Cohomology of group number 236 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    (2) · (t4  +  1/2·t3  +  1/2·t2  +  1/2·t  +  1/2)
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-4,-4. 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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_5, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. a_3_7, a nilpotent element of degree 3
  7. a_3_8, a nilpotent element of degree 3
  8. b_3_9, an element of degree 3
  9. a_4_13, a nilpotent element of degree 4
  10. a_4_15, a nilpotent element of degree 4
  11. b_4_14, an element of degree 4
  12. c_4_16, a Duflot regular element of degree 4
  13. c_4_17, a Duflot regular element of degree 4
  14. a_5_24, a nilpotent element of degree 5
  15. a_6_29, a nilpotent element of degree 6

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

Ring relations

There are 65 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_22
  4. a_2_5·a_1_0
  5. a_2_5·b_1_1
  6. b_2_4·b_1_1
  7. a_2_52
  8. a_1_0·a_3_7
  9. b_1_13·b_1_2 + b_1_1·a_3_7
  10. a_1_0·a_3_8
  11. b_1_1·a_3_8
  12. a_1_0·b_3_9
  13. a_2_5·b_3_9 + a_2_5·a_3_8
  14. b_1_22·b_3_9 + b_1_22·a_3_8 + a_2_5·a_3_7
  15. b_2_4·b_3_9 + b_2_4·a_3_8 + a_2_5·a_3_8 + a_2_5·a_3_7
  16. a_4_13·a_1_0 + a_2_5·a_3_7
  17. a_4_13·b_1_1
  18. a_4_15·a_1_0 + a_2_5·a_3_8
  19. b_1_1·b_1_2·b_3_9 + b_1_12·a_3_7 + a_4_15·b_1_1
  20. b_1_22·a_3_8 + b_1_22·a_3_7 + b_4_14·a_1_0 + b_2_4·a_3_7 + a_2_5·a_3_7
  21. b_4_14·b_1_1 + b_1_12·a_3_7
  22. a_3_72
  23. a_3_82
  24. a_3_7·a_3_8
  25. a_3_8·b_3_9
  26. b_3_92 + a_3_7·b_3_9 + c_4_16·b_1_12
  27. a_2_5·a_4_13
  28. a_2_5·b_1_2·a_3_7 + a_2_5·a_4_15
  29. a_3_7·b_3_9 + b_1_13·a_3_7 + a_4_15·b_1_12
  30. b_1_23·a_3_7 + b_4_14·a_1_0·b_1_2 + a_4_15·b_1_22 + a_4_13·b_1_22 + b_2_4·a_4_13
       + a_2_5·b_4_14
  31. a_1_0·a_5_24
  32. a_3_7·b_3_9 + b_1_1·a_5_24 + b_1_13·a_3_7
  33. a_4_13·b_3_9 + a_4_13·a_3_8
  34. a_4_15·a_3_8 + a_4_13·a_3_7 + a_2_5·b_2_4·a_3_8 + a_2_5·b_2_4·a_3_7
  35. a_4_15·a_3_7 + a_4_13·a_3_8
  36. a_4_13·a_3_7 + a_2_5·a_4_15·b_1_2
  37. b_1_14·a_3_7 + a_4_15·b_3_9 + a_4_15·b_1_13 + a_4_13·a_3_7 + a_2_5·b_2_4·a_3_8
       + a_2_5·b_2_4·a_3_7 + c_4_16·b_1_12·b_1_2
  38. b_4_14·b_3_9 + b_1_14·a_3_7 + b_4_14·a_3_8 + a_4_15·b_1_13 + a_4_13·a_3_8
       + a_4_13·a_3_7 + a_2_5·b_2_4·a_3_7
  39. a_4_13·a_3_8 + a_2_5·a_5_24 + a_2_5·b_2_4·a_3_7
  40. b_1_22·a_5_24 + b_4_14·a_3_7 + a_4_15·b_1_23 + a_4_13·b_1_23 + b_2_4·a_4_13·b_1_2
       + a_2_5·b_1_25 + a_2_5·b_4_14·b_1_2 + a_4_13·a_3_7 + b_2_4·c_4_17·a_1_0
       + b_2_4·c_4_16·a_1_0
  41. b_4_14·a_3_8 + b_4_14·a_3_7 + b_4_14·a_1_0·b_1_22 + b_2_4·a_5_24 + b_2_4·b_1_22·a_3_7
       + a_2_5·b_2_4·b_1_23 + a_4_13·a_3_8 + a_2_5·b_2_4·a_3_7 + c_4_17·a_1_0·b_1_22
  42. a_6_29·a_1_0 + a_4_13·a_3_8 + a_4_13·a_3_7 + a_2_5·b_2_4·a_3_7
  43. b_1_14·a_3_7 + a_6_29·b_1_1 + c_4_17·b_1_12·b_1_2 + c_4_16·b_1_12·b_1_2
  44. a_4_132
  45. a_4_152
  46. b_4_14·b_1_24 + b_4_142 + b_2_4·b_4_14·b_1_22 + a_4_13·b_1_24
       + b_2_4·a_4_13·b_1_22 + b_2_42·b_1_2·a_3_8 + a_2_5·a_4_15·b_1_22 + c_4_17·b_1_24
       + b_2_42·c_4_17 + b_2_42·c_4_16
  47. a_3_8·a_5_24 + a_2_5·a_4_15·b_1_22 + a_2_5·b_2_4·a_4_15
  48. a_3_7·a_5_24 + a_2_5·a_4_15·b_1_22
  49. a_4_13·a_4_15 + a_2_5·b_1_2·a_5_24
  50. b_3_9·a_5_24 + a_4_15·b_1_1·b_3_9 + a_2_5·a_4_15·b_1_22 + a_2_5·b_2_4·a_4_15
  51. a_4_13·a_4_15 + a_2_5·a_6_29 + a_2_5·a_4_15·b_1_22 + a_2_5·b_2_4·a_4_15
  52. a_6_29·b_1_22 + b_4_14·b_1_2·a_3_7 + b_4_14·a_1_0·b_1_23 + a_4_13·b_4_14
       + b_2_4·b_4_14·a_1_0·b_1_2 + b_2_4·a_4_15·b_1_22 + b_2_4·a_4_13·b_1_22
       + b_2_42·a_4_13 + a_2_5·b_1_26 + a_2_5·b_2_4·b_1_24 + a_2_5·b_2_4·b_4_14
       + a_2_5·b_2_4·b_1_2·a_3_8 + a_2_5·b_2_4·c_4_17 + a_2_5·b_2_4·c_4_16
  53. b_4_14·b_1_2·a_3_7 + b_4_14·a_1_0·b_1_23 + a_4_15·b_4_14 + a_4_13·b_4_14
       + b_2_4·b_1_2·a_5_24 + b_2_4·a_6_29 + b_2_4·a_4_15·b_1_22 + b_2_4·a_4_13·b_1_22
       + b_2_42·b_1_2·a_3_7 + b_2_42·a_4_13 + a_2_5·b_4_14·b_1_22 + a_2_5·b_2_42·b_1_22
       + a_2_5·a_4_15·b_1_22 + c_4_17·a_1_0·b_1_23 + a_2_5·c_4_17·b_1_22
  54. b_4_14·a_5_24 + b_4_142·a_1_0 + b_2_4·b_4_14·a_3_7 + a_2_5·b_4_14·b_1_23
       + a_2_5·b_4_14·a_3_7 + a_2_5·a_4_15·b_1_23 + a_2_5·b_2_4·a_4_15·b_1_2
       + c_4_17·b_1_22·a_3_7 + b_2_4·c_4_17·a_3_8 + b_2_4·c_4_17·a_3_7 + b_2_4·c_4_16·a_3_8
       + b_2_4·c_4_16·a_3_7 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_17·a_3_7 + a_2_5·c_4_16·a_3_8
       + a_2_5·c_4_16·a_3_7
  55. a_4_15·a_5_24 + a_2_5·b_2_4·a_5_24 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_16·a_3_8
       + a_2_5·c_4_16·a_3_7
  56. a_4_13·a_5_24 + a_2_5·a_4_15·b_1_23 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_17·a_3_7
       + a_2_5·c_4_16·a_3_8 + a_2_5·c_4_16·a_3_7
  57. a_6_29·a_3_8 + a_2_5·a_4_15·b_1_23 + a_2_5·b_2_42·a_3_7 + a_2_5·c_4_17·a_3_8
       + a_2_5·c_4_16·a_3_8 + a_2_5·c_4_16·a_3_7
  58. a_6_29·a_3_7 + a_2_5·b_4_14·a_3_7 + a_2_5·a_4_15·b_1_23 + a_2_5·b_2_4·a_4_15·b_1_2
       + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_17·a_3_7 + a_2_5·c_4_16·a_3_8 + a_2_5·c_4_16·a_3_7
  59. a_6_29·b_3_9 + a_4_15·b_1_12·b_3_9 + a_2_5·a_4_15·b_1_23 + a_2_5·b_2_42·a_3_7
       + c_4_17·b_1_12·a_3_7 + a_4_15·c_4_17·b_1_1 + a_4_15·c_4_16·b_1_1 + a_2_5·c_4_17·a_3_8
       + a_2_5·c_4_16·a_3_8 + a_2_5·c_4_16·a_3_7
  60. a_5_242
  61. b_4_14·a_6_29 + a_4_15·b_4_14·b_1_22 + a_4_15·a_6_29 + a_2_5·a_4_15·b_4_14
       + a_2_5·b_2_4·a_4_15·b_1_22 + a_2_5·b_2_42·a_4_15 + b_4_14·c_4_17·a_1_0·b_1_2
       + a_4_15·c_4_17·b_1_22 + b_2_4·c_4_17·b_1_2·a_3_8 + b_2_4·c_4_16·b_1_2·a_3_8
       + b_2_4·a_4_15·c_4_17 + b_2_4·a_4_15·c_4_16 + b_2_4·a_4_13·c_4_16
       + a_2_5·c_4_17·b_1_24 + a_2_5·b_4_14·c_4_17 + a_2_5·b_2_42·c_4_17
       + a_2_5·b_2_42·c_4_16
  62. b_4_14·a_6_29 + a_4_15·b_4_14·b_1_22 + a_2_5·a_4_15·b_1_24 + a_2_5·b_2_4·a_6_29
       + a_2_5·b_2_4·a_4_15·b_1_22 + a_2_5·b_2_42·a_4_15 + b_4_14·c_4_17·a_1_0·b_1_2
       + a_4_15·c_4_17·b_1_22 + b_2_4·c_4_17·b_1_2·a_3_8 + b_2_4·c_4_16·b_1_2·a_3_8
       + b_2_4·a_4_15·c_4_17 + b_2_4·a_4_15·c_4_16 + b_2_4·a_4_13·c_4_16
       + a_2_5·c_4_17·b_1_24 + a_2_5·b_4_14·c_4_17 + a_2_5·b_2_42·c_4_17
       + a_2_5·b_2_42·c_4_16 + a_2_5·c_4_17·b_1_2·a_3_8 + a_2_5·c_4_16·b_1_2·a_3_8
       + a_2_5·a_4_15·c_4_17 + a_2_5·a_4_15·c_4_16
  63. a_4_13·a_6_29 + a_2_5·b_2_4·a_4_15·b_1_22
  64. a_6_29·a_5_24 + a_2_5·a_4_15·b_1_25 + a_2_5·a_4_15·b_4_14·b_1_2
       + a_2_5·b_2_4·b_4_14·a_3_7 + a_2_5·b_2_4·c_4_17·a_3_7 + a_2_5·b_2_4·c_4_16·a_3_7
  65. a_6_292

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

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_4_16, a Duflot regular element of degree 4
    2. c_4_17, a Duflot regular element of degree 4
    3. b_1_24 + b_1_14 + b_2_4·b_1_22 + b_2_42, an element of degree 4
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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

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. b_1_2 → 0, an element of degree 1
  4. a_2_5 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. a_3_8 → 0, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. a_4_13 → 0, an element of degree 4
  10. a_4_15 → 0, an element of degree 4
  11. b_4_14 → 0, an element of degree 4
  12. c_4_16 → c_1_04, an element of degree 4
  13. c_4_17 → c_1_14, an element of degree 4
  14. a_5_24 → 0, an element of degree 5
  15. a_6_29 → 0, an element of degree 6

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_5 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. a_3_8 → 0, an element of degree 3
  8. b_3_9 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. a_4_13 → 0, an element of degree 4
  10. a_4_15 → 0, an element of degree 4
  11. b_4_14 → 0, an element of degree 4
  12. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. c_4_17 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. a_5_24 → 0, an element of degree 5
  15. a_6_29 → 0, an element of degree 6

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 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_5 → 0, an element of degree 2
  5. b_2_4 → c_1_32, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. a_3_8 → 0, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. a_4_13 → 0, an element of degree 4
  10. a_4_15 → 0, an element of degree 4
  11. b_4_14 → c_1_12·c_1_32 + c_1_12·c_1_22 + c_1_02·c_1_32, an element of degree 4
  12. c_4_16 → c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. c_4_17 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_32, an element of degree 4
  14. a_5_24 → 0, an element of degree 5
  15. a_6_29 → 0, an element of degree 6

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