Cohomology of group number 971 of order 128

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

General information on the group

  • The group has 3 minimal generators and exponent 16.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t7  −  t5  −  t4  +  t3  +  t2  −  t  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-6,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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

Ring generators

The cohomology ring has 13 minimal generators of maximal degree 10:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. a_3_3, a nilpotent element of degree 3
  5. a_4_3, a nilpotent element of degree 4
  6. b_4_4, an element of degree 4
  7. a_5_6, a nilpotent element of degree 5
  8. b_5_5, an element of degree 5
  9. b_6_8, an element of degree 6
  10. a_7_8, a nilpotent element of degree 7
  11. a_8_6, a nilpotent element of degree 8
  12. c_8_11, a Duflot regular element of degree 8
  13. a_10_15, a nilpotent element of degree 10

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

Ring relations

There are 59 minimal relations of maximal degree 20:

  1. a_1_0·a_1_2
  2. a_1_0·b_1_1 + a_1_22
  3. a_1_22·b_1_1
  4. a_1_04
  5. b_1_1·a_3_3
  6. a_1_2·a_3_3
  7. a_4_3·b_1_1 + a_1_02·a_3_3
  8. a_4_3·a_1_0 + a_1_02·a_3_3
  9. a_4_3·a_1_2
  10. b_4_4·a_1_2
  11. a_3_32 + b_4_4·a_1_02
  12. a_1_2·a_5_6
  13. a_1_0·b_5_5
  14. a_4_3·a_3_3 + b_4_4·a_1_03
  15. b_1_12·a_5_6 + a_4_3·a_3_3
  16. b_6_8·a_1_0 + b_4_4·a_3_3
  17. b_6_8·a_1_2
  18. a_4_32
  19. a_1_03·a_5_6
  20. a_3_3·b_5_5
  21. b_1_1·a_7_8 + a_4_3·b_4_4 + b_4_4·a_1_0·a_3_3
  22. a_3_3·a_5_6 + a_1_0·a_7_8
  23. a_1_2·a_7_8
  24. a_4_3·b_5_5 + b_4_4·a_1_02·a_3_3
  25. b_6_8·a_3_3 + b_4_42·a_1_0 + b_4_4·a_1_02·a_3_3
  26. a_4_3·a_5_6 + a_1_02·a_7_8
  27. b_6_8·b_1_13 + b_4_4·b_5_5 + b_4_4·b_1_15 + b_4_42·b_1_1 + a_1_2·b_1_13·b_5_5
       + a_8_6·b_1_1 + a_4_3·a_5_6 + b_4_4·a_1_02·a_3_3
  28. a_8_6·a_1_0 + a_4_3·a_5_6
  29. a_8_6·a_1_2
  30. a_5_6·b_5_5 + b_4_4·b_1_1·a_5_6
  31. b_4_4·b_1_1·a_5_6 + a_4_3·b_6_8 + b_4_42·a_1_02
  32. a_3_3·a_7_8 + b_4_4·a_1_0·a_5_6
  33. a_5_62 + b_4_4·a_1_0·a_5_6 + c_8_11·a_1_02
  34. b_5_52 + b_4_4·b_1_16 + b_4_42·b_1_12 + a_1_2·b_1_14·b_5_5 + c_8_11·a_1_22
  35. b_6_8·a_5_6 + b_4_4·a_7_8 + b_4_4·a_1_02·a_5_6
  36. a_4_3·a_7_8 + b_4_4·a_1_02·a_5_6
  37. a_8_6·a_3_3 + b_4_4·a_1_02·a_5_6
  38. b_6_8·b_5_5 + b_4_4·b_1_12·b_5_5 + b_4_4·b_6_8·b_1_1 + a_10_15·b_1_1
       + b_4_4·a_1_02·a_5_6 + b_4_42·a_1_03
  39. a_10_15·a_1_0 + b_4_4·a_1_02·a_5_6 + b_4_42·a_1_03
  40. a_10_15·a_1_2
  41. b_6_82 + b_4_42·b_1_14 + b_4_43 + b_4_42·a_1_0·a_3_3
  42. b_5_5·a_7_8 + a_4_3·b_4_42 + b_4_42·a_1_0·a_3_3
  43. a_5_6·a_7_8 + b_4_4·a_1_0·a_7_8 + c_8_11·a_1_0·a_3_3
  44. a_4_3·a_8_6
  45. b_6_8·a_7_8 + b_4_42·a_5_6
  46. a_8_6·a_5_6 + b_4_4·a_1_02·a_7_8 + c_8_11·a_1_02·a_3_3
  47. a_10_15·a_3_3 + b_4_4·a_1_02·a_7_8 + b_4_42·a_1_02·a_3_3
  48. a_10_15·b_1_13 + a_8_6·b_5_5 + b_4_4·a_8_6·b_1_1
  49. a_7_82 + b_4_42·a_1_0·a_5_6 + b_4_4·c_8_11·a_1_02
  50. b_6_8·a_8_6 + b_4_4·a_10_15 + b_4_4·a_8_6·b_1_12 + b_4_43·a_1_02
  51. a_4_3·a_10_15
  52. a_8_6·a_7_8 + b_4_42·a_1_02·a_5_6 + b_4_4·c_8_11·a_1_03
  53. a_10_15·a_5_6 + b_4_4·c_8_11·a_1_03
  54. a_10_15·b_5_5 + b_4_4·a_10_15·b_1_1 + b_4_4·a_8_6·b_1_13
  55. a_8_62
  56. b_6_8·a_10_15 + b_4_4·a_10_15·b_1_12 + b_4_42·a_8_6 + b_4_43·a_1_0·a_3_3
  57. a_10_15·a_7_8 + b_4_4·c_8_11·a_1_02·a_3_3
  58. a_8_6·a_10_15
  59. a_10_152

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 20.
  • 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_11, a Duflot regular element of degree 8
    2. b_1_14 + b_4_4, an element of degree 4
    3. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 2, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. a_4_3 → 0, an element of degree 4
  6. b_4_4 → 0, an element of degree 4
  7. a_5_6 → 0, an element of degree 5
  8. b_5_5 → 0, an element of degree 5
  9. b_6_8 → 0, an element of degree 6
  10. a_7_8 → 0, an element of degree 7
  11. a_8_6 → 0, an element of degree 8
  12. c_8_11 → c_1_08, an element of degree 8
  13. a_10_15 → 0, an element of degree 10

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_1, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. a_4_3 → 0, an element of degree 4
  6. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  7. a_5_6 → 0, an element of degree 5
  8. b_5_5 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  9. b_6_8 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_14·c_1_22, an element of degree 6
  10. a_7_8 → 0, an element of degree 7
  11. a_8_6 → 0, an element of degree 8
  12. c_8_11 → c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
       + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
  13. a_10_15 → 0, an element of degree 10

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