Cohomology of group number 197 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 all of rank 4.

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)
  • The a-invariants are -∞,-∞,-4,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_4, a nilpotent 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. c_4_17, a Duflot regular element of degree 4
  12. c_4_18, a Duflot regular element of degree 4
  13. a_5_11, a nilpotent element of degree 5
  14. b_5_29, 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 53 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. b_1_1·b_1_22 + b_2_4·a_1_0
  5. b_1_1·b_1_22 + b_2_4·b_1_1
  6. b_1_1·b_1_22 + b_2_5·a_1_0
  7. b_2_5·b_1_22 + b_2_42 + b_2_4·a_1_0·b_1_2
  8. a_1_0·a_3_3
  9. b_1_1·a_3_3 + b_2_4·a_1_0·b_1_2
  10. a_1_0·a_3_4
  11. b_2_5·b_1_1·b_1_2 + b_1_1·a_3_4 + b_2_4·a_1_0·b_1_2
  12. a_1_0·b_3_8
  13. a_1_0·b_3_10
  14. b_1_22·a_3_4 + b_2_5·a_3_3 + b_2_4·a_3_3
  15. b_1_22·b_3_8 + b_2_4·b_1_23 + b_2_4·a_3_3
  16. b_2_4·b_3_8 + b_2_42·b_1_2 + b_2_5·a_3_3
  17. b_1_22·b_3_10 + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_4·a_3_4
  18. b_2_52·b_1_2 + b_2_4·b_3_10 + b_2_4·b_2_5·b_1_2 + b_2_5·a_3_4
  19. b_4_16·a_1_0
  20. b_1_1·b_1_2·b_3_8 + b_4_16·b_1_1 + b_2_5·b_3_8 + b_2_4·b_2_5·b_1_2 + b_2_5·a_3_3
       + b_2_4·a_3_4
  21. a_3_32
  22. a_3_3·a_3_4
  23. a_3_42
  24. a_3_3·b_3_8 + b_2_4·b_1_2·a_3_3
  25. b_2_5·b_1_2·b_3_8 + b_2_43 + a_3_4·b_3_8 + b_2_5·b_1_2·a_3_3
  26. a_3_3·b_3_10 + b_2_4·b_1_2·a_3_4
  27. b_2_5·b_1_2·b_3_10 + b_2_4·b_2_52 + b_2_42·b_2_5 + a_3_4·b_3_10 + b_2_5·b_1_2·a_3_4
  28. b_3_102 + b_1_12·b_1_2·b_3_10 + b_2_5·b_1_1·b_3_10 + b_2_53 + b_2_42·b_2_5
       + c_4_17·b_1_12
  29. b_3_82 + b_2_42·b_1_22 + c_4_18·b_1_12
  30. a_1_0·a_5_11
  31. b_1_12·b_1_2·b_3_10 + b_2_5·b_1_2·b_3_8 + b_2_43 + b_1_1·a_5_11 + b_1_13·a_3_4
       + b_2_5·b_1_2·a_3_3 + b_2_5·b_1_1·a_3_4 + b_2_4·b_1_2·a_3_4
  32. a_1_0·b_5_29 + c_4_17·a_1_0·b_1_2
  33. b_3_8·b_3_10 + b_1_1·b_5_29 + b_2_42·b_2_5 + b_2_43 + c_4_17·b_1_1·b_1_2
  34. b_4_16·b_3_8 + b_2_4·b_4_16·b_1_2 + b_1_1·a_3_4·b_3_10 + b_4_16·a_3_4 + b_2_5·a_5_11
       + b_2_5·b_1_12·a_3_4 + b_2_52·a_3_4 + c_4_18·b_1_12·b_1_2 + b_2_5·c_4_18·b_1_1
       + b_2_4·c_4_18·a_1_0
  35. b_1_22·a_5_11 + b_4_16·a_3_3 + b_2_42·a_3_4
  36. b_4_16·b_3_8 + b_2_4·b_4_16·b_1_2 + b_2_4·a_5_11 + b_2_4·b_2_5·a_3_4
       + c_4_18·b_1_12·b_1_2 + b_2_5·c_4_18·b_1_1 + b_2_4·c_4_18·a_1_0
  37. b_1_22·b_5_29 + b_4_16·b_3_8 + c_4_18·b_1_12·b_1_2 + c_4_17·b_1_23
       + b_2_5·c_4_18·b_1_1 + b_2_4·c_4_18·a_1_0
  38. b_1_1·b_1_2·b_5_29 + b_4_16·b_3_10 + b_4_16·b_3_8 + b_2_5·b_5_29 + b_2_5·b_4_16·b_1_2
       + b_2_4·b_4_16·b_1_2 + b_4_16·a_3_4 + c_4_18·b_1_12·b_1_2 + b_2_5·c_4_18·b_1_1
       + b_2_5·c_4_17·b_1_2 + b_2_4·c_4_18·a_1_0
  39. b_4_16·b_3_8 + b_2_5·b_4_16·b_1_2 + b_2_4·b_5_29 + b_2_4·b_4_16·b_1_2 + b_4_16·a_3_4
       + c_4_18·b_1_12·b_1_2 + b_2_5·c_4_18·b_1_1 + b_2_4·c_4_17·b_1_2 + b_2_4·c_4_18·a_1_0
  40. b_4_162 + b_2_4·b_4_16·b_1_22 + b_2_42·b_1_24 + b_2_42·b_4_16
       + b_2_4·b_1_23·a_3_3 + b_2_4·b_2_5·b_1_2·a_3_3 + b_2_42·b_1_2·a_3_4 + c_4_17·b_1_24
       + b_2_52·c_4_18 + b_2_42·c_4_18
  41. a_3_3·a_5_11
  42. a_3_4·a_5_11
  43. a_3_3·b_5_29 + b_2_4·b_1_2·a_5_11 + b_2_4·b_2_5·b_1_2·a_3_4 + c_4_17·b_1_2·a_3_3
  44. b_3_10·b_5_29 + b_4_16·b_1_1·b_3_10 + b_2_52·b_4_16 + b_2_4·b_2_5·b_4_16
       + b_2_5·a_3_4·b_3_8 + b_2_4·b_2_5·b_1_2·a_3_4 + c_4_17·b_1_2·b_3_10
       + c_4_17·b_1_1·b_3_8
  45. b_2_5·b_1_2·b_5_29 + b_2_4·b_2_5·b_4_16 + b_3_10·a_5_11 + b_2_5·a_3_4·b_3_10
       + b_2_5·b_1_2·a_5_11 + b_2_52·b_1_1·a_3_4 + b_2_4·a_3_4·b_3_10
       + b_2_4·b_2_5·b_1_2·a_3_4 + c_4_17·b_1_13·b_1_2 + b_2_42·c_4_17
  46. b_4_16·b_1_1·b_3_10 + b_2_5·b_1_1·b_5_29 + b_3_8·a_5_11 + b_1_12·a_3_4·b_3_8
       + b_2_5·a_3_4·b_3_8 + b_2_4·b_1_2·a_5_11 + b_2_4·b_2_5·b_1_2·a_3_4 + c_4_18·b_1_1·a_3_4
       + c_4_17·b_1_1·a_3_4 + b_2_4·c_4_17·a_1_0·b_1_2
  47. b_3_10·a_5_11 + a_3_4·b_5_29 + b_2_5·a_3_4·b_3_10 + b_2_52·b_1_1·a_3_4
       + c_4_17·b_1_13·b_1_2 + c_4_17·b_1_2·a_3_4 + b_2_4·c_4_17·a_1_0·b_1_2
  48. b_3_8·b_5_29 + b_2_42·b_4_16 + c_4_18·b_1_1·b_3_10 + c_4_17·b_1_2·b_3_8
  49. b_1_1·a_3_4·b_5_29 + b_4_16·a_5_11 + b_2_5·b_1_1·a_3_4·b_3_8 + b_2_5·b_4_16·a_3_4
       + b_2_4·b_4_16·a_3_3 + b_2_42·a_5_11 + b_2_42·b_1_22·a_3_3 + b_2_42·b_2_5·a_3_4
       + c_4_17·b_1_22·a_3_3 + b_2_5·c_4_18·a_3_4 + b_2_4·c_4_18·a_3_4
  50. b_4_16·b_5_29 + b_2_42·b_5_29 + b_2_42·b_4_16·b_1_2 + b_2_43·b_1_23
       + b_2_42·a_5_11 + b_2_42·b_1_22·a_3_3 + b_2_42·b_2_5·a_3_4 + b_2_42·b_2_5·a_3_3
       + b_2_43·a_3_4 + b_2_43·a_3_3 + c_4_18·b_1_1·b_1_2·b_3_10 + b_4_16·c_4_17·b_1_2
       + b_2_5·c_4_18·b_3_10 + b_2_4·c_4_18·b_3_10 + b_2_4·c_4_17·b_1_23
       + b_2_42·c_4_17·b_1_2 + b_2_4·c_4_17·a_3_3
  51. a_5_112
  52. b_5_292 + b_2_42·b_2_5·b_4_16 + b_2_43·b_4_16 + b_2_44·b_1_22
       + b_2_42·b_2_5·b_1_2·a_3_4 + b_2_42·b_2_5·b_1_2·a_3_3 + b_2_43·b_1_2·a_3_4
       + b_2_43·b_1_2·a_3_3 + b_2_5·c_4_18·b_1_1·b_3_10 + b_2_53·c_4_18
       + b_2_42·c_4_17·b_1_22 + b_2_42·b_2_5·c_4_18 + c_4_18·a_3_4·b_3_8
       + c_4_18·b_1_1·a_5_11 + c_4_18·b_1_13·a_3_4 + b_2_5·c_4_18·b_1_1·a_3_4
       + b_2_4·c_4_18·b_1_2·a_3_4 + c_4_17·c_4_18·b_1_12 + c_4_172·b_1_22
  53. a_5_11·b_5_29 + b_2_5·a_3_4·b_5_29 + b_2_52·a_3_4·b_3_8 + b_2_4·b_2_5·b_1_2·a_5_11
       + b_2_42·b_1_2·a_5_11 + b_2_42·b_2_5·b_1_2·a_3_4 + b_2_43·b_1_2·a_3_3
       + b_4_16·c_4_17·b_1_12 + b_2_5·c_4_17·b_1_1·b_3_8 + c_4_18·a_3_4·b_3_10
       + c_4_17·b_1_2·a_5_11 + b_2_5·c_4_17·b_1_2·a_3_4 + b_2_4·c_4_17·b_1_2·a_3_3

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_17, a Duflot regular element of degree 4
    2. c_4_18, a Duflot regular element of degree 4
    3. b_1_22 + b_1_12 + b_2_5 + b_2_4, an element of degree 2
    4. b_2_5·b_1_2 + b_2_5·b_1_1 + b_2_4·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 9].
  • 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. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 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. c_4_17 → c_1_04, an element of degree 4
  12. c_4_18 → c_1_14, an element of degree 4
  13. a_5_11 → 0, an element of degree 5
  14. b_5_29 → 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 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. b_3_8 → c_1_12·c_1_2, an element of degree 3
  9. b_3_10 → c_1_33 + c_1_22·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  10. b_4_16 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3, an element of degree 4
  11. c_4_17 → 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
  12. c_4_18 → c_1_14, an element of degree 4
  13. a_5_11 → 0, an element of degree 5
  14. b_5_29 → c_1_12·c_1_33 + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_12·c_1_22
       + c_1_02·c_1_12·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_3, an element of degree 1
  4. b_2_4 → c_1_2·c_1_3, an element of degree 2
  5. b_2_5 → c_1_22, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. b_3_8 → c_1_2·c_1_32, an element of degree 3
  9. b_3_10 → c_1_22·c_1_3 + c_1_23, an element of degree 3
  10. b_4_16 → c_1_2·c_1_33 + c_1_22·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_02·c_1_32, an element of degree 4
  11. c_4_17 → c_1_22·c_1_32 + 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
  12. c_4_18 → c_1_12·c_1_32 + c_1_14, an element of degree 4
  13. a_5_11 → 0, an element of degree 5
  14. b_5_29 → c_1_23·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23 + c_1_0·c_1_22·c_1_32
       + c_1_02·c_1_33 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, 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