Cohomology of group number 51 of order 243

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General information on the group

  • The group is also known as 81gp7xC3, the Direct product 81gp7 x C_3.
  • The group has 3 minimal generators and exponent 9.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    1
    (t  −  1)4 · (t2  +  t  +  1)
  • The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 18 minimal generators of maximal degree 7:

  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. a_2_3, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_4, an element of degree 2
  7. c_2_5, a Duflot regular element of degree 2
  8. a_3_8, a nilpotent element of degree 3
  9. a_3_9, a nilpotent element of degree 3
  10. a_3_10, a nilpotent element of degree 3
  11. a_4_16, a nilpotent element of degree 4
  12. b_4_17, an element of degree 4
  13. a_5_25, a nilpotent element of degree 5
  14. a_5_26, a nilpotent element of degree 5
  15. a_6_36, a nilpotent element of degree 6
  16. b_6_37, an element of degree 6
  17. c_6_38, a Duflot regular element of degree 6
  18. a_7_53, a nilpotent element of degree 7

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Ring relations

There are 9 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_82, a_3_92, a_3_102, a_5_252, a_5_262, a_7_532

Apart from that, there are 80 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. a_2_3·a_1_1
  3. a_2_3·a_1_0
  4. b_2_2·a_1_1
  5. b_2_4·a_1_0
  6. a_2_32
  7. a_2_3·b_2_2
  8. b_2_2·b_2_4
  9.  − a_2_3·b_2_4 + a_1_1·a_3_8
  10. a_1_0·a_3_8
  11. a_1_1·a_3_9
  12. a_1_0·a_3_10
  13. b_2_2·a_3_8
  14. a_2_3·a_3_8
  15. a_2_3·a_3_9
  16. b_2_2·a_3_10
  17.  − b_2_4·a_3_9 + a_2_3·a_3_10
  18.  − b_2_4·a_3_9 + a_4_16·a_1_1
  19. a_4_16·a_1_0
  20. b_4_17·a_1_0
  21. a_3_8·a_3_9
  22. a_3_9·a_3_10
  23. b_2_2·a_4_16
  24. a_2_3·a_4_16
  25. b_2_2·b_4_17
  26.  − b_2_4·a_4_16 + a_2_3·b_4_17 + a_3_8·a_3_10
  27.  − b_2_4·a_4_16 + a_3_8·a_3_10 + a_1_1·a_5_25
  28. a_1_0·a_5_25
  29. a_1_0·a_5_26
  30. a_4_16·a_3_9
  31. a_4_16·a_3_8
  32. b_4_17·a_3_9 − a_4_16·a_3_10
  33. b_2_2·a_5_25
  34. a_2_3·a_5_25
  35. b_2_2·a_5_26
  36.  − a_4_16·a_3_10 + a_2_3·a_5_26
  37. a_6_36·a_1_1 − a_4_16·a_3_10
  38. a_6_36·a_1_0
  39. b_6_37·a_1_1 + b_4_17·a_3_8 − b_2_4·a_5_25
  40. b_6_37·a_1_0
  41. a_4_162
  42. a_3_9·a_5_25
  43. a_3_9·a_5_26
  44.  − a_4_16·b_4_17 + a_3_8·a_5_26
  45. a_3_8·a_5_25 − b_4_17·a_1_1·a_3_10 + b_2_4·a_1_1·a_5_26 + b_2_4·a_1_1·a_5_25
  46. b_2_2·a_6_36
  47. b_2_4·a_6_36 + a_3_10·a_5_25 + a_3_8·a_5_25
  48. a_2_3·a_6_36
  49. b_2_2·b_6_37
  50. a_2_3·b_6_37 + a_3_8·a_5_25
  51. a_4_16·b_4_17 + a_3_10·a_5_25 + a_3_8·a_5_25 + a_1_1·a_7_53
  52. a_1_0·a_7_53
  53. a_4_16·a_5_26
  54. a_4_16·a_5_25 + a_1_1·a_3_10·a_5_26 − a_1_1·a_3_8·a_5_26
  55. a_6_36·a_3_9
  56. a_6_36·a_3_8 + a_4_16·a_5_25
  57. a_6_36·a_3_10 − a_4_16·a_5_25
  58. b_6_37·a_3_9 − a_4_16·a_5_25
  59. b_6_37·a_3_8 − b_4_172·a_1_1 + b_2_4·b_4_17·a_3_10 − b_2_4·b_4_17·a_3_8
       − b_2_42·a_5_26
  60. b_2_2·a_7_53
  61.  − b_6_37·a_3_10 + b_4_17·a_5_25 − b_4_172·a_1_1 + b_2_4·a_7_53 + b_2_4·b_4_17·a_3_10
       − b_2_4·b_4_17·a_3_8 − b_2_42·a_5_26
  62.  − a_4_16·a_5_25 + a_2_3·a_7_53
  63. a_4_16·a_6_36
  64. b_4_17·a_6_36 − a_5_25·a_5_26 − c_6_38·a_1_1·a_3_8
  65. a_4_16·b_6_37 − b_4_17·a_1_1·a_5_26 + b_2_4·a_3_10·a_5_26 − b_2_4·a_3_8·a_5_26
  66. a_3_9·a_7_53
  67. a_3_8·a_7_53 − b_4_17·a_1_1·a_5_26 + b_2_4·a_3_10·a_5_26 − b_2_4·a_3_8·a_5_26
  68.  − a_5_25·a_5_26 + a_3_10·a_7_53 + b_4_17·a_1_1·a_5_26 − b_2_4·a_3_10·a_5_26
       + b_2_4·a_3_8·a_5_26 − c_6_38·a_1_1·a_3_8
  69. a_6_36·a_5_25
  70. a_6_36·a_5_26 − a_2_3·c_6_38·a_3_10
  71. b_6_37·a_5_25 + b_4_172·a_3_10 + b_4_172·a_3_8 − b_2_4·b_4_17·a_5_26
       + b_2_4·b_4_17·a_5_25 + b_2_42·c_6_38·a_1_1
  72. a_4_16·a_7_53
  73.  − b_6_37·a_5_26 + b_4_17·a_7_53 − b_2_4·c_6_38·a_3_8
  74. a_6_362
  75. b_6_372 + b_4_173 + b_2_4·b_4_17·b_6_37 + b_2_43·c_6_38
  76.  − a_6_36·b_6_37 + a_5_25·a_7_53
  77.  − a_6_36·b_6_37 − b_4_17·a_3_10·a_5_26 + b_4_17·a_1_1·a_7_53 + b_2_4·a_3_10·a_7_53
       + b_2_4·b_4_17·a_1_1·a_5_26 − b_2_42·a_3_10·a_5_26 + b_2_42·a_3_8·a_5_26
       − b_2_4·c_6_38·a_1_1·a_3_10 − b_2_4·c_6_38·a_1_1·a_3_8
  78. a_5_26·a_7_53 + c_6_38·a_3_8·a_3_10 + c_6_38·a_1_1·a_5_25
  79. a_6_36·a_7_53
  80. b_6_37·a_7_53 + b_4_172·a_5_26 + b_2_4·b_4_17·a_7_53 − b_2_4·b_4_17·c_6_38·a_1_1
       + b_2_42·c_6_38·a_3_10 + b_2_42·c_6_38·a_3_8

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 13.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_5, a Duflot regular element of degree 2
    2. c_6_38, a Duflot regular element of degree 6
    3. b_4_173 − b_2_42·b_4_172 − b_2_46 − b_2_26 + b_2_43·c_6_38, an element of degree 12
    4. b_2_4, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 16, 18].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms 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 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 → a_1_0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. c_2_5 → c_2_1, an element of degree 2
  8. a_3_8 → 0, an element of degree 3
  9. a_3_9 → 0, an element of degree 3
  10. a_3_10 → 0, an element of degree 3
  11. a_4_16 → 0, an element of degree 4
  12. b_4_17 → 0, an element of degree 4
  13. a_5_25 → 0, an element of degree 5
  14. a_5_26 → 0, an element of degree 5
  15. a_6_36 → 0, an element of degree 6
  16. b_6_37 → 0, an element of degree 6
  17. c_6_38 → c_2_23, an element of degree 6
  18. a_7_53 → 0, an element of degree 7

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_0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_2 → c_2_5, an element of degree 2
  6. b_2_4 → 0, an element of degree 2
  7. c_2_5 → c_2_3, an element of degree 2
  8. a_3_8 → 0, an element of degree 3
  9. a_3_9 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
  10. a_3_10 → 0, an element of degree 3
  11. a_4_16 → 0, an element of degree 4
  12. b_4_17 → 0, an element of degree 4
  13. a_5_25 → 0, an element of degree 5
  14. a_5_26 → 0, an element of degree 5
  15. a_6_36 → 0, an element of degree 6
  16. b_6_37 → 0, an element of degree 6
  17. c_6_38 →  − c_2_4·c_2_52 + c_2_43, an element of degree 6
  18. a_7_53 → 0, an element of degree 7

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → a_1_2, an element of degree 1
  3. a_1_2 → a_1_0, an element of degree 1
  4. a_2_3 →  − a_1_2·a_1_3, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_4 → c_2_8, an element of degree 2
  7. c_2_5 → c_2_6, an element of degree 2
  8. a_3_8 → c_2_9·a_1_2 − c_2_8·a_1_3, an element of degree 3
  9. a_3_9 →  − a_1_1·a_1_2·a_1_3, an element of degree 3
  10. a_3_10 →  − c_2_9·a_1_3 − c_2_9·a_1_2 − c_2_8·a_1_3 + c_2_8·a_1_1 + c_2_7·a_1_2, an element of degree 3
  11. a_4_16 →  − c_2_9·a_1_1·a_1_2 + c_2_8·a_1_1·a_1_3 − c_2_7·a_1_2·a_1_3, an element of degree 4
  12. b_4_17 →  − c_2_92 + c_2_8·c_2_9 − c_2_7·c_2_8, an element of degree 4
  13. a_5_25 → c_2_92·a_1_3 − c_2_8·c_2_9·a_1_3 − c_2_7·c_2_9·a_1_2 + c_2_7·c_2_8·a_1_3
       − c_2_7·c_2_8·a_1_2, an element of degree 5
  14. a_5_26 →  − c_2_92·a_1_1 + c_2_8·c_2_9·a_1_1 + c_2_7·c_2_9·a_1_3 + c_2_7·c_2_9·a_1_2
       + c_2_7·c_2_8·a_1_3 − c_2_7·c_2_8·a_1_1 + c_2_72·a_1_2, an element of degree 5
  15. a_6_36 →  − c_2_92·a_1_1·a_1_3 + c_2_8·c_2_9·a_1_1·a_1_3 + c_2_7·c_2_9·a_1_2·a_1_3
       + c_2_7·c_2_9·a_1_1·a_1_2 − c_2_7·c_2_8·a_1_2·a_1_3 − c_2_7·c_2_8·a_1_1·a_1_3
       + c_2_7·c_2_8·a_1_1·a_1_2 − c_2_72·a_1_2·a_1_3, an element of degree 6
  16. b_6_37 → c_2_93 − c_2_8·c_2_92 − c_2_7·c_2_82, an element of degree 6
  17. c_6_38 →  − c_2_7·c_2_92 + c_2_7·c_2_8·c_2_9 + c_2_72·c_2_8 + c_2_73, an element of degree 6
  18. a_7_53 → c_2_93·a_1_1 − c_2_8·c_2_92·a_1_1 − c_2_7·c_2_92·a_1_3 − c_2_7·c_2_92·a_1_2
       − c_2_7·c_2_8·c_2_9·a_1_3 + c_2_7·c_2_8·c_2_9·a_1_2 − c_2_7·c_2_82·a_1_3
       − c_2_7·c_2_82·a_1_1 − c_2_72·c_2_9·a_1_2 + c_2_72·c_2_8·a_1_3 + c_2_72·c_2_8·a_1_2, an element of degree 7

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