Cohomology of group number 742 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 1.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t5  −  3·t4  +  3·t3  −  2·t2  +  t  −  1)
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-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 14 minimal generators of maximal degree 8:

  1. b_1_0, an 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_3, a nilpotent element of degree 2
  5. a_2_4, a nilpotent element of degree 2
  6. b_2_5, an element of degree 2
  7. b_2_6, an element of degree 2
  8. b_3_11, an element of degree 3
  9. b_5_23, an element of degree 5
  10. b_5_24, an element of degree 5
  11. b_5_25, an element of degree 5
  12. a_6_26, a nilpotent element of degree 6
  13. a_6_24, a nilpotent element of degree 6
  14. c_8_66, a Duflot regular element of degree 8

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

Ring relations

There are 53 minimal relations of maximal degree 12:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_1_1·b_1_2
  4. a_2_3·b_1_1
  5. a_2_4·b_1_2
  6. a_2_4·b_1_0 + a_2_3·b_1_0
  7. b_2_6·b_1_0 + b_2_5·b_1_1
  8. a_2_32
  9. a_2_3·a_2_4
  10. a_2_42
  11. b_1_0·b_3_11 + a_2_4·b_2_5
  12. b_1_1·b_3_11 + a_2_4·b_2_6
  13. b_2_5·b_2_6·b_1_1 + b_2_52·b_1_1
  14. a_2_4·b_3_11
  15. b_1_2·b_5_23 + b_2_5·b_2_6·b_1_22 + a_2_4·b_2_5·b_2_6 + a_2_4·b_2_52
       + a_2_3·b_1_24 + a_2_3·b_2_5·b_1_22 + a_2_3·b_2_5·b_2_6 + a_2_3·b_2_52
  16. b_1_1·b_5_23 + a_2_4·b_2_6·b_1_12
  17. b_3_112 + b_1_2·b_5_24 + b_2_6·b_1_2·b_3_11 + b_2_5·b_1_2·b_3_11 + b_2_5·b_2_62
       + b_2_52·b_2_6 + a_2_4·b_2_52 + a_2_3·b_1_2·b_3_11 + a_2_3·b_2_5·b_2_6
       + a_2_3·b_2_52
  18. b_1_0·b_5_24 + b_1_0·b_5_23 + a_2_4·b_2_5·b_2_6 + a_2_3·b_1_04
  19. b_1_1·b_5_24 + a_2_4·b_2_5·b_2_6
  20. b_1_2·b_5_25 + b_2_5·b_2_6·b_1_22 + a_2_3·b_2_6·b_1_22 + a_2_3·b_2_62
       + a_2_3·b_2_5·b_1_22 + a_2_3·b_2_5·b_2_6
  21. b_1_0·b_5_25 + a_2_3·b_1_04 + a_2_3·b_2_5·b_1_02
  22. a_2_4·b_5_23 + a_2_3·b_5_23 + a_2_3·b_2_5·b_2_6·b_1_2
  23. a_2_4·b_5_24 + a_2_3·b_5_23 + a_2_3·b_2_5·b_2_6·b_1_2
  24. b_2_6·b_5_23 + b_2_5·b_5_25 + b_2_5·b_2_62·b_1_2 + b_2_52·b_2_6·b_1_2 + b_2_53·b_1_1
       + a_2_4·b_2_62·b_1_1 + a_2_3·b_2_6·b_1_23 + a_2_3·b_2_5·b_1_03
       + a_2_3·b_2_52·b_1_2 + a_2_3·b_2_52·b_1_0
  25. a_2_3·b_5_25 + a_2_3·b_2_5·b_2_6·b_1_2
  26. a_6_26·b_1_2 + a_2_3·b_5_24 + a_2_3·b_5_23 + a_2_3·b_1_22·b_3_11 + a_2_3·b_1_25
       + a_2_3·b_2_6·b_1_23 + a_2_3·b_2_62·b_1_2 + a_2_3·b_2_5·b_3_11
       + a_2_3·b_2_5·b_1_23 + a_2_3·b_2_5·b_2_6·b_1_2 + a_2_3·b_2_52·b_1_2
  27. a_6_26·b_1_0 + a_2_3·b_5_23 + a_2_3·b_1_05 + a_2_3·b_2_5·b_1_03
       + a_2_3·b_2_5·b_2_6·b_1_2
  28. a_6_26·b_1_1
  29. a_6_24·b_1_2 + a_2_3·b_1_22·b_3_11 + a_2_3·b_2_6·b_3_11 + a_2_3·b_2_6·b_1_23
       + a_2_3·b_2_5·b_3_11 + a_2_3·b_2_5·b_2_6·b_1_2
  30. a_6_24·b_1_0 + a_2_3·b_5_23 + a_2_3·b_1_05 + a_2_3·b_2_5·b_1_03
       + a_2_3·b_2_5·b_2_6·b_1_2 + a_2_3·b_2_52·b_1_0
  31. a_6_24·b_1_1 + a_2_4·b_5_25 + a_2_4·b_2_6·b_1_13
  32. a_2_3·a_6_26
  33. a_2_4·a_6_26
  34. b_3_11·b_5_23 + b_2_5·b_2_6·b_1_2·b_3_11 + b_2_5·a_6_24 + a_2_4·b_2_52·b_2_6
       + a_2_4·b_2_53 + a_2_3·b_1_23·b_3_11 + a_2_3·b_2_5·b_1_04
       + a_2_3·b_2_5·b_2_6·b_1_22 + a_2_3·b_2_52·b_1_02 + a_2_3·b_2_52·b_2_6
  35. b_3_11·b_5_25 + b_2_5·b_2_6·b_1_2·b_3_11 + b_2_6·a_6_24 + a_2_4·b_2_62·b_1_12
       + a_2_4·b_2_52·b_2_6 + a_2_3·b_2_62·b_1_22 + a_2_3·b_2_5·b_1_2·b_3_11
       + a_2_3·b_2_5·b_2_62
  36. a_2_3·a_6_24
  37. a_2_4·a_6_24
  38. b_2_5·b_2_6·b_5_25 + b_2_52·b_5_25 + b_2_52·b_2_62·b_1_2 + b_2_53·b_2_6·b_1_2
       + a_6_24·b_3_11 + a_2_3·b_1_22·b_5_24 + a_2_3·b_2_6·b_5_24 + a_2_3·b_2_62·b_3_11
       + a_2_3·b_2_5·b_5_24 + a_2_3·b_2_5·b_5_23 + a_2_3·b_2_5·b_1_22·b_3_11
       + a_2_3·b_2_5·b_2_6·b_3_11 + a_2_3·b_2_52·b_3_11 + a_2_3·b_2_52·b_1_03
       + a_2_3·b_2_53·b_1_2 + a_2_3·b_2_53·b_1_0
  39. b_5_23·b_5_25 + b_2_52·b_2_62·b_1_22 + a_2_4·b_2_6·b_1_1·b_5_25
       + a_2_3·b_1_03·b_5_23 + a_2_3·b_2_5·b_1_0·b_5_23 + a_2_3·b_2_5·b_2_6·b_1_24
       + a_2_3·b_2_5·b_2_62·b_1_22 + a_2_3·b_2_5·b_2_63 + a_2_3·b_2_53·b_2_6
  40. b_5_23·b_5_24 + b_5_232 + b_2_5·b_2_6·b_1_2·b_5_24 + b_2_52·b_2_62·b_1_22
       + b_2_5·b_2_6·a_6_26 + b_2_52·a_6_24 + b_2_52·a_6_26 + a_2_3·b_1_23·b_5_24
       + a_2_3·b_1_03·b_5_23 + a_2_3·b_2_5·b_1_2·b_5_24 + a_2_3·b_2_5·b_2_6·b_1_2·b_3_11
       + a_2_3·b_2_5·b_2_6·b_1_24 + a_2_3·b_2_5·b_2_62·b_1_22 + a_2_3·b_2_5·b_2_63
       + a_2_3·b_2_52·b_1_24 + a_2_3·b_2_52·b_2_6·b_1_22 + a_2_3·b_2_52·b_2_62
       + a_2_3·b_2_53·b_1_22 + a_2_3·b_2_54
  41. b_5_24·b_5_25 + b_2_5·b_2_6·b_1_2·b_5_24 + b_2_62·a_6_26 + b_2_5·b_2_6·a_6_24
       + b_2_5·b_2_6·a_6_26 + a_2_4·b_2_53·b_2_6 + a_2_3·b_1_03·b_5_23
       + a_2_3·b_2_6·b_1_2·b_5_24 + a_2_3·b_2_62·b_1_2·b_3_11 + a_2_3·b_2_62·b_1_24
       + a_2_3·b_2_63·b_1_22 + a_2_3·b_2_64 + a_2_3·b_2_5·b_1_2·b_5_24
       + a_2_3·b_2_5·b_1_0·b_5_23 + a_2_3·b_2_5·b_2_6·b_1_24
       + a_2_3·b_2_5·b_2_62·b_1_22 + a_2_3·b_2_5·b_2_63 + a_2_3·b_2_52·b_2_6·b_1_22
       + a_2_3·b_2_53·b_2_6
  42. b_5_242 + b_5_232 + b_1_22·b_3_11·b_5_24 + b_1_25·b_5_24 + b_1_27·b_3_11
       + b_2_6·b_1_23·b_5_24 + b_2_62·b_1_23·b_3_11 + b_2_5·b_1_25·b_3_11
       + b_2_5·b_2_6·b_1_2·b_5_24 + b_2_5·b_2_62·b_1_24 + b_2_5·b_2_63·b_1_22
       + b_2_52·b_1_23·b_3_11 + b_2_52·b_2_6·b_1_24 + b_2_52·b_2_63
       + b_2_53·b_2_6·b_1_22 + b_2_53·b_2_62 + a_2_4·b_2_53·b_2_6
       + a_2_3·b_1_23·b_5_24 + a_2_3·b_2_6·b_1_23·b_3_11 + a_2_3·b_2_6·b_1_26
       + a_2_3·b_2_63·b_1_22 + a_2_3·b_2_5·b_1_2·b_5_24 + a_2_3·b_2_5·b_1_23·b_3_11
       + a_2_3·b_2_5·b_2_6·b_1_24 + a_2_3·b_2_5·b_2_62·b_1_22 + a_2_3·b_2_52·b_1_24
       + a_2_3·b_2_52·b_2_6·b_1_22 + a_2_3·b_2_52·b_2_62 + a_2_3·b_2_53·b_1_22
       + a_2_3·b_2_53·b_2_6 + c_8_66·b_1_22
  43. b_5_232 + b_2_52·b_1_0·b_5_23 + b_2_52·b_2_62·b_1_22 + a_2_4·b_2_53·b_2_6
       + a_2_4·b_2_54 + a_2_3·b_2_5·b_1_0·b_5_23 + a_2_3·b_2_5·b_1_06
       + a_2_3·b_2_53·b_1_02 + c_8_66·b_1_02
  44. b_5_252 + b_2_6·b_1_13·b_5_25 + b_2_62·b_1_1·b_5_25 + b_2_52·b_2_62·b_1_22
       + a_2_4·b_1_13·b_5_25 + a_2_4·b_2_6·b_1_16 + a_2_4·b_2_62·b_1_14
       + a_2_4·b_2_63·b_1_12 + a_2_4·b_2_64 + a_2_4·b_2_53·b_2_6 + c_8_66·b_1_12
  45. a_6_26·b_5_25 + a_2_3·b_2_5·b_2_6·b_5_24 + a_2_3·b_2_5·b_2_6·b_1_22·b_3_11
       + a_2_3·b_2_5·b_2_6·b_1_25 + a_2_3·b_2_5·b_2_62·b_1_23
       + a_2_3·b_2_5·b_2_63·b_1_2 + a_2_3·b_2_52·b_2_6·b_3_11
       + a_2_3·b_2_52·b_2_6·b_1_23 + a_2_3·b_2_53·b_2_6·b_1_2
  46. a_6_24·b_5_23 + a_6_26·b_5_23 + a_2_3·b_2_5·b_2_6·b_5_24 + a_2_3·b_2_5·b_2_6·b_1_25
       + a_2_3·b_2_5·b_2_62·b_3_11 + a_2_3·b_2_5·b_2_63·b_1_2 + a_2_3·b_2_52·b_5_23
       + a_2_3·b_2_52·b_2_6·b_1_23 + a_2_3·b_2_52·b_2_62·b_1_2
  47. a_6_24·b_5_24 + a_6_26·b_5_23 + b_2_6·a_6_26·b_3_11 + b_2_5·a_6_24·b_3_11
       + b_2_5·a_6_26·b_3_11 + a_2_3·b_1_2·b_3_11·b_5_24 + a_2_3·b_2_6·b_1_24·b_3_11
       + a_2_3·b_2_63·b_3_11 + a_2_3·b_2_5·b_1_24·b_3_11
       + a_2_3·b_2_5·b_2_6·b_1_22·b_3_11 + a_2_3·b_2_5·b_2_6·b_1_25
       + a_2_3·b_2_5·b_2_62·b_3_11 + a_2_3·b_2_5·b_2_62·b_1_23 + a_2_3·b_2_52·b_5_23
       + a_2_3·b_2_52·b_1_22·b_3_11 + a_2_3·b_2_52·b_2_6·b_3_11
       + a_2_3·b_2_52·b_2_6·b_1_23 + a_2_3·b_2_52·b_2_62·b_1_2 + a_2_3·b_2_53·b_3_11
  48. a_6_24·b_5_24 + a_6_26·b_5_24 + b_2_6·a_6_26·b_3_11 + a_2_3·b_1_2·b_3_11·b_5_24
       + a_2_3·b_1_26·b_3_11 + a_2_3·b_2_6·b_1_24·b_3_11 + a_2_3·b_2_62·b_5_24
       + a_2_3·b_2_62·b_1_22·b_3_11 + a_2_3·b_2_63·b_3_11 + a_2_3·b_2_5·b_1_22·b_5_24
       + a_2_3·b_2_5·b_1_24·b_3_11 + a_2_3·b_2_5·b_2_6·b_5_24 + a_2_3·b_2_5·b_2_62·b_3_11
       + a_2_3·b_2_5·b_2_62·b_1_23 + a_2_3·b_2_52·b_5_24 + a_2_3·b_2_52·b_1_22·b_3_11
       + a_2_3·b_2_52·b_2_6·b_1_23 + a_2_3·b_2_53·b_2_6·b_1_2 + a_2_3·c_8_66·b_1_2
  49. a_6_26·b_5_23 + a_2_3·b_1_04·b_5_23 + a_2_3·b_2_5·b_1_02·b_5_23
       + a_2_3·b_2_5·b_2_6·b_5_24 + a_2_3·b_2_5·b_2_6·b_1_22·b_3_11
       + a_2_3·b_2_5·b_2_6·b_1_25 + a_2_3·b_2_5·b_2_62·b_1_23
       + a_2_3·b_2_5·b_2_63·b_1_2 + a_2_3·b_2_52·b_5_23 + a_2_3·b_2_52·b_2_6·b_3_11
       + a_2_3·b_2_52·b_2_6·b_1_23 + a_2_3·c_8_66·b_1_0
  50. a_6_24·b_5_25 + a_2_4·b_2_62·b_5_25 + a_2_3·b_2_5·b_2_6·b_1_22·b_3_11
       + a_2_3·b_2_5·b_2_62·b_3_11 + a_2_3·b_2_5·b_2_62·b_1_23
       + a_2_3·b_2_52·b_2_6·b_3_11 + a_2_3·b_2_52·b_2_62·b_1_2 + a_2_4·c_8_66·b_1_1
  51. a_6_262
  52. a_6_242
  53. a_6_26·a_6_24

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 17.
  • However, the last relation was already found in degree 12 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_66, a Duflot regular element of degree 8
    2. b_1_24 + b_1_14 + b_1_04 + b_2_62 + b_2_5·b_2_6 + b_2_52, an element of degree 4
    3. b_2_62·b_1_22 + b_2_62·b_1_12 + b_2_5·b_2_6·b_1_22 + b_2_5·b_2_62
         + b_2_52·b_1_22 + b_2_52·b_1_02 + b_2_52·b_2_6, an element of degree 6
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 14, 16].
  • 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 1

  1. b_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_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → 0, an element of degree 2
  8. b_3_11 → 0, an element of degree 3
  9. b_5_23 → 0, an element of degree 5
  10. b_5_24 → 0, an element of degree 5
  11. b_5_25 → 0, an element of degree 5
  12. a_6_26 → 0, an element of degree 6
  13. a_6_24 → 0, an element of degree 6
  14. c_8_66 → c_1_08, an element of degree 8

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

  1. b_1_0 → c_1_1, 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_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  7. b_2_6 → 0, an element of degree 2
  8. b_3_11 → 0, an element of degree 3
  9. b_5_23 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_24 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. b_5_25 → 0, an element of degree 5
  12. a_6_26 → 0, an element of degree 6
  13. a_6_24 → 0, an element of degree 6
  14. c_8_66 → 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

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  8. b_3_11 → 0, an element of degree 3
  9. b_5_23 → 0, an element of degree 5
  10. b_5_24 → 0, an element of degree 5
  11. b_5_25 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  12. a_6_26 → 0, an element of degree 6
  13. a_6_24 → 0, an element of degree 6
  14. c_8_66 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_1, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  7. b_2_6 → c_1_32 + c_1_1·c_1_3, an element of degree 2
  8. b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1, an element of degree 3
  9. b_5_23 → c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3
       + c_1_13·c_1_2·c_1_3, an element of degree 5
  10. b_5_24 → c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_1·c_1_22·c_1_32 + c_1_02·c_1_1·c_1_32
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_3 + c_1_02·c_1_12·c_1_2
       + c_1_04·c_1_1, an element of degree 5
  11. b_5_25 → c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3
       + c_1_13·c_1_2·c_1_3, an element of degree 5
  12. a_6_26 → 0, an element of degree 6
  13. a_6_24 → 0, an element of degree 6
  14. c_8_66 → c_1_24·c_1_34 + c_1_1·c_1_24·c_1_33 + c_1_1·c_1_25·c_1_32
       + c_1_12·c_1_23·c_1_33 + c_1_13·c_1_2·c_1_34 + c_1_13·c_1_22·c_1_33
       + c_1_13·c_1_23·c_1_32 + c_1_14·c_1_22·c_1_32 + c_1_15·c_1_22·c_1_3
       + c_1_16·c_1_2·c_1_3 + c_1_02·c_1_22·c_1_34 + c_1_02·c_1_24·c_1_32
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_2·c_1_32
       + c_1_02·c_1_13·c_1_22·c_1_3 + c_1_02·c_1_14·c_1_32
       + c_1_02·c_1_14·c_1_2·c_1_3 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_3
       + c_1_02·c_1_16 + c_1_04·c_1_34 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_32 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14
       + c_1_06·c_1_12 + c_1_08, 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