Cohomology of group number 242 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 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 all of rank 4.

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
    t4  +  t3  +  t2  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-4,-4. 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 15 minimal generators of maximal degree 5:

  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_9, a nilpotent element of degree 3
  7. b_3_7, an element of degree 3
  8. b_3_8, an element of degree 3
  9. b_3_10, an element of degree 3
  10. b_4_16, an element of degree 4
  11. b_4_17, an element of degree 4
  12. c_4_18, a Duflot regular element of degree 4
  13. c_4_19, a Duflot regular element of degree 4
  14. b_5_30, an element of degree 5
  15. b_5_31, an element of degree 5

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 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·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. b_2_4·a_1_0·b_1_2 + a_2_52
  8. a_1_0·a_3_9
  9. b_1_1·a_3_9
  10. a_1_0·b_3_7
  11. b_1_1·b_3_7
  12. a_1_0·b_3_8
  13. a_1_0·b_3_10
  14. b_1_22·a_3_9 + a_2_5·b_3_7
  15. b_1_22·a_3_9 + a_2_5·b_3_8
  16. b_2_4·b_3_8 + b_2_4·b_3_7 + b_1_22·a_3_9
  17. a_2_5·b_3_10
  18. b_2_4·b_3_10 + a_2_5·a_3_9
  19. b_4_16·a_1_0
  20. b_1_22·b_3_10 + b_4_16·b_1_1
  21. b_4_17·a_1_0 + a_2_5·a_3_9
  22. b_4_17·b_1_1
  23. a_3_92 + a_2_52·b_2_4
  24. a_3_9·b_3_7 + a_2_5·b_2_4·b_1_22 + a_2_5·b_1_2·a_3_9
  25. b_3_72 + b_2_4·b_1_24
  26. a_3_9·b_3_8 + a_2_5·b_2_4·b_1_22 + a_2_5·b_1_2·a_3_9
  27. b_3_7·b_3_8 + b_2_4·b_1_24 + a_2_5·b_1_24
  28. a_3_9·b_3_10
  29. b_3_7·b_3_10
  30. b_3_102 + c_4_18·b_1_12
  31. b_3_82 + b_1_1·b_1_22·b_3_8 + b_1_12·b_1_2·b_3_10 + b_2_4·b_1_24 + c_4_19·b_1_12
  32. b_2_4·b_1_2·a_3_9 + a_2_5·b_4_17 + a_2_52·b_2_4
  33. b_4_17·b_1_22 + b_2_4·b_1_2·b_3_7 + a_2_5·b_4_16
  34. a_1_0·b_5_30 + a_2_5·b_1_2·a_3_9
  35. b_3_82 + b_1_1·b_5_30 + b_1_1·b_1_22·b_3_8 + b_1_12·b_1_2·b_3_10
       + b_4_16·b_1_1·b_1_2 + b_4_16·b_1_12 + b_2_4·b_1_24
  36. a_1_0·b_5_31 + a_2_5·b_1_2·a_3_9 + c_4_18·a_1_0·b_1_2
  37. b_3_8·b_3_10 + b_1_1·b_5_31 + b_4_16·b_1_12 + c_4_18·b_1_1·b_1_2
  38. b_4_16·b_3_10 + c_4_18·b_1_1·b_1_22
  39. b_4_17·a_3_9 + a_2_5·b_2_42·b_1_2 + a_2_5·b_2_4·a_3_9
  40. b_4_17·b_3_10
  41. b_4_17·b_3_7 + b_2_42·b_1_23 + b_4_16·a_3_9
  42. b_4_17·b_3_8 + b_2_42·b_1_23 + b_4_16·a_3_9 + a_2_5·b_2_4·b_1_23
  43. b_4_16·a_3_9 + a_2_5·b_5_30 + a_2_5·b_1_22·b_3_7 + a_2_5·b_4_16·b_1_2
       + a_2_5·b_2_4·b_3_7 + a_2_5·b_2_4·b_1_23
  44. b_1_22·b_5_30 + b_1_24·b_3_7 + b_4_16·b_3_7 + b_4_16·b_1_23 + b_4_16·b_1_1·b_1_22
       + b_2_4·b_1_22·b_3_7 + b_2_4·b_1_25 + a_2_5·b_2_4·b_1_23 + c_4_19·b_1_1·b_1_22
  45. b_4_16·a_3_9 + a_2_5·b_5_31 + a_2_5·b_2_4·b_3_7 + a_2_5·b_2_4·b_1_23
       + a_2_5·c_4_18·b_1_2
  46. b_1_22·b_5_31 + b_4_16·b_3_8 + b_4_16·b_1_1·b_1_22 + b_2_4·b_1_22·b_3_7
       + b_2_4·b_1_25 + c_4_18·b_1_23
  47. b_2_4·b_5_31 + b_2_4·b_5_30 + b_2_4·b_1_22·b_3_7 + b_2_4·b_4_16·b_1_2 + b_4_16·a_3_9
       + a_2_5·b_2_42·b_1_2 + b_2_4·c_4_18·b_1_2 + b_2_4·c_4_19·a_1_0
  48. b_4_162 + b_2_4·b_1_23·b_3_7 + b_2_4·b_1_26 + a_2_5·b_2_4·b_1_24
       + c_4_18·b_1_24 + a_2_52·c_4_19
  49. b_4_172 + b_2_43·b_1_22 + a_2_52·b_2_42 + a_2_52·c_4_18
  50. b_3_10·b_5_30 + c_4_19·b_1_1·b_3_10 + c_4_18·b_1_1·b_1_23 + c_4_18·b_1_12·b_1_22
  51. b_3_7·b_5_30 + b_4_16·b_1_2·b_3_7 + b_2_4·b_1_23·b_3_7 + b_2_4·b_1_26
       + b_2_4·b_4_16·b_1_22 + b_2_42·b_1_24 + a_2_5·b_2_4·b_1_2·b_3_7
  52. b_4_16·b_4_17 + b_2_4·b_1_2·b_5_30 + b_2_4·b_1_23·b_3_7 + b_2_4·b_4_16·b_1_22
       + b_2_42·b_1_2·b_3_7 + b_2_42·b_1_24 + a_2_5·b_2_4·b_1_2·b_3_7
       + a_2_5·b_2_4·b_1_24 + a_2_5·b_2_42·b_1_22 + a_2_5·c_4_18·b_1_22
       + a_2_52·c_4_19
  53. a_3_9·b_5_30 + a_2_5·b_1_2·b_5_30 + a_2_5·b_1_23·b_3_7 + a_2_5·b_4_16·b_1_22
       + a_2_5·b_2_4·b_4_16 + a_2_5·b_2_42·b_1_22
  54. b_3_8·b_5_30 + b_4_16·b_1_2·b_3_8 + b_4_16·b_1_1·b_3_8 + b_2_4·b_1_23·b_3_7
       + b_2_4·b_1_26 + b_2_4·b_4_16·b_1_22 + b_2_42·b_1_24 + a_2_5·b_1_23·b_3_7
       + a_2_5·b_1_26 + a_2_5·b_4_16·b_1_22 + a_2_5·b_2_4·b_1_2·b_3_7
       + a_2_5·b_2_4·b_1_24 + c_4_19·b_1_1·b_3_8
  55. a_3_9·b_5_31 + a_2_5·b_2_4·b_1_2·b_3_7 + a_2_5·b_2_4·b_4_16 + a_2_5·b_2_42·b_1_22
       + a_2_52·b_4_17 + c_4_18·b_1_2·a_3_9
  56. b_3_10·b_5_31 + c_4_18·b_1_2·b_3_10 + c_4_18·b_1_1·b_3_8 + c_4_18·b_1_12·b_1_22
  57. b_3_7·b_5_31 + b_2_4·b_1_23·b_3_7 + b_2_4·b_4_16·b_1_22 + b_2_42·b_1_24
       + a_2_5·b_4_16·b_1_22 + c_4_18·b_1_2·b_3_7
  58. b_3_8·b_5_31 + b_2_4·b_1_23·b_3_7 + b_2_4·b_4_16·b_1_22 + b_2_42·b_1_24
       + a_2_5·b_1_23·b_3_7 + a_2_5·b_2_4·b_1_24 + c_4_19·b_1_1·b_3_10 + c_4_18·b_1_2·b_3_8
       + c_4_18·b_1_13·b_1_2
  59. b_4_17·b_5_30 + b_2_4·b_4_16·b_3_7 + b_2_42·b_1_22·b_3_7 + b_2_42·b_1_25
       + b_2_42·b_4_16·b_1_2 + b_2_43·b_1_23 + a_2_5·b_4_16·b_3_7 + a_2_5·b_2_4·b_5_30
       + a_2_5·b_2_4·b_1_22·b_3_7 + a_2_5·b_2_4·b_1_25 + a_2_5·c_4_18·b_3_7
       + a_2_5·c_4_18·b_1_23 + a_2_5·c_4_19·a_3_9
  60. b_4_16·b_5_30 + b_4_16·b_1_22·b_3_7 + b_2_4·b_1_27 + b_2_4·b_4_16·b_3_7
       + b_2_4·b_4_16·b_1_23 + b_2_42·b_1_25 + a_2_5·b_2_4·b_1_22·b_3_7
       + a_2_5·b_2_4·b_1_25 + a_2_5·b_2_4·b_4_16·b_1_2 + c_4_18·b_1_22·b_3_7
       + c_4_18·b_1_25 + c_4_18·b_1_1·b_1_24 + b_4_16·c_4_19·b_1_1 + a_2_5·c_4_19·a_3_9
  61. b_4_17·b_5_31 + b_2_42·b_1_22·b_3_7 + b_2_42·b_4_16·b_1_2 + b_2_43·b_1_23
       + a_2_5·b_2_4·b_5_30 + a_2_5·b_2_4·b_4_16·b_1_2 + a_2_5·b_2_42·b_3_7
       + b_4_17·c_4_18·b_1_2 + a_2_5·c_4_18·b_3_7
  62. b_4_16·b_5_31 + b_2_4·b_1_24·b_3_7 + b_2_4·b_4_16·b_3_7 + b_2_4·b_4_16·b_1_23
       + b_2_42·b_1_25 + a_2_5·b_1_24·b_3_7 + a_2_5·b_2_4·b_1_22·b_3_7
       + a_2_5·b_2_4·b_1_25 + c_4_18·b_1_22·b_3_8 + c_4_18·b_1_1·b_1_24
       + b_4_16·c_4_18·b_1_2 + a_2_5·c_4_19·a_3_9
  63. b_5_302 + b_2_4·b_1_25·b_3_7 + b_2_42·b_1_23·b_3_7 + b_2_43·b_1_24
       + a_2_5·b_2_4·b_1_26 + a_2_5·b_2_42·b_1_24 + c_4_18·b_1_26
       + c_4_18·b_1_12·b_1_24 + b_2_4·c_4_18·b_1_24 + a_2_52·b_2_4·c_4_19
       + c_4_192·b_1_12
  64. b_5_312 + b_2_42·b_1_23·b_3_7 + b_2_43·b_1_24 + a_2_5·b_2_42·b_1_24
       + c_4_18·b_1_1·b_1_22·b_3_8 + c_4_18·b_1_12·b_1_2·b_3_10 + c_4_18·b_1_12·b_1_24
       + b_2_4·c_4_18·b_1_24 + a_2_52·b_2_4·c_4_19 + c_4_18·c_4_19·b_1_12
       + c_4_182·b_1_22
  65. b_5_30·b_5_31 + b_2_4·b_4_16·b_1_2·b_3_7 + b_2_42·b_1_23·b_3_7 + b_2_43·b_1_24
       + a_2_5·b_1_25·b_3_7 + a_2_5·b_4_16·b_1_2·b_3_7 + a_2_5·b_4_16·b_1_24
       + a_2_5·b_2_4·b_1_2·b_5_30 + a_2_5·b_2_4·b_1_23·b_3_7 + a_2_5·b_2_42·b_1_24
       + c_4_19·b_1_1·b_5_31 + c_4_18·b_1_2·b_5_30 + c_4_18·b_1_23·b_3_8
       + c_4_18·b_1_1·b_1_22·b_3_8 + c_4_18·b_1_1·b_1_25 + c_4_18·b_1_12·b_1_24
       + b_2_4·c_4_18·b_1_24 + a_2_5·c_4_18·b_1_24 + a_2_52·b_2_4·c_4_19
       + c_4_18·c_4_19·b_1_1·b_1_2

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 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_18, a Duflot regular element of degree 4
    2. c_4_19, a Duflot regular element of degree 4
    3. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_4·b_1_22 + b_2_42, an element of degree 4
    4. b_3_7 + b_1_1·b_1_22 + b_1_12·b_1_2 + b_2_4·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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_9 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_3_8 → 0, an element of degree 3
  9. b_3_10 → 0, an element of degree 3
  10. b_4_16 → 0, an element of degree 4
  11. b_4_17 → 0, an element of degree 4
  12. c_4_18 → c_1_04, an element of degree 4
  13. c_4_19 → c_1_14 + c_1_04, an element of degree 4
  14. b_5_30 → 0, an element of degree 5
  15. b_5_31 → 0, an element of degree 5

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. b_1_2 → c_1_3, 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_9 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  10. b_4_16 → c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32, an element of degree 4
  11. b_4_17 → 0, an element of degree 4
  12. c_4_18 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. c_4_19 → c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_22 + c_1_14
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. b_5_30 → c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_23 + c_1_14·c_1_2
       + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_33 + c_1_02·c_1_22·c_1_3
       + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  15. b_5_31 → c_1_0·c_1_22·c_1_32 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_23
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_3 + c_1_04·c_1_2, an element of degree 5

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_9 → 0, an element of degree 3
  7. b_3_7 → c_1_22·c_1_3, an element of degree 3
  8. b_3_8 → c_1_22·c_1_3, an element of degree 3
  9. b_3_10 → 0, an element of degree 3
  10. b_4_16 → c_1_23·c_1_3 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22, an element of degree 4
  11. b_4_17 → c_1_2·c_1_33, an element of degree 4
  12. c_4_18 → c_1_2·c_1_33 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. c_4_19 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. b_5_30 → c_1_22·c_1_33 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_23·c_1_3
       + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_23, an element of degree 5
  15. b_5_31 → c_1_0·c_1_22·c_1_32 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3
       + c_1_04·c_1_2, 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