Cohomology of group number 55 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 16.
  • 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
    t5  −  t4  −  t2  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  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 12 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. a_3_1, a nilpotent element of degree 3
  6. a_3_2, a nilpotent element of degree 3
  7. b_3_3, an element of degree 3
  8. a_4_2, a nilpotent element of degree 4
  9. b_4_4, an element of degree 4
  10. c_4_5, a Duflot regular element of degree 4
  11. c_4_6, a Duflot regular element of degree 4
  12. b_5_8, an element of degree 5

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Ring relations

There are 44 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_1·a_1_1 + a_1_13
  4. a_2_1·a_1_0 + a_1_13
  5. b_2_2·a_1_1 + a_1_13
  6. b_2_2·a_1_0 + a_1_13
  7. a_2_12
  8. a_1_1·a_3_1
  9. a_1_0·a_3_1
  10. a_1_0·a_3_2
  11. a_1_1·b_3_3
  12. a_1_0·b_3_3
  13. b_2_2·a_3_2 + b_2_2·a_3_1 + a_2_1·a_3_2 + a_2_1·a_3_1
  14. a_2_1·a_3_1 + a_1_12·a_3_2
  15. b_2_2·a_3_2 + a_2_1·b_3_3
  16. b_2_2·a_3_2 + b_2_2·a_3_1 + a_4_2·a_1_1 + a_2_1·a_3_1
  17. a_4_2·a_1_0 + a_2_1·a_3_1
  18. b_4_4·a_1_1 + b_2_2·a_3_2 + b_2_2·a_3_1
  19. b_4_4·a_1_0 + b_2_2·a_3_2 + b_2_2·a_3_1
  20. a_3_12
  21. a_3_1·a_3_2
  22. a_3_2·b_3_3 + a_2_1·b_2_22
  23. a_3_1·b_3_3 + a_2_1·b_2_22
  24. b_3_32 + b_2_23
  25. a_3_22 + c_4_5·a_1_12
  26. a_2_1·a_4_2
  27. b_2_2·a_4_2 + a_2_1·b_4_4
  28. a_1_1·b_5_8 + a_3_22
  29. a_1_0·b_5_8
  30. a_4_2·a_3_2 + c_4_5·a_1_13
  31. a_4_2·a_3_1 + c_4_5·a_1_13
  32. b_4_4·a_3_2 + a_4_2·b_3_3 + c_4_6·a_1_13 + c_4_5·a_1_13
  33. b_4_4·a_3_1 + a_4_2·b_3_3 + c_4_5·a_1_13
  34. b_4_4·b_3_3 + b_2_2·b_5_8 + b_2_22·b_3_3 + c_4_6·a_1_13
  35. a_4_2·b_3_3 + a_2_1·b_5_8 + a_2_1·b_2_2·b_3_3 + c_4_5·a_1_13
  36. a_4_22
  37. b_4_42 + b_2_22·b_4_4 + a_2_1·b_2_23 + b_2_22·c_4_5
  38. a_4_2·b_4_4 + a_2_1·b_2_2·b_4_4 + a_2_1·b_2_2·c_4_5
  39. a_3_2·b_5_8 + a_2_1·b_2_2·b_4_4 + a_2_1·b_2_23 + c_4_5·a_1_1·a_3_2
  40. a_3_1·b_5_8 + a_2_1·b_2_2·b_4_4 + a_2_1·b_2_23
  41. b_3_3·b_5_8 + b_2_22·b_4_4 + b_2_24
  42. b_4_4·b_5_8 + a_2_1·b_2_22·b_3_3 + b_2_2·c_4_5·b_3_3 + a_2_1·c_4_5·a_3_2
       + c_4_6·a_1_12·a_3_2
  43. a_4_2·b_5_8 + a_2_1·c_4_5·b_3_3 + a_2_1·c_4_5·a_3_2
  44. b_5_82 + b_2_23·b_4_4 + b_2_25 + a_2_1·b_2_24 + b_2_23·c_4_5 + c_4_52·a_1_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 10.
  • 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_4_5, a Duflot regular element of degree 4
    2. c_4_6, a Duflot regular element of degree 4
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • 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_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. b_3_3 → 0, an element of degree 3
  8. a_4_2 → 0, an element of degree 4
  9. b_4_4 → 0, an element of degree 4
  10. c_4_5 → c_1_04, an element of degree 4
  11. c_4_6 → c_1_14, an element of degree 4
  12. b_5_8 → 0, an element of degree 5

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_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. b_3_3 → c_1_23, an element of degree 3
  8. a_4_2 → 0, an element of degree 4
  9. b_4_4 → c_1_24 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  10. c_4_5 → c_1_0·c_1_23 + c_1_04, an element of degree 4
  11. c_4_6 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  12. b_5_8 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


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