Cohomology of group number 846 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 16.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t4  −  t3  +  t2  −  t  +  1
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-6,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. c_1_2, a Duflot regular element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_9, a nilpotent element of degree 3
  7. a_4_13, a nilpotent element of degree 4
  8. a_5_19, a nilpotent element of degree 5
  9. b_5_21, an element of degree 5
  10. a_6_29, a nilpotent element of degree 6
  11. a_7_41, a nilpotent element of degree 7
  12. a_8_48, a nilpotent element of degree 8
  13. c_8_55, a Duflot regular element of degree 8

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Ring relations

There are 52 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_4·a_1_0
  4. b_2_5·a_1_0 + a_2_4·b_1_1
  5. a_2_42
  6. a_1_0·a_3_9
  7. b_1_1·a_3_9 + a_2_4·b_2_5
  8. a_2_4·b_2_5·b_1_1
  9. a_2_4·a_3_9
  10. a_4_13·a_1_0
  11. a_3_92
  12. a_2_4·a_4_13
  13. a_1_0·a_5_19
  14. b_1_1·a_5_19
  15. a_1_0·b_5_21
  16. a_4_13·a_3_9
  17. a_2_4·a_5_19
  18. b_2_5·a_5_19 + a_2_4·b_5_21
  19. a_6_29·a_1_0
  20. a_6_29·b_1_1 + b_2_5·a_5_19
  21. a_4_132
  22. a_3_9·a_5_19
  23. a_3_9·b_5_21 + b_2_5·a_6_29 + a_2_4·b_2_53
  24. a_2_4·a_6_29
  25. a_1_0·a_7_41
  26. a_3_9·b_5_21 + b_1_1·a_7_41 + b_2_52·a_4_13 + a_2_4·b_2_53
  27. a_4_13·a_5_19
  28. a_6_29·a_3_9
  29. a_2_4·a_7_41
  30. a_8_48·a_1_0
  31. a_8_48·b_1_1 + a_4_13·b_5_21 + b_2_53·a_3_9
  32. a_5_192
  33. a_5_19·b_5_21 + a_2_4·b_2_54
  34. a_4_13·a_6_29
  35. a_3_9·a_7_41
  36. b_5_212 + b_1_15·b_5_21 + b_2_52·b_1_1·b_5_21 + b_2_52·b_1_16 + b_2_53·b_1_14
       + b_2_54·b_1_12 + b_2_55 + a_4_13·b_1_1·b_5_21 + b_2_5·a_4_13·b_1_14
       + a_2_4·b_2_54 + c_8_55·b_1_12
  37. a_2_4·a_8_48
  38. a_6_29·a_5_19
  39. a_4_13·a_7_41
  40. a_6_29·b_5_21 + b_2_54·a_3_9 + a_2_4·c_8_55·b_1_1
  41. a_8_48·a_3_9
  42. a_6_292
  43. a_5_19·a_7_41
  44. b_5_21·a_7_41 + b_2_52·a_8_48 + a_2_4·b_2_55 + a_2_4·b_2_5·c_8_55
  45. a_4_13·a_8_48
  46. a_6_29·a_7_41
  47. a_8_48·b_5_21 + a_4_13·b_1_14·b_5_21 + b_2_52·a_4_13·b_5_21
       + b_2_52·a_4_13·b_1_15 + b_2_53·a_7_41 + b_2_53·a_4_13·b_1_13
       + b_2_54·a_4_13·b_1_1 + b_2_55·a_3_9 + a_4_13·c_8_55·b_1_1
  48. a_8_48·a_5_19
  49. a_7_412
  50. a_6_29·a_8_48
  51. a_8_48·a_7_41
  52. a_8_482

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 16.
  • 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_1_2, a Duflot regular element of degree 1
    2. c_8_55, a Duflot regular element of degree 8
    3. b_1_12 + b_2_5, an element of degree 2
    4. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 7, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. b_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. a_3_9 → 0, an element of degree 3
  7. a_4_13 → 0, an element of degree 4
  8. a_5_19 → 0, an element of degree 5
  9. b_5_21 → 0, an element of degree 5
  10. a_6_29 → 0, an element of degree 6
  11. a_7_41 → 0, an element of degree 7
  12. a_8_48 → 0, an element of degree 8
  13. c_8_55 → c_1_18, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. a_3_9 → 0, an element of degree 3
  7. a_4_13 → 0, an element of degree 4
  8. a_5_19 → 0, an element of degree 5
  9. b_5_21 → c_1_35 + c_1_22·c_1_33 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. a_6_29 → 0, an element of degree 6
  11. a_7_41 → 0, an element of degree 7
  12. a_8_48 → 0, an element of degree 8
  13. c_8_55 → c_1_38 + c_1_22·c_1_36 + c_1_24·c_1_34 + c_1_26·c_1_32
       + c_1_12·c_1_22·c_1_34 + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_26
       + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32 + c_1_18, an element of degree 8

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