Cohomology of group number 52 of order 243

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General information on the group

  • The group is also known as 81gp8xC3, the Direct product 81gp8 x C_3.
  • 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 2 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) · (t4  −  t3  +  t2  +  1)
    (t  −  1)3 · (t2  −  t  +  1) · (t2  +  t  +  1)
  • The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 15 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. a_2_4, a nilpotent element of degree 2
  6. b_2_2, 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_4_10, a nilpotent element of degree 4
  10. b_4_11, an element of degree 4
  11. a_5_14, a nilpotent element of degree 5
  12. a_5_15, a nilpotent element of degree 5
  13. b_6_19, an element of degree 6
  14. c_6_20, a Duflot regular element of degree 6
  15. a_7_26, a nilpotent element of degree 7

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Ring relations

There are 7 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_82, a_5_142, a_5_152, a_7_262

Apart from that, there are 59 minimal relations of maximal degree 13:

  1. a_1_0·a_1_1
  2. a_2_3·a_1_0
  3. a_2_4·a_1_1
  4. a_2_4·a_1_0 − a_2_3·a_1_1
  5. b_2_2·a_1_1
  6. a_2_32
  7. a_2_3·a_2_4
  8. a_2_42
  9. a_2_3·b_2_2
  10. a_2_4·b_2_2
  11. a_1_1·a_3_8
  12. a_1_0·a_3_8
  13. b_2_2·a_3_8
  14. a_2_3·a_3_8
  15. a_2_4·a_3_8
  16. a_4_10·a_1_1
  17. a_4_10·a_1_0
  18. b_4_11·a_1_0
  19. a_2_3·a_4_10
  20. a_2_4·a_4_10
  21. b_2_2·b_4_11 − b_2_2·a_4_10
  22. a_2_3·b_4_11
  23. a_1_1·a_5_14
  24.  − b_2_2·a_4_10 + a_1_0·a_5_14
  25.  − a_2_4·b_4_11 + a_1_1·a_5_15
  26.  − b_2_2·a_4_10 + a_1_0·a_5_15
  27. a_4_10·a_3_8
  28. a_2_3·a_5_14
  29. a_2_4·a_5_14
  30. b_2_2·a_5_15 − b_2_2·a_5_14
  31. a_2_3·a_5_15
  32. a_2_4·a_5_15
  33. b_6_19·a_1_1 + b_4_11·a_3_8
  34. b_6_19·a_1_0
  35. a_4_102
  36. a_3_8·a_5_14
  37.  − a_4_10·b_4_11 + a_3_8·a_5_15
  38. b_2_2·b_6_19 + b_2_2·a_1_0·a_5_14
  39. a_2_3·b_6_19
  40. a_4_10·b_4_11 + a_2_4·b_6_19
  41. a_4_10·b_4_11 + a_1_1·a_7_26
  42. a_1_0·a_7_26 − b_2_2·a_1_0·a_5_14
  43. b_4_11·a_5_14 − b_4_112·a_1_1
  44. a_4_10·a_5_14
  45. a_4_10·a_5_15
  46. b_6_19·a_3_8 − b_4_112·a_1_1
  47. b_2_2·a_7_26 − b_2_22·a_5_14 + b_2_2·c_6_20·a_1_0
  48. a_2_3·a_7_26 − a_2_3·c_6_20·a_1_1
  49. a_2_4·a_7_26 + a_2_3·c_6_20·a_1_1
  50. a_5_14·a_5_15 − b_4_11·a_1_1·a_5_15
  51. a_4_10·b_6_19 − b_4_11·a_1_1·a_5_15
  52. a_3_8·a_7_26 − b_4_11·a_1_1·a_5_15
  53. b_6_19·a_5_14 + b_4_112·a_3_8
  54.  − b_6_19·a_5_15 + b_4_11·a_7_26 − b_4_11·c_6_20·a_1_1
  55. a_4_10·a_7_26
  56. b_6_192 + b_4_113
  57. a_5_14·a_7_26 − b_4_11·a_1_1·a_7_26 − c_6_20·a_1_0·a_5_14
  58. a_5_15·a_7_26 + c_6_20·a_1_1·a_5_15 − c_6_20·a_1_0·a_5_14
  59. b_6_19·a_7_26 + b_4_112·a_5_15 + b_4_11·c_6_20·a_3_8

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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_20, a Duflot regular element of degree 6
    3. b_4_11 − b_2_22, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_2_4 → 0, an element of degree 2
  6. b_2_2 → 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_4_10 → 0, an element of degree 4
  10. b_4_11 → 0, an element of degree 4
  11. a_5_14 → 0, an element of degree 5
  12. a_5_15 → 0, an element of degree 5
  13. b_6_19 → 0, an element of degree 6
  14. c_6_20 → c_2_23, an element of degree 6
  15. a_7_26 → 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. a_2_4 → 0, an element of degree 2
  6. b_2_2 → c_2_5, 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_4_10 → c_2_5·a_1_1·a_1_2, an element of degree 4
  10. b_4_11 → c_2_5·a_1_1·a_1_2, an element of degree 4
  11. a_5_14 →  − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  12. a_5_15 →  − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  13. b_6_19 →  − c_2_52·a_1_1·a_1_2, an element of degree 6
  14. c_6_20 →  − c_2_4·c_2_52 + c_2_43, an element of degree 6
  15. a_7_26 →  − c_2_53·a_1_1 − c_2_4·c_2_52·a_1_2 − c_2_43·a_1_2, an element of degree 7

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_0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_2 → 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_4_10 → 0, an element of degree 4
  10. b_4_11 →  − c_2_52, an element of degree 4
  11. a_5_14 → 0, an element of degree 5
  12. a_5_15 → c_2_52·a_1_2, an element of degree 5
  13. b_6_19 → c_2_53, an element of degree 6
  14. c_6_20 →  − c_2_53 − c_2_4·c_2_52 + c_2_43, an element of degree 6
  15. a_7_26 →  − c_2_53·a_1_2, an element of degree 7

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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