Cohomology of group number 45 of order 243

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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 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t8  −  t7  +  2·t6  −  2·t5  +  t4  −  t3  −  1

    (t  −  1)3 · (t2  −  t  +  1)2 · (t2  +  t  +  1)2
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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. b_2_3, an element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_4, a nilpotent element of degree 3
  8. a_5_3, a nilpotent element of degree 5
  9. a_5_4, a nilpotent element of degree 5
  10. a_5_5, a nilpotent element of degree 5
  11. a_5_6, a nilpotent element of degree 5
  12. c_6_8, a Duflot regular element of degree 6
  13. c_6_9, a Duflot regular element of degree 6
  14. a_7_12, a nilpotent element of degree 7

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Ring relations

There are 11 "obvious" relations:
   a_1_02, a_1_12, a_1_22, a_3_22, a_3_32, a_3_42, a_5_32, a_5_42, a_5_52, a_5_62, a_7_122

Apart from that, there are 44 minimal relations of maximal degree 12:

  1. b_2_3·a_1_1
  2. b_2_3·a_1_2
  3. a_1_1·a_3_2
  4. a_1_2·a_3_2 + a_1_1·a_3_3
  5. a_1_2·a_3_3 − a_1_2·a_3_2 + a_1_1·a_3_4
  6. a_1_2·a_3_4 + a_1_2·a_3_3 + a_1_2·a_3_2
  7. b_2_3·a_3_2
  8. b_2_3·a_3_3
  9. b_2_3·a_3_4
  10. a_1_1·a_5_3
  11.  − a_3_2·a_3_3 + a_1_2·a_5_3
  12. a_3_2·a_3_3 + a_1_1·a_5_4
  13. a_3_2·a_3_4 − a_3_2·a_3_3 + a_1_2·a_5_4
  14.  − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_1·a_5_5
  15.  − a_3_3·a_3_4 − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_2·a_5_5
  16. a_3_3·a_3_4 − a_3_2·a_3_3 + a_1_1·a_5_6
  17. a_1_2·a_5_6
  18. b_2_3·a_5_4 + b_2_3·a_5_3
  19. b_2_3·a_5_5 + b_2_3·a_5_3
  20. a_3_2·a_5_3
  21.  − a_3_4·a_5_3 + a_3_3·a_5_4
  22. a_3_3·a_5_3 + a_3_2·a_5_4
  23.  − a_3_4·a_5_4 − a_3_4·a_5_3 + a_3_3·a_5_5 − a_3_3·a_5_3
  24.  − a_3_4·a_5_3 + a_3_3·a_5_3 + a_3_2·a_5_5
  25. a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_4·a_5_3 + a_3_3·a_5_6 + a_3_3·a_5_3
  26.  − a_3_4·a_5_4 + a_3_4·a_5_3 + a_3_2·a_5_6
  27. a_3_4·a_5_6 − a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_3·a_5_3
  28.  − a_3_4·a_5_5 + a_3_4·a_5_4 + a_3_4·a_5_3 − a_3_3·a_5_3 + c_6_8·a_1_1·a_1_2
  29. a_3_4·a_5_5 − a_3_4·a_5_4 − a_3_4·a_5_3 − a_3_3·a_5_3 + c_6_9·a_1_1·a_1_2
  30. a_3_4·a_5_5 − a_3_4·a_5_4 + a_3_3·a_5_3 + a_1_1·a_7_12 − c_6_8·a_1_0·a_1_1
  31.  − a_3_4·a_5_5 − a_3_4·a_5_4 + a_1_2·a_7_12 − c_6_8·a_1_0·a_1_2
  32. b_2_3·a_7_12 + b_2_32·a_5_6 + b_2_3·c_6_8·a_1_0
  33. a_5_4·a_5_5 + a_5_3·a_5_5 − a_5_3·a_5_4
  34. a_5_4·a_5_6 + a_5_3·a_5_6 + c_6_8·a_1_1·a_3_3
  35.  − a_5_5·a_5_6 + a_5_4·a_5_6 + c_6_8·a_1_1·a_3_4
  36.  − a_5_4·a_5_6 − a_5_3·a_5_6 + a_5_3·a_5_4 + c_6_9·a_1_1·a_3_3
  37. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_5 + a_5_3·a_5_4 + c_6_9·a_1_1·a_3_4
  38. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_5 − a_5_3·a_5_4 + a_3_3·a_7_12 − c_6_8·a_1_0·a_3_3
  39.  − a_5_4·a_5_6 − a_5_3·a_5_6 − a_5_3·a_5_5 + a_5_3·a_5_4 + a_3_2·a_7_12 − c_6_8·a_1_0·a_3_2
  40. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4 + a_3_4·a_7_12 − c_6_8·a_1_0·a_3_4
  41. a_5_3·a_7_12 + b_2_3·a_5_3·a_5_6 + c_6_9·a_1_1·a_5_5 − c_6_9·a_1_1·a_5_4
       + c_6_8·a_1_1·a_5_5 − c_6_8·a_1_0·a_5_3
  42. a_5_5·a_7_12 − b_2_3·a_5_3·a_5_6 − c_6_9·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5
       − c_6_8·a_1_0·a_5_5
  43. a_5_4·a_7_12 − b_2_3·a_5_3·a_5_6 − c_6_9·a_1_1·a_5_6 + c_6_9·a_1_1·a_5_4
       − c_6_8·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5 − c_6_8·a_1_0·a_5_4
  44. a_5_6·a_7_12 − c_6_9·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_6 + c_6_8·a_1_1·a_5_5
       + c_6_8·a_1_1·a_5_4 − c_6_8·a_1_0·a_5_6


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 12.
  • 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_8, a Duflot regular element of degree 6
    2. c_6_9, a Duflot regular element of degree 6
    3. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • 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_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_1_20, an element of degree 1
  4. b_2_30, an element of degree 2
  5. a_3_20, an element of degree 3
  6. a_3_30, an element of degree 3
  7. a_3_40, an element of degree 3
  8. a_5_30, an element of degree 5
  9. a_5_40, an element of degree 5
  10. a_5_50, an element of degree 5
  11. a_5_60, an element of degree 5
  12. c_6_8 − c_2_23, an element of degree 6
  13. c_6_9c_2_23 + c_2_13, an element of degree 6
  14. a_7_120, an element of degree 7

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

  1. a_1_0a_1_2, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_1_20, an element of degree 1
  4. b_2_3c_2_5, an element of degree 2
  5. a_3_20, an element of degree 3
  6. a_3_30, an element of degree 3
  7. a_3_40, an element of degree 3
  8. a_5_3c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
  9. a_5_4 − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  10. a_5_5 − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  11. a_5_6 − c_2_52·a_1_0 + c_2_3·c_2_5·a_1_2, an element of degree 5
  12. c_6_8 − c_2_52·a_1_1·a_1_2 + c_2_4·c_2_52 − c_2_43, an element of degree 6
  13. c_6_9 − c_2_52·a_1_0·a_1_2 − c_2_4·c_2_52 + c_2_43 − c_2_3·c_2_52 + c_2_33, an element of degree 6
  14. a_7_12c_2_53·a_1_0 − c_2_4·c_2_52·a_1_2 + c_2_43·a_1_2 − c_2_3·c_2_52·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