2·b_2_25·b_2_3·a_3_5
       + 3·b_2_25·b_2_32·a_1_1 + 2·b_2_26·a_3_5 + 3·b_2_26·a_3_4 − 2·b_2_26·b_2_3·a_1_1
       − b_2_27·a_1_1
  72. a_8_52
  73. a_7_9·a_9_11 + 2·b_2_22·b_2_34·a_1_1·a_3_5 + b_2_23·b_2_33·a_1_1·a_3_5
       + 3·b_2_24·b_2_32·a_1_1·a_3_5 − 2·b_2_26·a_1_1·a_3_5 − b_2_26·a_1_0·a_3_5
       + 2·b_2_26·a_1_0·a_3_4
  74. a_3_5·a_13_15 + 2·b_2_22·b_2_34·a_1_1·a_3_5 − 3·b_2_23·b_2_33·a_1_1·a_3_5
       + 2·b_2_24·b_2_32·a_1_1·a_3_5 − 3·b_2_25·b_2_3·a_1_1·a_3_5 + 2·b_2_26·a_1_1·a_3_5
       − b_2_26·a_1_0·a_3_5 + 2·b_2_26·a_1_0·a_3_4
  75. a_3_4·a_13_15 − b_2_22·b_2_34·a_1_1·a_3_5 − 2·b_2_23·b_2_33·a_1_1·a_3_5
       + b_2_24·b_2_32·a_1_1·a_3_5 − 2·b_2_25·b_2_3·a_1_1·a_3_5 − 3·b_2_26·a_1_1·a_3_5
       + 3·b_2_26·a_1_0·a_3_5 + b_2_26·a_1_0·a_3_4
  76. a_8_5·a_9_11
  77. a_10_6·a_7_9
  78. a_8_5·a_10_6
  79. a_7_9·a_11_13 − 3·b_2_23·b_2_34·a_1_1·a_3_5 − b_2_24·b_2_33·a_1_1·a_3_5
       − 3·b_2_26·b_2_3·a_1_1·a_3_5 + b_2_27·a_1_1·a_3_5 + b_2_27·a_1_0·a_3_5
       − 2·b_2_27·a_1_0·a_3_4
  80. a_10_6·a_9_11
  81. a_8_5·a_11_13
  82. b_12_13·a_7_9 − 2·b_2_24·b_2_34·a_3_5 − 3·b_2_24·b_2_35·a_1_1
       − b_2_25·b_2_33·a_3_5 + 2·b_2_25·b_2_34·a_1_1 − 2·b_2_26·b_2_32·a_3_5
       + 3·b_2_26·b_2_33·a_1_1 − 2·b_2_27·b_2_3·a_3_5 + b_2_27·b_2_32·a_1_1
       − 2·b_2_28·a_3_5 − 3·b_2_28·a_3_4 + b_2_28·b_2_3·a_1_1 + b_2_29·a_1_1
  83. a_10_62
  84. a_9_11·a_11_13 − b_2_24·b_2_34·a_1_1·a_3_5 + b_2_25·b_2_33·a_1_1·a_3_5
       − 3·b_2_26·b_2_32·a_1_1·a_3_5 − 3·b_2_27·b_2_3·a_1_1·a_3_5 + 2·b_2_28·a_1_1·a_3_5
       + 3·b_2_28·a_1_0·a_3_5 + b_2_28·a_1_0·a_3_4
  85. a_8_5·b_12_13 + 2·b_2_24·b_2_34·a_1_1·a_3_5 − b_2_25·b_2_33·a_1_1·a_3_5
       − 2·b_2_26·b_2_32·a_1_1·a_3_5 − 2·b_2_27·b_2_3·a_1_1·a_3_5 + b_2_28·a_1_1·a_3_5
  86. a_7_9·a_13_15 − 2·b_2_24·b_2_34·a_1_1·a_3_5 + 2·b_2_26·b_2_32·a_1_1·a_3_5
       + b_2_27·b_2_3·a_1_1·a_3_5 − b_2_28·a_1_0·a_3_5 + 2·b_2_28·a_1_0·a_3_4
  87. a_10_6·a_11_13
  88. b_12_13·a_9_11 + 3·b_2_24·b_2_35·a_3_5 − 3·b_2_25·b_2_34·a_3_5
       − 3·b_2_25·b_2_35·a_1_1 − b_2_26·b_2_34·a_1_1 − 3·b_2_27·b_2_33·a_1_1
       + 3·b_2_28·b_2_3·a_3_5 + 2·b_2_28·b_2_32·a_1_1 + b_2_29·a_3_5 − 2·b_2_29·a_3_4
       + 2·b_2_29·b_2_3·a_1_1 + b_2_210·a_1_1
  89. a_8_5·a_13_15
  90. a_10_6·b_12_13 + 3·b_2_26·b_2_33·a_1_1·a_3_5 + b_2_27·b_2_32·a_1_1·a_3_5
       − b_2_28·b_2_3·a_1_1·a_3_5 − 2·b_2_29·a_1_1·a_3_5 + 2·b_2_29·a_1_0·a_3_5
       + 3·b_2_29·a_1_0·a_3_4
  91. a_9_11·a_13_15 + b_2_25·b_2_34·a_1_1·a_3_5 + 3·b_2_26·b_2_33·a_1_1·a_3_5
       + 3·b_2_28·b_2_3·a_1_1·a_3_5 + 3·b_2_29·a_1_1·a_3_5 − b_2_29·a_1_0·a_3_5
       + 2·b_2_29·a_1_0·a_3_4
  92. b_12_13·a_11_13 + 3·b_2_26·b_2_34·a_3_5 − 2·b_2_26·b_2_35·a_1_1
       − 3·b_2_27·b_2_33·a_3_5 − 3·b_2_27·b_2_34·a_1_1 + 3·b_2_28·b_2_32·a_3_5
       − 2·b_2_28·b_2_33·a_1_1 − b_2_29·b_2_32·a_1_1 − 2·b_2_210·b_2_3·a_1_1
  93. a_10_6·a_13_15
  94. b_12_132 + 2·b_2_26·b_2_36 + 2·b_2_27·b_2_35 + 2·b_2_28·b_2_34
       + 3·b_2_29·b_2_33 − 2·b_2_210·b_2_32 − 2·b_2_211·b_2_3
       + b_2_26·b_2_34·a_1_1·a_3_5 − 3·b_2_27·b_2_33·a_1_1·a_3_5
       + 3·b_2_28·b_2_32·a_1_1·a_3_5 − 2·b_2_29·b_2_3·a_1_1·a_3_5
       + 3·b_2_210·a_1_1·a_3_5 − b_2_210·a_1_0·a_3_5 + 2·b_2_210·a_1_0·a_3_4
  95. a_11_13·a_13_15 − b_2_26·b_2_34·a_1_1·a_3_5 − b_2_27·b_2_33·a_1_1·a_3_5
       − b_2_28·b_2_32·a_1_1·a_3_5 + b_2_29·b_2_3·a_1_1·a_3_5 + b_2_210·a_1_1·a_3_5
       − 2·b_2_210·a_1_0·a_3_5 − 3·b_2_210·a_1_0·a_3_4
  96. b_12_13·a_13_15 + b_2_27·b_2_34·a_3_5 − 2·b_2_27·b_2_35·a_1_1
       − b_2_29·b_2_32·a_3_5 + b_2_29·b_2_33·a_1_1 + 2·b_2_210·b_2_3·a_3_5
       + 3·b_2_210·b_2_32·a_1_1 + 3·b_2_211·a_3_5 + b_2_211·a_3_4 + b_2_211·b_2_3·a_1_1
       + b_2_212·a_1_1

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 25.
  • 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_14_16, a Duflot regular element of degree 14
    2.  − b_2_36·b_12_136 + b_2_342 + b_2_2·b_2_35·b_12_136
         − b_2_22·b_2_34·b_12_136 + b_2_23·b_2_33·b_12_136
         − b_2_24·b_2_32·b_12_136 + b_2_25·b_2_3·b_12_136 − b_2_26·b_12_136
         + b_2_26·b_2_36·b_12_135 − b_2_27·b_2_35·b_12_135 − b_2_27·b_2_335
         + b_2_28·b_2_34·b_12_135 − b_2_29·b_2_33·b_12_135
         + b_2_210·b_2_32·b_12_135 − b_2_211·b_2_3·b_12_135 + b_2_212·b_12_135
         − b_2_212·b_2_36·b_12_134 + b_2_213·b_2_35·b_12_134
         − b_2_214·b_2_34·b_12_134 + b_2_214·b_2_328 + b_2_215·b_2_33·b_12_134
         − b_2_216·b_2_32·b_12_134 + b_2_217·b_2_3·b_12_134 − b_2_218·b_12_134
         + b_2_218·b_2_36·b_12_133 − b_2_219·b_2_35·b_12_133
         + b_2_220·b_2_34·b_12_133 − b_2_221·b_2_33·b_12_133 − b_2_221·b_2_321
         + b_2_222·b_2_32·b_12_133 − b_2_223·b_2_3·b_12_133 + b_2_224·b_12_133
         − b_2_224·b_2_36·b_12_132 + b_2_225·b_2_35·b_12_132
         − b_2_226·b_2_34·b_12_132 + b_2_227·b_2_33·b_12_132
         − b_2_228·b_2_32·b_12_132 + b_2_228·b_2_314 + b_2_229·b_2_3·b_12_132
         − b_2_230·b_12_132 + b_2_230·b_2_36·b_12_13 − b_2_231·b_2_35·b_12_13
         + b_2_232·b_2_34·b_12_13 − b_2_233·b_2_33·b_12_13 + b_2_234·b_2_32·b_12_13
         − b_2_235·b_2_3·b_12_13 − b_2_235·b_2_37 + b_2_236·b_12_13 − b_2_236·b_2_36
         + b_2_237·b_2_35 − b_2_238·b_2_34 + b_2_239·b_2_33 − b_2_240·b_2_32
         + b_2_241·b_2_3 + b_2_242, an element of degree 84
  • The Raw Filter Degree Type of that HSOP is [-1, 10, 96].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 2.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. a_3_5 → 0, an element of degree 3
  9. a_7_9 → 0, an element of degree 7
  10. a_8_5 → 0, an element of degree 8
  11. a_9_11 → 0, an element of degree 9
  12. a_10_6 → 0, an element of degree 10
  13. a_11_13 → 0, an element of degree 11
  14. b_12_13 → 0, an element of degree 12
  15. a_13_15 → 0, an element of degree 13
  16. c_14_16 →  − c_2_07, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 →  − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
  9. a_7_9 →  − 3·c_2_23·a_1_0 + 3·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 →  − 3·c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 → 3·c_2_24·a_1_0 − 3·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 → 2·c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 → 3·c_2_25·a_1_0 − 3·c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 →  − 3·c_2_25·a_1_0·a_1_1, an element of degree 12
  15. a_13_15 → 0, an element of degree 13
  16. c_14_16 → c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 → 0, an element of degree 3
  8. a_3_5 →  − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  9. a_7_9 →  − c_2_23·a_1_0 + c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 →  − c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 →  − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 →  − c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 →  − c_2_25·a_1_0 + c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 →  − 3·c_2_25·a_1_0·a_1_1, an element of degree 12
  15. a_13_15 →  − c_2_26·a_1_0 + c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 →  − 2·c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 2·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 → 3·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → 3·c_2_2·a_1_1 − 3·c_2_2·a_1_0 + 3·c_2_1·a_1_1, an element of degree 3
  9. a_7_9 →  − c_2_23·a_1_1 + 3·c_2_23·a_1_0 − 3·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 →  − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 →  − c_2_24·a_1_1 − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 →  − 3·c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 → 2·c_2_25·a_1_1 − 3·c_2_25·a_1_0 + 3·c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 → 2·c_2_25·a_1_0·a_1_1 − 3·c_2_26, an element of degree 12
  15. a_13_15 →  − 2·c_2_26·a_1_1 − 3·c_2_26·a_1_0 + 3·c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 → 2·c_2_26·a_1_0·a_1_1 − c_2_27 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → 2·a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 →  − 2·a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 3·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → 2·c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_1 − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
  8. a_3_5 →  − c_2_2·a_1_1 + 2·c_2_2·a_1_0 − 2·c_2_1·a_1_1, an element of degree 3
  9. a_7_9 → c_2_23·a_1_1 + 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 → c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 →  − 3·c_2_24·a_1_1 + c_2_24·a_1_0 − c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 →  − 2·c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 → 3·c_2_25·a_1_1 + c_2_25·a_1_0 − c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 → 2·c_2_25·a_1_0·a_1_1 + 3·c_2_26, an element of degree 12
  15. a_13_15 → 2·c_2_26·a_1_1 + 3·c_2_26·a_1_0 − 3·c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 → 3·c_2_26·a_1_0·a_1_1 + 3·c_2_27 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 2·a_1_1, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 3·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → 2·c_2_2, an element of degree 2
  7. a_3_4 →  − c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 →  − c_2_2·a_1_1 + 3·c_2_2·a_1_0 − 3·c_2_1·a_1_1, an element of degree 3
  9. a_7_9 → 3·c_2_23·a_1_1 − 3·c_2_23·a_1_0 + 3·c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 →  − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 →  − 3·c_2_24·a_1_1 − 2·c_2_24·a_1_0 + 2·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 → 0, an element of degree 10
  13. a_11_13 →  − c_2_25·a_1_0 + c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 → c_2_25·a_1_0·a_1_1 + 3·c_2_26, an element of degree 12
  15. a_13_15 →  − c_2_26·a_1_1, an element of degree 13
  16. c_14_16 →  − c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → 3·a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 →  − 3·a_1_0·a_1_1, an element of degree 2
  4. a_2_1 →  − 3·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → 3·c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 → 2·c_2_2·a_1_1 − 3·c_2_2·a_1_0 + 3·c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → 2·c_2_2·a_1_1, an element of degree 3
  9. a_7_9 → 2·c_2_23·a_1_1 − c_2_23·a_1_0 + c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 → 0, an element of degree 8
  11. a_9_11 → c_2_24·a_1_1 − 2·c_2_24·a_1_0 + 2·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 →  − 2·c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 → 2·c_2_25·a_1_1, an element of degree 11
  14. b_12_13 →  − 2·c_2_25·a_1_0·a_1_1 − 3·c_2_26, an element of degree 12
  15. a_13_15 →  − 3·c_2_26·a_1_1 + 2·c_2_26·a_1_0 − 2·c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 → c_2_26·a_1_0·a_1_1 + 2·c_2_27 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → a_1_1, an element of degree 1
  2. a_1_1 → 3·a_1_1, an element of degree 1
  3. a_2_0 →  − a_1_0·a_1_1, an element of degree 2
  4. a_2_1 →  − 3·a_1_0·a_1_1, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. b_2_3 → 3·c_2_2, an element of degree 2
  7. a_3_4 → 2·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
  8. a_3_5 → 2·c_2_2·a_1_1 + 2·c_2_2·a_1_0 − 2·c_2_1·a_1_1, an element of degree 3
  9. a_7_9 → 3·c_2_23·a_1_1 + c_2_23·a_1_0 − c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 → 0, an element of degree 8
  11. a_9_11 →  − 3·c_2_24·a_1_1 − 3·c_2_24·a_1_0 + 3·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 → 3·c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 →  − 2·c_2_25·a_1_1 + c_2_25·a_1_0 − c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 →  − 2·c_2_25·a_1_0·a_1_1, an element of degree 12
  15. a_13_15 → 3·c_2_26·a_1_1 + 2·c_2_26·a_1_0 − 2·c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 →  − 2·c_2_26·a_1_0·a_1_1 − 2·c_2_27 + c_2_1·c_2_26 − c_2_17, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 →  − a_1_1, an element of degree 1
  2. a_1_1 → a_1_1, an element of degree 1
  3. a_2_0 → a_1_0·a_1_1, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 →  − c_2_2, an element of degree 2
  6. b_2_3 → c_2_2, an element of degree 2
  7. a_3_4 →  − 3·c_2_2·a_1_1 + c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
  8. a_3_5 →  − 3·c_2_2·a_1_1 + c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
  9. a_7_9 → c_2_23·a_1_1 + c_2_23·a_1_0 − c_2_1·c_2_22·a_1_1, an element of degree 7
  10. a_8_5 → 3·c_2_23·a_1_0·a_1_1, an element of degree 8
  11. a_9_11 →  − 3·c_2_24·a_1_1 + 3·c_2_24·a_1_0 − 3·c_2_1·c_2_23·a_1_1, an element of degree 9
  12. a_10_6 →  − c_2_24·a_1_0·a_1_1, an element of degree 10
  13. a_11_13 →  − c_2_25·a_1_1 + 2·c_2_25·a_1_0 − 2·c_2_1·c_2_24·a_1_1, an element of degree 11
  14. b_12_13 →  − 2·c_2_25·a_1_0·a_1_1 − c_2_26, an element of degree 12
  15. a_13_15 → 3·c_2_26·a_1_1 − 2·c_2_26·a_1_0 + 2·c_2_1·c_2_25·a_1_1, an element of degree 13
  16. c_14_16 →  − 2·c_2_26·a_1_0·a_1_1 − 2·c_2_27 + c_2_1·c_2_26 − c_2_17, an element of degree 14

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343


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