Cohomology of group number 387 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 of rank 3 and 4, respectively.

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
    t2  −  t  +  1
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-6,-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 13 minimal generators of maximal degree 6:

  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_4, a nilpotent element of degree 2
  5. b_3_5, an element of degree 3
  6. b_3_6, an element of degree 3
  7. b_3_7, an element of degree 3
  8. b_4_10, an element of degree 4
  9. b_4_11, an element of degree 4
  10. c_4_12, a Duflot regular element of degree 4
  11. c_4_13, a Duflot regular element of degree 4
  12. b_5_21, an element of degree 5
  13. b_6_29, an 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 52 minimal relations of maximal degree 12:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. a_2_4·b_1_2
  4. a_2_4·b_1_0
  5. a_2_4·b_1_1
  6. a_2_42
  7. b_1_2·b_3_5
  8. b_1_1·b_3_5
  9. b_1_2·b_3_6
  10. b_1_1·b_3_6
  11. b_1_0·b_3_7 + b_1_0·b_3_6
  12. a_2_4·b_3_5
  13. a_2_4·b_3_6
  14. a_2_4·b_3_7
  15. b_1_02·b_3_6 + b_4_10·b_1_0
  16. b_4_11·b_1_0
  17. b_1_22·b_3_7 + b_1_1·b_1_2·b_3_7 + b_4_11·b_1_1 + b_4_10·b_1_2
  18. b_3_5·b_3_7 + b_3_5·b_3_6
  19. b_3_6·b_3_7 + b_3_62
  20. b_3_52 + c_4_12·b_1_02
  21. a_2_4·b_4_10
  22. b_3_62 + b_4_10·b_1_02 + c_4_13·b_1_02
  23. b_3_72 + b_3_62 + b_4_11·b_1_22 + b_4_10·b_1_22 + b_4_10·b_1_1·b_1_2
       + b_4_10·b_1_12 + c_4_13·b_1_12 + c_4_12·b_1_22
  24. a_2_4·b_4_11
  25. b_1_2·b_5_21 + b_4_10·b_1_22 + b_4_10·b_1_1·b_1_2 + c_4_13·b_1_22
       + c_4_13·b_1_1·b_1_2 + c_4_12·b_1_22
  26. b_3_62 + b_3_5·b_3_6 + b_1_0·b_5_21
  27. b_1_1·b_5_21 + b_4_10·b_1_1·b_1_2 + b_4_10·b_1_12 + c_4_13·b_1_1·b_1_2
       + c_4_13·b_1_12 + c_4_12·b_1_1·b_1_2
  28. b_4_10·b_3_6 + b_4_10·b_1_03 + c_4_13·b_1_03
  29. b_4_11·b_3_5
  30. b_4_11·b_3_6
  31. a_2_4·b_5_21
  32. b_1_02·b_5_21 + b_4_10·b_3_5 + b_4_10·b_1_03 + c_4_13·b_1_03
  33. b_6_29·b_1_2 + b_4_11·b_3_7 + b_4_11·b_1_12·b_1_2 + b_4_10·b_1_23
       + b_4_10·b_1_1·b_1_22 + b_4_10·b_1_12·b_1_2 + c_4_13·b_1_23 + c_4_13·b_1_12·b_1_2
       + c_4_12·b_1_23 + c_4_12·b_1_1·b_1_22 + c_4_12·b_1_12·b_1_2
  34. b_6_29·b_1_0 + b_4_10·b_3_5 + b_4_10·b_1_03 + c_4_13·b_1_03 + c_4_12·b_1_03
  35. b_6_29·b_1_1 + b_4_11·b_1_23 + b_4_11·b_1_1·b_1_22 + b_4_11·b_1_13 + b_4_10·b_3_7
       + b_4_10·b_1_23 + b_4_10·b_1_1·b_1_22 + b_4_10·b_1_12·b_1_2 + b_4_10·b_1_03
       + c_4_13·b_1_1·b_1_22 + c_4_13·b_1_12·b_1_2 + c_4_13·b_1_03 + c_4_12·b_1_23
       + c_4_12·b_1_12·b_1_2 + c_4_12·b_1_13
  36. b_4_11·b_1_1·b_1_23 + b_4_11·b_1_12·b_1_22 + b_4_11·b_1_13·b_1_2
       + b_4_11·b_1_14 + b_4_112 + b_4_10·b_1_1·b_1_23 + b_4_10·b_1_12·b_1_22
       + b_4_10·b_1_13·b_1_2 + b_4_10·b_1_14 + b_4_10·b_1_04 + b_4_102 + c_4_13·b_1_24
       + c_4_13·b_1_12·b_1_22 + c_4_13·b_1_04 + c_4_12·b_1_24 + c_4_12·b_1_14
  37. b_4_11·b_1_2·b_3_7 + b_4_11·b_1_12·b_1_22 + b_4_11·b_1_14 + b_4_10·b_1_2·b_3_7
       + b_4_10·b_1_12·b_1_22 + b_4_10·b_1_14 + b_4_10·b_1_04 + b_4_10·b_4_11 + b_4_102
       + c_4_13·b_1_1·b_1_23 + c_4_13·b_1_12·b_1_22 + c_4_13·b_1_04
       + c_4_12·b_1_1·b_1_23 + c_4_12·b_1_12·b_1_22 + c_4_12·b_1_13·b_1_2
       + c_4_12·b_1_14
  38. b_4_11·b_1_1·b_3_7 + b_4_11·b_1_1·b_1_23 + b_4_11·b_1_13·b_1_2 + b_4_11·b_1_14
       + b_4_10·b_1_2·b_3_7 + b_4_10·b_1_1·b_1_23 + b_4_10·b_1_12·b_1_22
       + b_4_10·b_1_14 + b_4_10·b_1_04 + b_4_102 + c_4_13·b_1_12·b_1_22
       + c_4_13·b_1_13·b_1_2 + c_4_13·b_1_04 + c_4_12·b_1_1·b_1_23
       + c_4_12·b_1_12·b_1_22 + c_4_12·b_1_14
  39. b_3_5·b_5_21 + b_4_10·b_1_0·b_3_5 + c_4_13·b_1_0·b_3_5 + c_4_12·b_1_0·b_3_6
  40. b_3_6·b_5_21 + b_4_10·b_1_0·b_3_5 + b_4_10·b_1_04 + c_4_13·b_1_0·b_3_6
       + c_4_13·b_1_0·b_3_5 + c_4_13·b_1_04
  41. b_3_7·b_5_21 + b_4_10·b_1_2·b_3_7 + b_4_10·b_1_1·b_3_7 + b_4_10·b_1_0·b_3_5
       + b_4_10·b_1_04 + c_4_13·b_1_2·b_3_7 + c_4_13·b_1_1·b_3_7 + c_4_13·b_1_0·b_3_6
       + c_4_13·b_1_0·b_3_5 + c_4_13·b_1_04 + c_4_12·b_1_2·b_3_7
  42. a_2_4·b_6_29
  43. b_4_11·b_5_21 + b_4_10·b_4_11·b_1_2 + b_4_10·b_4_11·b_1_1 + b_4_11·c_4_13·b_1_2
       + b_4_11·c_4_13·b_1_1 + b_4_11·c_4_12·b_1_2
  44. b_4_10·b_5_21 + b_4_10·b_1_02·b_3_5 + b_4_102·b_1_2 + b_4_102·b_1_1 + b_4_102·b_1_0
       + c_4_13·b_1_02·b_3_5 + b_4_10·c_4_13·b_1_2 + b_4_10·c_4_13·b_1_1
       + b_4_10·c_4_13·b_1_0 + b_4_10·c_4_12·b_1_2
  45. b_6_29·b_3_5 + b_4_10·b_1_02·b_3_5 + c_4_13·b_1_02·b_3_5 + c_4_12·b_1_02·b_3_5
       + b_4_10·c_4_12·b_1_0
  46. b_6_29·b_3_6 + b_4_10·b_1_02·b_3_5 + b_4_102·b_1_0 + c_4_13·b_1_02·b_3_5
       + b_4_10·c_4_13·b_1_0 + b_4_10·c_4_12·b_1_0
  47. b_6_29·b_3_7 + b_4_11·b_1_13·b_1_22 + b_4_112·b_1_2 + b_4_112·b_1_1
       + b_4_10·b_1_14·b_1_2 + b_4_10·b_1_02·b_3_5 + b_4_10·b_4_11·b_1_2 + b_4_102·b_1_2
       + b_4_102·b_1_1 + b_4_102·b_1_0 + c_4_13·b_1_1·b_1_24 + c_4_13·b_1_14·b_1_2
       + c_4_13·b_1_02·b_3_5 + c_4_12·b_1_1·b_1_24 + c_4_12·b_1_12·b_3_7
       + c_4_12·b_1_12·b_1_23 + c_4_12·b_1_13·b_1_22 + b_4_11·c_4_13·b_1_1
       + b_4_11·c_4_12·b_1_2 + b_4_11·c_4_12·b_1_1 + b_4_10·c_4_13·b_1_2 + b_4_10·c_4_13·b_1_1
       + b_4_10·c_4_13·b_1_0 + b_4_10·c_4_12·b_1_2 + b_4_10·c_4_12·b_1_0
  48. b_5_212 + b_4_102·b_1_22 + b_4_102·b_1_12 + b_4_102·b_1_02
       + b_4_10·c_4_12·b_1_02 + c_4_132·b_1_22 + c_4_132·b_1_12 + c_4_132·b_1_02
       + c_4_12·c_4_13·b_1_02 + c_4_122·b_1_22
  49. b_4_11·b_1_16 + b_4_11·b_6_29 + b_4_112·b_1_22 + b_4_112·b_1_12
       + b_4_10·b_1_12·b_1_24 + b_4_10·b_1_13·b_1_23 + b_4_10·b_1_16
       + b_4_10·b_4_11·b_1_12 + b_4_102·b_1_1·b_1_2 + b_4_102·b_1_12 + c_4_13·b_1_26
       + c_4_13·b_1_1·b_1_25 + c_4_13·b_1_12·b_1_24 + c_4_13·b_1_14·b_1_22
       + c_4_12·b_1_26 + c_4_12·b_1_12·b_1_2·b_3_7 + c_4_12·b_1_15·b_1_2 + c_4_12·b_1_16
       + b_4_11·c_4_13·b_1_22 + b_4_11·c_4_13·b_1_1·b_1_2 + b_4_11·c_4_12·b_1_22
       + b_4_11·c_4_12·b_1_1·b_1_2 + b_4_11·c_4_12·b_1_12 + b_4_10·c_4_13·b_1_22
       + b_4_10·c_4_13·b_1_1·b_1_2
  50. b_4_10·b_1_12·b_1_2·b_3_7 + b_4_10·b_1_12·b_1_24 + b_4_10·b_1_13·b_3_7
       + b_4_10·b_1_13·b_1_23 + b_4_10·b_1_14·b_1_22 + b_4_10·b_1_15·b_1_2
       + b_4_10·b_1_03·b_3_5 + b_4_10·b_6_29 + b_4_10·b_4_11·b_1_22
       + b_4_10·b_4_11·b_1_1·b_1_2 + b_4_102·b_1_02 + c_4_13·b_1_26
       + c_4_13·b_1_1·b_1_25 + c_4_13·b_1_13·b_1_23 + c_4_13·b_1_15·b_1_2
       + c_4_13·b_1_03·b_3_5 + c_4_12·b_1_26 + c_4_12·b_1_12·b_1_24
       + c_4_12·b_1_13·b_3_7 + b_4_11·c_4_12·b_1_22 + b_4_11·c_4_12·b_1_1·b_1_2
       + b_4_10·c_4_13·b_1_22 + b_4_10·c_4_13·b_1_1·b_1_2 + b_4_10·c_4_13·b_1_02
       + b_4_10·c_4_12·b_1_22 + b_4_10·c_4_12·b_1_1·b_1_2 + b_4_10·c_4_12·b_1_12
       + b_4_10·c_4_12·b_1_02
  51. b_6_29·b_5_21 + b_4_10·b_4_11·b_3_7 + b_4_10·b_4_11·b_1_23
       + b_4_10·b_4_11·b_1_1·b_1_22 + b_4_10·b_4_11·b_1_12·b_1_2 + b_4_10·b_4_11·b_1_13
       + b_4_102·b_3_7 + b_4_11·c_4_13·b_3_7 + b_4_11·c_4_13·b_1_23
       + b_4_11·c_4_13·b_1_1·b_1_22 + b_4_11·c_4_13·b_1_12·b_1_2 + b_4_11·c_4_13·b_1_13
       + b_4_11·c_4_12·b_3_7 + b_4_11·c_4_12·b_1_12·b_1_2 + b_4_10·c_4_13·b_3_7
       + b_4_10·c_4_13·b_1_23 + b_4_10·c_4_13·b_1_1·b_1_22 + b_4_10·c_4_12·b_3_5
       + b_4_10·c_4_12·b_1_23 + b_4_10·c_4_12·b_1_12·b_1_2 + b_4_10·c_4_12·b_1_13
       + c_4_132·b_1_23 + c_4_132·b_1_1·b_1_22 + c_4_12·c_4_13·b_1_23
       + c_4_12·c_4_13·b_1_1·b_1_22 + c_4_12·c_4_13·b_1_12·b_1_2 + c_4_12·c_4_13·b_1_13
       + c_4_122·b_1_23 + c_4_122·b_1_1·b_1_22 + c_4_122·b_1_12·b_1_2
  52. b_6_292 + b_4_112·b_1_14 + b_4_113 + b_4_10·b_4_11·b_1_12·b_1_22
       + b_4_10·b_4_11·b_1_13·b_1_2 + b_4_102·b_1_24 + b_4_102·b_1_12·b_1_22
       + b_4_102·b_1_13·b_1_2 + b_4_103 + b_4_11·c_4_13·b_1_12·b_1_22
       + b_4_11·c_4_13·b_1_13·b_1_2 + b_4_11·c_4_13·b_1_14 + b_4_112·c_4_12
       + b_4_10·c_4_13·b_1_24 + b_4_10·c_4_13·b_1_1·b_1_23
       + b_4_10·c_4_13·b_1_12·b_1_22 + b_4_10·c_4_13·b_1_13·b_1_2
       + b_4_10·c_4_13·b_1_14 + b_4_10·c_4_13·b_1_04 + b_4_10·c_4_12·b_1_24
       + b_4_10·c_4_12·b_1_13·b_1_2 + b_4_10·c_4_12·b_1_04 + c_4_132·b_1_24
       + c_4_132·b_1_12·b_1_22 + c_4_132·b_1_04 + c_4_12·c_4_13·b_1_14
       + c_4_12·c_4_13·b_1_04 + c_4_122·b_1_24 + c_4_122·b_1_12·b_1_22
       + c_4_122·b_1_14 + c_4_122·b_1_04

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_12, a Duflot regular element of degree 4
    2. c_4_13, a Duflot regular element of degree 4
    3. b_1_22 + b_1_1·b_1_2 + b_1_12 + b_1_02, an element of degree 2
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 2, 6, 8].
  • 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. 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_4 → 0, an element of degree 2
  5. b_3_5 → 0, an element of degree 3
  6. b_3_6 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_4_10 → 0, an element of degree 4
  9. b_4_11 → 0, an element of degree 4
  10. c_4_12 → c_1_14 + c_1_04, an element of degree 4
  11. c_4_13 → c_1_14, an element of degree 4
  12. b_5_21 → 0, an element of degree 5
  13. b_6_29 → 0, an element of degree 6

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

  1. b_1_0 → c_1_2, 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_4 → 0, an element of degree 2
  5. b_3_5 → 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
  6. b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_3_7 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_10 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  9. b_4_11 → 0, an element of degree 4
  10. c_4_12 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  11. c_4_13 → c_1_1·c_1_23 + c_1_14, an element of degree 4
  12. b_5_21 → c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2, an element of degree 5
  13. b_6_29 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23
       + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22
       + c_1_04·c_1_22, an element of degree 6

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 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. b_3_5 → 0, an element of degree 3
  6. b_3_6 → 0, an element of degree 3
  7. b_3_7 → c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_32
       + c_1_02·c_1_3, an element of degree 3
  8. b_4_10 → c_1_1·c_1_33 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  9. b_4_11 → c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_2·c_1_3, an element of degree 4
  10. c_4_12 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_33
       + c_1_0·c_1_23 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  11. c_4_13 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_33
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32
       + c_1_02·c_1_22, an element of degree 4
  12. b_5_21 → c_1_1·c_1_34 + c_1_1·c_1_2·c_1_33 + c_1_12·c_1_33 + 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_34 + c_1_0·c_1_22·c_1_32
       + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, an element of degree 5
  13. b_6_29 → c_1_1·c_1_35 + c_1_1·c_1_2·c_1_34 + c_1_1·c_1_24·c_1_3 + c_1_1·c_1_25
       + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32 + c_1_12·c_1_24
       + c_1_13·c_1_2·c_1_32 + c_1_13·c_1_22·c_1_3 + c_1_14·c_1_32 + c_1_0·c_1_35
       + c_1_0·c_1_22·c_1_33 + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_25
       + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_22·c_1_32 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_33 + c_1_0·c_1_12·c_1_22·c_1_3 + c_1_0·c_1_12·c_1_23
       + c_1_02·c_1_2·c_1_33 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_1·c_1_33
       + c_1_02·c_1_1·c_1_2·c_1_32 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_32
       + c_1_02·c_1_12·c_1_2·c_1_3 + c_1_02·c_1_12·c_1_22 + c_1_03·c_1_2·c_1_32
       + c_1_03·c_1_22·c_1_3 + c_1_04·c_1_32 + c_1_04·c_1_22, 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