Cohomology of group number 16 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 2 minimal generators and exponent 27.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t4  −  t3  +  t2  −  t  +  1)
    (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-3,-3,-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 17 minimal generators of maximal degree 8:

  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. a_3_2, a nilpotent element of degree 3
  7. a_3_3, a nilpotent element of degree 3
  8. a_4_3, a nilpotent element of degree 4
  9. b_4_4, an element of degree 4
  10. a_5_5, a nilpotent element of degree 5
  11. a_5_6, a nilpotent element of degree 5
  12. a_6_7, a nilpotent element of degree 6
  13. b_6_8, an element of degree 6
  14. c_6_9, a Duflot regular element of degree 6
  15. a_7_11, a nilpotent element of degree 7
  16. a_7_12, a nilpotent element of degree 7
  17. a_8_14, a nilpotent element of degree 8

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

Ring relations

There are 8 "obvious" relations:
   a_1_02, a_1_12, a_3_22, a_3_32, a_5_52, a_5_62, a_7_112, a_7_122

Apart from that, there are 95 minimal relations of maximal degree 16:

  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_2·a_1_0
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9. a_2_0·b_2_2
  10.  − a_2_1·b_2_2 + a_1_1·a_3_2
  11. a_1_0·a_3_2
  12. a_1_0·a_3_3
  13. a_2_0·a_3_2
  14. a_2_1·a_3_2
  15. a_2_0·a_3_3
  16. a_4_3·a_1_1 − a_2_1·a_3_3
  17. a_4_3·a_1_0
  18. b_4_4·a_1_0
  19. b_2_2·a_4_3 − a_3_2·a_3_3 − b_2_2·a_1_1·a_3_3
  20. a_2_0·a_4_3
  21. a_2_1·a_4_3
  22. a_2_0·b_4_4
  23.  − a_2_1·b_4_4 + a_1_1·a_5_5 − b_2_2·a_1_1·a_3_3
  24. a_1_0·a_5_5
  25. a_1_0·a_5_6
  26. a_4_3·a_3_3
  27. a_4_3·a_3_2 + a_1_1·a_3_2·a_3_3
  28. a_2_0·a_5_5
  29. a_2_1·a_5_5 − a_1_1·a_3_2·a_3_3
  30.  − b_4_4·a_3_3 − b_4_4·a_3_2 + b_2_2·a_5_6 − b_2_22·a_3_2 − a_1_1·a_3_2·a_3_3
  31. a_2_0·a_5_6
  32. a_6_7·a_1_1 − a_2_1·a_5_6
  33. a_6_7·a_1_0
  34. b_6_8·a_1_1 + b_4_4·a_3_2 − b_2_2·a_5_5 + b_2_22·a_3_3 − a_2_1·a_5_6 + a_1_1·a_3_2·a_3_3
  35. b_6_8·a_1_0
  36. a_4_32
  37. a_3_3·a_5_6 + a_3_2·a_5_6 + b_2_2·a_3_2·a_3_3
  38.  − a_4_3·b_4_4 − a_3_3·a_5_6 − b_2_2·a_3_2·a_3_3 + b_2_2·a_1_1·a_5_6 − b_2_2·a_1_1·a_5_5
       + b_2_22·a_1_1·a_3_3 − b_2_22·a_1_1·a_3_2
  39. b_2_2·a_6_7 + a_3_3·a_5_5 + a_3_2·a_5_5 − b_2_2·a_3_2·a_3_3
  40. a_2_0·a_6_7
  41. a_2_1·a_6_7
  42. a_2_0·b_6_8
  43. a_2_1·b_6_8 + a_3_2·a_5_5 − b_2_2·a_3_2·a_3_3
  44. a_3_2·a_5_5 + a_1_1·a_7_11 − b_2_2·a_3_2·a_3_3
  45. a_1_0·a_7_11
  46.  − a_3_3·a_5_6 + a_3_3·a_5_5 + a_1_1·a_7_12 − b_2_2·a_3_2·a_3_3
  47. a_1_0·a_7_12
  48. a_4_3·a_5_6 + a_1_1·a_3_2·a_5_6 + b_2_2·a_1_1·a_3_2·a_3_3
  49. a_6_7·a_3_3 + a_4_3·a_5_6 − a_4_3·a_5_5 + b_2_2·a_1_1·a_3_2·a_3_3
  50. a_6_7·a_3_2 − a_4_3·a_5_6 + a_4_3·a_5_5 − b_2_2·a_1_1·a_3_2·a_3_3
  51.  − b_6_8·a_3_2 + b_4_42·a_1_1 + b_2_2·a_7_11 + b_2_2·b_4_4·a_3_2 − b_2_22·a_5_5
       + b_2_22·b_4_4·a_1_1 + b_2_23·a_3_3 − a_4_3·a_5_5
  52. a_2_0·a_7_11
  53. a_2_1·a_7_11
  54.  − b_6_8·a_3_3 − b_4_42·a_1_1 + b_2_2·a_7_12 − b_2_2·b_4_4·a_3_2 + b_2_22·a_5_5
       − b_2_22·b_4_4·a_1_1 − b_2_23·a_3_3 + a_4_3·a_5_6 + a_4_3·a_5_5
       + b_2_2·a_1_1·a_3_2·a_3_3
  55. a_2_0·a_7_12
  56. a_4_3·a_5_6 − a_4_3·a_5_5 + a_2_1·a_7_12 + b_2_2·a_1_1·a_3_2·a_3_3
  57. a_8_14·a_1_1 + a_4_3·a_5_6 − a_4_3·a_5_5 + b_2_2·a_1_1·a_3_2·a_3_3
  58. a_8_14·a_1_0
  59. a_4_3·a_6_7
  60. a_4_3·b_6_8 + a_3_3·a_7_11 − b_4_4·a_1_1·a_5_6 + b_4_4·a_1_1·a_5_5 + b_2_22·a_1_1·a_5_6
       − b_2_23·a_1_1·a_3_2
  61.  − b_4_4·a_6_7 + a_5_5·a_5_6 + a_3_2·a_7_11 − b_4_4·a_1_1·a_5_5 + b_2_2·a_3_2·a_5_6
       + b_2_22·a_3_2·a_3_3 + b_2_22·a_1_1·a_5_6 + b_2_22·a_1_1·a_5_5 − b_2_23·a_1_1·a_3_3
       − b_2_23·a_1_1·a_3_2
  62. b_4_4·a_6_7 − a_5_5·a_5_6 − b_2_2·a_3_2·a_5_6 + b_2_2·a_1_1·a_7_11 − b_2_22·a_3_2·a_3_3
  63. b_4_4·a_6_7 − a_4_3·b_6_8 − a_5_5·a_5_6 + a_3_3·a_7_12 + a_3_2·a_7_12 + b_4_4·a_1_1·a_5_6
       − b_2_2·a_3_2·a_5_6 − b_2_22·a_3_2·a_3_3 + b_2_22·a_1_1·a_5_6 − b_2_22·a_1_1·a_5_5
       + b_2_23·a_1_1·a_3_3 − b_2_23·a_1_1·a_3_2
  64.  − a_3_3·a_7_12 − b_4_4·a_1_1·a_5_6 + b_4_4·a_1_1·a_5_5 + b_2_2·a_1_1·a_7_12
       + b_2_22·a_1_1·a_5_6 − b_2_23·a_1_1·a_3_2
  65.  − a_4_3·b_6_8 + b_2_2·a_8_14 − a_3_3·a_7_12 + b_4_4·a_1_1·a_5_5 − b_2_22·a_1_1·a_5_5
       − b_2_23·a_1_1·a_3_3 − b_2_23·a_1_1·a_3_2
  66. a_2_0·a_8_14
  67. a_2_1·a_8_14
  68. a_6_7·a_5_6 + a_6_7·a_5_5
  69. a_4_3·a_7_11 + a_1_1·a_5_5·a_5_6 − b_2_2·a_1_1·a_3_2·a_5_6 + b_2_22·a_1_1·a_3_2·a_3_3
  70. a_6_7·a_5_5 + a_4_3·a_7_12 − a_1_1·a_5_5·a_5_6 + b_2_2·a_1_1·a_3_2·a_5_6
       − b_2_22·a_1_1·a_3_2·a_3_3
  71.  − a_6_7·a_5_5 + a_1_1·a_3_2·a_7_12
  72.  − b_6_8·a_5_5 + b_4_4·a_7_11 + b_4_42·a_3_2 − b_2_2·b_4_4·a_5_5 + b_2_2·b_4_42·a_1_1
       + b_2_22·a_7_12 + b_2_23·a_5_6 − b_2_23·a_5_5 + b_2_23·b_4_4·a_1_1 + b_2_24·a_3_3
       − b_2_24·a_3_2 − a_6_7·a_5_5 + a_1_1·a_5_5·a_5_6 − b_2_2·a_1_1·a_3_2·a_5_6
       + b_2_22·a_1_1·a_3_2·a_3_3 − b_2_22·c_6_9·a_1_1
  73.  − b_6_8·a_5_6 + b_4_4·a_7_12 + b_4_4·a_7_11 + b_2_2·b_4_42·a_1_1 + b_2_22·a_7_11
       + b_2_22·b_4_4·a_3_2 − b_2_23·a_5_5 + b_2_23·b_4_4·a_1_1 + b_2_24·a_3_3
       − a_1_1·a_5_5·a_5_6 − b_2_22·a_1_1·a_3_2·a_3_3
  74. a_8_14·a_3_3 + a_1_1·a_5_5·a_5_6
  75. a_8_14·a_3_2 − a_6_7·a_5_5 − a_1_1·a_5_5·a_5_6 + b_2_22·a_1_1·a_3_2·a_3_3
  76. a_6_72
  77.  − b_6_82 − b_4_43 + b_2_22·b_4_42 + b_2_23·b_6_8 − b_2_24·b_4_4 − a_6_7·b_6_8
       + a_5_6·a_7_11 + a_5_5·a_7_12 − b_4_4·a_1_1·a_7_11 − b_2_2·b_4_4·a_1_1·a_5_6
       − b_2_22·a_3_2·a_5_6 − b_2_22·a_1_1·a_7_11 − b_2_23·a_3_2·a_3_3 − b_2_23·c_6_9
       + b_2_2·c_6_9·a_1_1·a_3_3
  78. b_6_82 + b_4_43 − b_2_22·b_4_42 − b_2_23·b_6_8 + b_2_24·b_4_4 + a_5_6·a_7_12
       + a_5_5·a_7_11 + b_4_4·a_1_1·a_7_11 + b_2_2·b_4_4·a_1_1·a_5_6 + b_2_22·a_3_2·a_5_6
       + b_2_22·a_1_1·a_7_11 + b_2_23·a_3_2·a_3_3 + b_2_23·c_6_9 − b_2_2·c_6_9·a_1_1·a_3_3
  79. b_6_82 + b_4_43 − b_2_22·b_4_42 − b_2_23·b_6_8 + b_2_24·b_4_4 − a_5_6·a_7_11
       + a_5_5·a_7_11 + b_4_4·a_1_1·a_7_11 + b_2_2·a_3_2·a_7_12 + b_2_2·b_4_4·a_1_1·a_5_6
       + b_2_2·b_4_4·a_1_1·a_5_5 + b_2_22·a_3_2·a_5_6 + b_2_23·a_3_2·a_3_3
       − b_2_23·a_1_1·a_5_6 − b_2_23·a_1_1·a_5_5 + b_2_24·a_1_1·a_3_3 + b_2_24·a_1_1·a_3_2
       + b_2_23·c_6_9 − b_2_2·c_6_9·a_1_1·a_3_3
  80.  − b_6_82 − b_4_43 + b_2_22·b_4_42 + b_2_23·b_6_8 − b_2_24·b_4_4 + a_5_6·a_7_11
       + b_2_2·b_4_4·a_1_1·a_5_6 − b_2_2·b_4_4·a_1_1·a_5_5 − b_2_22·a_3_2·a_5_6
       + b_2_22·a_1_1·a_7_12 + b_2_22·a_1_1·a_7_11 − b_2_23·a_3_2·a_3_3
       − b_2_23·a_1_1·a_5_5 + b_2_24·a_1_1·a_3_3 − b_2_23·c_6_9 + b_2_2·c_6_9·a_1_1·a_3_3
       + b_2_2·c_6_9·a_1_1·a_3_2
  81.  − b_6_82 − b_4_43 + b_2_22·b_4_42 + b_2_23·b_6_8 − b_2_24·b_4_4 + a_6_7·b_6_8
       − a_5_6·a_7_11 + b_4_4·a_1_1·a_7_12 + b_4_4·a_1_1·a_7_11 − b_2_2·b_4_4·a_1_1·a_5_6
       + b_2_2·b_4_4·a_1_1·a_5_5 − b_2_22·a_3_2·a_5_6 + b_2_22·a_1_1·a_7_11
       − b_2_23·a_3_2·a_3_3 − b_2_23·a_1_1·a_5_6 − b_2_23·a_1_1·a_5_5 + b_2_24·a_1_1·a_3_3
       + b_2_24·a_1_1·a_3_2 − b_2_23·c_6_9 − b_2_2·c_6_9·a_1_1·a_3_3
       − b_2_2·c_6_9·a_1_1·a_3_2
  82. a_4_3·a_8_14
  83.  − b_6_82 − b_4_43 + b_2_22·b_4_42 + b_2_23·b_6_8 − b_2_24·b_4_4 + a_6_7·b_6_8
       + b_4_4·a_8_14 − b_4_4·a_1_1·a_7_11 − b_2_22·a_3_2·a_5_6 − b_2_22·a_1_1·a_7_11
       − b_2_23·a_3_2·a_3_3 + b_2_23·a_1_1·a_5_6 − b_2_24·a_1_1·a_3_2 − b_2_23·c_6_9
       − b_2_2·c_6_9·a_1_1·a_3_3 − b_2_2·c_6_9·a_1_1·a_3_2
  84.  − b_6_8·a_7_11 − b_4_42·a_5_5 − b_4_43·a_1_1 + b_2_2·b_4_42·a_3_2
       + b_2_22·b_4_4·a_5_6 − b_2_22·b_4_4·a_5_5 − b_2_22·b_4_42·a_1_1 + b_2_23·a_7_11
       + b_2_24·a_5_6 − b_2_25·a_3_2 + a_6_7·a_7_11 + b_2_2·a_1_1·a_5_5·a_5_6
       + b_2_2·a_1_1·a_3_2·a_7_12 + b_2_22·a_1_1·a_3_2·a_5_6 + b_2_23·a_1_1·a_3_2·a_3_3
       − b_2_22·c_6_9·a_3_2 − b_2_23·c_6_9·a_1_1
  85.  − b_6_8·a_7_12 − b_6_8·a_7_11 − b_4_42·a_5_6 + b_2_2·b_4_42·a_3_2 + b_2_22·b_4_4·a_5_6
       + b_2_23·a_7_12 + b_2_23·a_7_11 − b_2_23·b_4_4·a_3_2 − b_2_24·a_5_6 + b_2_25·a_3_2
       + a_6_7·a_7_11 − b_2_22·a_1_1·a_3_2·a_5_6 + b_2_23·a_1_1·a_3_2·a_3_3
       − b_2_22·c_6_9·a_3_3 − b_2_22·c_6_9·a_3_2 − c_6_9·a_1_1·a_3_2·a_3_3
  86.  − b_6_8·a_7_11 − b_4_42·a_5_5 − b_4_43·a_1_1 + b_2_2·b_4_42·a_3_2
       + b_2_22·b_4_4·a_5_6 − b_2_22·b_4_4·a_5_5 − b_2_22·b_4_42·a_1_1 + b_2_23·a_7_11
       + b_2_24·a_5_6 − b_2_25·a_3_2 + a_1_1·a_5_5·a_7_12 − b_2_2·a_1_1·a_5_5·a_5_6
       − b_2_23·a_1_1·a_3_2·a_3_3 − b_2_22·c_6_9·a_3_2 − b_2_23·c_6_9·a_1_1
       + c_6_9·a_1_1·a_3_2·a_3_3
  87. a_6_7·a_7_12 + a_6_7·a_7_11
  88. b_6_8·a_7_11 + b_4_42·a_5_5 + b_4_43·a_1_1 − b_2_2·b_4_42·a_3_2 − b_2_22·b_4_4·a_5_6
       + b_2_22·b_4_4·a_5_5 + b_2_22·b_4_42·a_1_1 − b_2_23·a_7_11 − b_2_24·a_5_6
       + b_2_25·a_3_2 + a_8_14·a_5_5 + a_6_7·a_7_11 + b_2_2·a_1_1·a_5_5·a_5_6
       + b_2_22·a_1_1·a_3_2·a_5_6 + b_2_22·c_6_9·a_3_2 + b_2_23·c_6_9·a_1_1
       − c_6_9·a_1_1·a_3_2·a_3_3
  89. b_6_8·a_7_11 + b_4_42·a_5_5 + b_4_43·a_1_1 − b_2_2·b_4_42·a_3_2 − b_2_22·b_4_4·a_5_6
       + b_2_22·b_4_4·a_5_5 + b_2_22·b_4_42·a_1_1 − b_2_23·a_7_11 − b_2_24·a_5_6
       + b_2_25·a_3_2 + a_8_14·a_5_6 + a_6_7·a_7_11 − b_2_2·a_1_1·a_5_5·a_5_6
       − b_2_22·a_1_1·a_3_2·a_5_6 + b_2_23·a_1_1·a_3_2·a_3_3 + b_2_22·c_6_9·a_3_2
       + b_2_23·c_6_9·a_1_1 + c_6_9·a_1_1·a_3_2·a_3_3
  90. a_7_11·a_7_12 + b_4_4·a_5_5·a_5_6 + b_4_42·a_1_1·a_5_6 + b_2_2·b_4_4·a_1_1·a_7_12
       − b_2_22·a_3_2·a_7_12 − b_2_22·b_4_4·a_1_1·a_5_6 + b_2_22·b_4_4·a_1_1·a_5_5
       − b_2_23·a_3_2·a_5_6 − b_2_23·a_1_1·a_7_12 + b_2_24·a_1_1·a_5_5
       − b_2_25·a_1_1·a_3_3 + b_2_2·c_6_9·a_3_2·a_3_3 + b_2_22·c_6_9·a_1_1·a_3_3
       + b_2_22·c_6_9·a_1_1·a_3_2
  91. b_6_8·a_8_14 + a_7_11·a_7_12 − b_4_4·a_5_5·a_5_6 + b_4_42·a_1_1·a_5_5
       + b_2_2·b_4_4·a_1_1·a_7_11 + b_2_22·a_3_2·a_7_12 − b_2_22·b_4_4·a_1_1·a_5_6
       + b_2_22·b_4_4·a_1_1·a_5_5 + b_2_23·a_3_2·a_5_6 + b_2_23·a_1_1·a_7_12
       − b_2_24·a_1_1·a_5_6 + b_2_24·a_1_1·a_5_5 − b_2_25·a_1_1·a_3_3 + b_2_25·a_1_1·a_3_2
       − b_2_2·c_6_9·a_3_2·a_3_3 + b_2_22·c_6_9·a_1_1·a_3_2
  92. a_6_7·a_8_14
  93. a_8_14·a_7_12 − b_4_4·a_1_1·a_5_5·a_5_6 + b_2_2·a_1_1·a_5_5·a_7_12
       + b_2_23·a_1_1·a_3_2·a_5_6 − b_2_2·c_6_9·a_1_1·a_3_2·a_3_3
  94. a_8_14·a_7_11 − b_4_4·a_1_1·a_5_5·a_5_6 − b_2_2·a_1_1·a_5_5·a_7_12
       + b_2_23·a_1_1·a_3_2·a_5_6 − b_2_2·c_6_9·a_1_1·a_3_2·a_3_3
  95. a_8_142

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 16.
  • 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_9, a Duflot regular element of degree 6
    2. b_4_43 − b_2_22·b_4_42 − b_2_23·b_6_8 + b_2_24·b_4_4 − b_2_26 + b_2_23·c_6_9, an element of degree 12
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 15, 17].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 4.

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 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. a_3_2 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_4_3 → 0, an element of degree 4
  9. b_4_4 → 0, an element of degree 4
  10. a_5_5 → 0, an element of degree 5
  11. a_5_6 → 0, an element of degree 5
  12. a_6_7 → 0, an element of degree 6
  13. b_6_8 → 0, an element of degree 6
  14. c_6_9 → c_2_03, an element of degree 6
  15. a_7_11 → 0, an element of degree 7
  16. a_7_12 → 0, an element of degree 7
  17. a_8_14 → 0, an element of degree 8

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 → a_1_1, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 →  − a_1_1·a_1_2, an element of degree 2
  5. b_2_2 → c_2_4, an element of degree 2
  6. a_3_2 → c_2_5·a_1_1 − c_2_4·a_1_2, an element of degree 3
  7. a_3_3 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  8. a_4_3 → c_2_4·a_1_1·a_1_2, an element of degree 4
  9. b_4_4 → c_2_4·a_1_1·a_1_2 − c_2_52 + c_2_4·c_2_5 − c_2_3·c_2_4, an element of degree 4
  10. a_5_5 → c_2_52·a_1_2 + c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2 + c_2_4·c_2_5·a_1_1 + c_2_42·a_1_2
       − c_2_3·c_2_5·a_1_1 + c_2_3·c_2_4·a_1_2, an element of degree 5
  11. a_5_6 → c_2_4·c_2_5·a_1_1 − c_2_42·a_1_2, an element of degree 5
  12. a_6_7 → 0, an element of degree 6
  13. b_6_8 →  − c_2_42·a_1_1·a_1_2 + c_2_53 − c_2_42·c_2_5, an element of degree 6
  14. c_6_9 → c_2_42·a_1_1·a_1_2 + c_2_53 + c_2_4·c_2_52 + c_2_42·c_2_5 − c_2_3·c_2_52
       + c_2_3·c_2_4·c_2_5 + c_2_3·c_2_42 + c_2_32·c_2_4 + c_2_33, an element of degree 6
  15. a_7_11 →  − c_2_53·a_1_2 − c_2_4·c_2_52·a_1_1 + c_2_42·c_2_5·a_1_2 + c_2_42·c_2_5·a_1_1
       + c_2_3·c_2_52·a_1_1 − c_2_3·c_2_4·c_2_5·a_1_1 + c_2_3·c_2_42·a_1_1
       − c_2_32·c_2_4·a_1_1, an element of degree 7
  16. a_7_12 → c_2_53·a_1_2 + c_2_4·c_2_52·a_1_1 − c_2_42·c_2_5·a_1_2 − c_2_42·c_2_5·a_1_1
       − c_2_3·c_2_52·a_1_1 + c_2_3·c_2_4·c_2_5·a_1_1 − c_2_3·c_2_42·a_1_1
       + c_2_32·c_2_4·a_1_1, an element of degree 7
  17. a_8_14 →  − c_2_53·a_1_1·a_1_2 + c_2_42·c_2_5·a_1_1·a_1_2 + c_2_43·a_1_1·a_1_2, an element of degree 8

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