Cohomology of group number 37 of order 64

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General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t5  +  t2  +  1
    (t  −  1)2 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-2,-2. 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 8:

  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. a_2_2, a nilpotent 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. b_4_3, an element of degree 4
  8. a_5_2, a nilpotent element of degree 5
  9. a_5_4, a nilpotent element of degree 5
  10. b_6_5, an element of degree 6
  11. a_7_5, a nilpotent element of degree 7
  12. c_8_6, a Duflot regular element of degree 8

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Ring relations

There are 47 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_1·a_1_0
  4. a_2_2·a_1_1
  5. a_2_2·a_1_0 + a_2_1·a_1_1
  6. a_1_14
  7. a_2_22 + a_2_1·a_2_2 + a_2_12
  8. a_1_0·a_3_2
  9. a_1_1·a_3_3 + a_2_22
  10. a_1_0·a_3_3 + a_2_12
  11. a_2_2·a_3_2
  12. a_2_1·a_3_2
  13. a_2_1·a_3_3 + a_1_12·a_3_2
  14. b_4_3·a_1_0
  15. a_3_2·a_3_3
  16. a_2_1·b_4_3 + a_3_32 + a_1_13·a_3_2
  17. a_3_22 + b_4_3·a_1_12 + a_1_13·a_3_2
  18. a_3_22 + a_1_1·a_5_2
  19. a_1_0·a_5_2
  20. a_1_0·a_5_4 + a_1_13·a_3_2
  21. a_2_1·a_5_2 + b_4_3·a_1_13
  22. a_2_2·a_5_4 + b_4_3·a_1_13
  23. a_2_2·a_5_2 + a_2_1·a_5_4 + b_4_3·a_1_13
  24. b_6_5·a_1_1 + b_4_3·a_3_2 + a_2_2·a_5_2 + b_4_3·a_1_13
  25. b_6_5·a_1_0
  26. a_3_2·a_5_2 + b_4_3·a_1_1·a_3_2 + a_2_2·a_3_32
  27. a_2_2·a_3_32 + a_1_13·a_5_4
  28. a_2_2·b_6_5 + a_3_3·a_5_4 + a_3_3·a_5_2
  29. a_2_1·b_6_5 + a_3_3·a_5_2
  30. a_3_2·a_5_4 + a_1_1·a_7_5 + b_4_3·a_1_1·a_3_2 + a_2_2·a_3_32
  31. a_1_0·a_7_5
  32. b_6_5·a_3_2 + b_4_32·a_1_1 + a_2_2·b_4_3·a_3_3 + b_4_3·a_1_12·a_3_2
  33. a_2_2·a_7_5 + a_2_2·b_4_3·a_3_3 + b_4_3·a_1_12·a_3_2
  34. a_2_2·b_4_3·a_3_3 + a_2_1·a_7_5
  35. b_6_5·a_3_3 + b_4_3·a_5_2 + b_4_32·a_1_1 + a_2_2·b_4_3·a_3_3 + a_1_12·a_7_5
       + b_4_3·a_1_12·a_3_2
  36. a_5_22 + b_4_3·a_3_32 + b_4_32·a_1_12
  37. a_2_2·b_4_32 + a_5_2·a_5_4 + b_4_3·a_3_32 + b_4_3·a_1_1·a_5_4
  38. a_2_2·b_4_32 + a_3_3·a_7_5 + a_1_13·a_7_5
  39. a_3_2·a_7_5 + b_4_3·a_1_1·a_5_4 + b_4_32·a_1_12
  40. a_2_2·b_4_32 + a_5_42 + b_4_3·a_1_1·a_5_4 + b_4_32·a_1_12 + a_1_13·a_7_5
       + c_8_6·a_1_12
  41. b_6_5·a_5_2 + b_4_32·a_3_3 + b_4_32·a_3_2 + b_4_3·a_1_12·a_5_4
  42. b_6_5·a_5_4 + b_4_3·a_7_5 + b_4_32·a_3_3 + b_4_32·a_3_2 + b_4_3·a_1_12·a_5_4
       + b_4_32·a_1_13 + a_2_1·c_8_6·a_1_1
  43. a_5_2·a_7_5 + b_4_3·a_3_3·a_5_4 + b_4_3·a_3_3·a_5_2 + b_4_3·a_1_1·a_7_5
  44. a_5_4·a_7_5 + b_4_3·a_3_3·a_5_2 + b_4_32·a_1_1·a_3_2 + c_8_6·a_1_1·a_3_2
  45. b_6_52 + b_4_33 + b_4_3·a_3_3·a_5_2 + b_4_3·a_1_13·a_5_4 + a_2_12·c_8_6
  46. b_6_5·a_7_5 + b_4_32·a_5_4 + b_4_32·a_5_2 + a_3_32·a_7_5 + b_4_32·a_1_12·a_3_2
  47. a_7_52 + b_4_3·a_3_3·a_7_5 + b_4_32·a_3_32 + b_4_32·a_1_1·a_5_4
       + b_4_3·a_1_13·a_7_5 + b_4_3·c_8_6·a_1_12 + c_8_6·a_1_13·a_3_2

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 14.
  • 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_8_6, a Duflot regular element of degree 8
    2. b_4_3, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, 6, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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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_1 → 0, an element of degree 2
  4. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. b_4_3 → 0, an element of degree 4
  8. a_5_2 → 0, an element of degree 5
  9. a_5_4 → 0, an element of degree 5
  10. b_6_5 → 0, an element of degree 6
  11. a_7_5 → 0, an element of degree 7
  12. c_8_6 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. 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. a_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. b_4_3 → c_1_14, an element of degree 4
  8. a_5_2 → 0, an element of degree 5
  9. a_5_4 → 0, an element of degree 5
  10. b_6_5 → c_1_16, an element of degree 6
  11. a_7_5 → 0, an element of degree 7
  12. c_8_6 → c_1_04·c_1_14 + c_1_08, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64


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