Cohomology of group number 340 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 a unique conjugacy class of maximal elementary abelian subgroups, which is 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
    ( − 1) · (t6  −  t5  −  t  −  1)
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-5,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 7:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_4, an element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. a_3_6, a nilpotent element of degree 3
  7. b_4_6, an element of degree 4
  8. b_4_8, an element of degree 4
  9. c_4_9, a Duflot regular element of degree 4
  10. c_4_10, a Duflot regular element of degree 4
  11. a_5_17, a nilpotent element of degree 5
  12. b_5_16, an element of degree 5
  13. b_7_25, an element of degree 7

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Ring relations

There are 52 minimal relations of maximal degree 14:

  1. a_1_0·a_1_1
  2. a_1_0·b_1_2
  3. a_1_03
  4. a_1_13
  5. b_2_4·a_1_1 + a_1_12·b_1_2
  6. a_1_12·b_1_22
  7. b_1_2·a_3_5
  8. a_1_1·a_3_5
  9. b_1_2·a_3_6 + b_2_4·a_1_02
  10. a_1_1·a_3_6
  11. a_1_02·a_3_5
  12. a_1_02·a_3_6
  13. b_4_6·a_1_0
  14. b_4_8·a_1_1
  15. b_4_8·a_1_0 + b_2_4·a_3_5
  16. a_3_62 + b_2_4·a_1_0·a_3_6 + b_2_42·a_1_02 + c_4_9·a_1_02
  17. a_3_52 + c_4_10·a_1_02
  18. b_4_6·a_1_12
  19. b_4_8·b_1_22 + b_2_4·b_4_6 + b_4_6·a_1_1·b_1_2 + b_2_4·a_1_0·a_3_5 + b_2_42·a_1_02
  20. b_1_2·a_5_17 + b_2_4·a_1_0·a_3_5
  21. a_1_1·a_5_17
  22. a_3_5·a_3_6 + a_3_52 + a_1_0·a_5_17
  23. a_1_1·b_5_16 + b_4_6·a_1_1·b_1_2 + c_4_9·a_1_1·b_1_2
  24. a_1_0·b_5_16 + b_2_4·a_1_0·a_3_6
  25. b_4_6·a_3_6
  26. b_4_6·a_3_5
  27. b_4_8·a_3_5 + b_2_4·c_4_10·a_1_0
  28. a_1_02·a_5_17
  29. b_4_8·a_3_6 + b_2_4·a_5_17 + b_2_4·c_4_10·a_1_0
  30. b_4_62 + c_4_10·b_1_24
  31. b_4_6·b_4_8 + b_2_42·a_1_0·a_3_5 + b_2_4·c_4_10·b_1_22 + c_4_10·a_1_1·b_1_23
       + b_2_4·c_4_10·a_1_02
  32. b_4_82 + b_2_42·c_4_10
  33. a_3_5·a_5_17 + c_4_10·a_1_0·a_3_6 + c_4_10·a_1_0·a_3_5
  34. a_3_6·a_5_17 + b_2_4·a_1_0·a_5_17 + b_2_42·a_1_0·a_3_5 + c_4_10·a_1_0·a_3_6
       + c_4_9·a_1_0·a_3_5 + b_2_4·c_4_10·a_1_02
  35. a_3_6·b_5_16 + b_2_42·a_1_0·a_3_6 + b_2_42·a_1_0·a_3_5 + b_2_43·a_1_02
  36. a_3_5·b_5_16 + a_3_6·a_5_17 + b_2_42·a_1_0·a_3_5 + c_4_10·a_1_0·a_3_6
       + c_4_9·a_1_0·a_3_5
  37. a_1_1·b_7_25 + b_4_6·a_1_1·b_1_23 + c_4_10·a_1_1·b_1_23
  38. a_1_0·b_7_25 + b_2_42·a_1_0·a_3_5 + b_2_4·c_4_9·a_1_02
  39. b_4_6·a_5_17
  40. b_4_8·a_5_17 + b_2_4·c_4_10·a_3_6 + b_2_4·c_4_10·a_3_5
  41. b_1_22·b_7_25 + b_1_24·b_5_16 + b_4_6·b_5_16 + b_2_4·b_4_6·b_1_23 + b_2_42·b_1_25
       + b_2_42·b_4_6·b_1_2 + b_2_43·b_1_23 + c_4_9·b_1_25 + b_4_6·c_4_9·b_1_2
       + b_2_4·c_4_10·b_1_23 + c_4_10·a_1_1·b_1_24 + c_4_9·a_1_1·b_1_24
  42. b_4_8·b_5_16 + b_2_4·b_7_25 + b_2_4·b_1_22·b_5_16 + b_2_42·b_4_8·b_1_2
       + b_2_42·b_4_6·b_1_2 + b_2_43·b_1_23 + b_2_44·b_1_2 + b_2_42·a_5_17 + b_2_43·a_3_5
       + b_4_8·c_4_9·b_1_2 + b_2_4·c_4_9·b_1_23 + b_2_42·c_4_10·b_1_2
       + c_4_10·a_1_1·b_1_24 + b_2_42·c_4_10·a_1_0 + b_2_42·c_4_9·a_1_0
  43. a_5_172 + b_2_4·c_4_10·a_1_0·a_3_6 + b_2_42·c_4_10·a_1_02 + c_4_102·a_1_02
       + c_4_9·c_4_10·a_1_02
  44. b_5_162 + b_2_42·b_4_6·b_1_22 + b_2_43·a_1_0·a_3_6 + b_2_44·a_1_02
       + c_4_10·b_1_26 + b_2_42·c_4_10·b_1_22 + b_2_42·c_4_9·a_1_02 + c_4_92·b_1_22
  45. a_5_17·b_5_16 + b_2_42·a_1_0·a_5_17 + b_2_43·a_1_0·a_3_5 + b_2_4·c_4_10·a_1_0·a_3_6
  46. a_3_6·b_7_25 + b_2_42·a_1_0·a_5_17 + b_2_43·a_1_0·a_3_5 + b_2_44·a_1_02
       + b_2_4·c_4_9·a_1_0·a_3_6 + b_2_42·c_4_10·a_1_02
  47. a_3_5·b_7_25 + b_2_4·c_4_9·a_1_0·a_3_5 + b_2_42·c_4_10·a_1_02
  48. b_4_6·b_7_25 + b_4_6·b_1_22·b_5_16 + b_2_42·b_4_6·b_1_23 + b_2_43·b_4_6·b_1_2
       + c_4_10·b_1_22·b_5_16 + b_4_6·c_4_9·b_1_23 + b_2_4·c_4_10·b_1_25
       + b_2_4·b_4_6·c_4_10·b_1_2 + b_2_42·c_4_10·b_1_23 + b_4_6·c_4_10·a_1_1·b_1_22
       + b_4_6·c_4_9·a_1_1·b_1_22 + c_4_9·c_4_10·b_1_23
  49. b_4_8·b_7_25 + b_2_4·b_4_6·b_5_16 + b_2_43·b_4_8·b_1_2 + b_2_43·b_4_6·b_1_2
       + b_2_4·c_4_10·b_5_16 + b_2_4·b_4_8·c_4_10·b_1_2 + b_2_4·b_4_6·c_4_9·b_1_2
       + b_2_42·c_4_10·b_1_23 + b_2_43·c_4_10·b_1_2 + c_4_10·a_1_1·b_1_26
       + b_4_6·c_4_10·a_1_1·b_1_22 + b_2_42·c_4_10·a_3_6 + b_2_42·c_4_9·a_3_5
       + b_2_43·c_4_10·a_1_0 + b_2_4·c_4_9·c_4_10·b_1_2
  50. b_5_16·b_7_25 + b_2_4·b_4_6·b_1_2·b_5_16 + b_2_42·b_1_2·b_7_25
       + b_2_42·b_4_6·b_1_24 + b_2_43·b_1_2·b_5_16 + b_2_43·b_4_6·b_1_22
       + b_2_44·b_1_24 + b_2_44·b_4_6 + b_2_45·b_1_22 + b_2_43·a_1_0·a_5_17
       + b_2_45·a_1_02 + c_4_10·b_1_28 + c_4_9·b_1_2·b_7_25 + b_4_6·c_4_10·b_1_24
       + b_2_4·c_4_10·b_1_2·b_5_16 + b_2_4·b_4_6·c_4_9·b_1_22 + b_2_42·b_4_6·c_4_10
       + b_2_43·c_4_10·b_1_22 + b_2_43·c_4_9·b_1_22 + b_4_6·c_4_10·a_1_1·b_1_23
       + b_4_6·c_4_9·a_1_1·b_1_23 + b_2_42·c_4_10·a_1_0·a_3_5 + b_2_42·c_4_9·a_1_0·a_3_6
       + b_2_43·c_4_10·a_1_02 + b_2_4·c_4_9·c_4_10·b_1_22
  51. a_5_17·b_7_25 + b_2_44·a_1_0·a_3_5 + b_2_4·c_4_9·a_1_0·a_5_17
       + b_2_42·c_4_10·a_1_0·a_3_6 + b_2_42·c_4_10·a_1_0·a_3_5 + b_2_43·c_4_10·a_1_02
  52. b_7_252 + b_2_42·b_4_6·b_1_26 + b_2_44·b_1_26 + b_2_46·b_1_22
       + c_4_10·b_1_210 + b_2_42·b_4_6·c_4_10·b_1_22 + b_2_44·c_4_10·b_1_22
       + b_2_44·c_4_10·a_1_02 + c_4_102·b_1_26 + b_2_42·c_4_92·a_1_02

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 14.
  • 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_9, a Duflot regular element of degree 4
    2. c_4_10, a Duflot regular element of degree 4
    3. b_1_22 + b_2_4, 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, 3, 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. a_1_0 → 0, an element of degree 1
  2. a_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. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. b_4_6 → 0, an element of degree 4
  8. b_4_8 → 0, an element of degree 4
  9. c_4_9 → c_1_14, an element of degree 4
  10. c_4_10 → c_1_04, an element of degree 4
  11. a_5_17 → 0, an element of degree 5
  12. b_5_16 → 0, an element of degree 5
  13. b_7_25 → 0, an element of degree 7

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. b_4_6 → c_1_02·c_1_22, an element of degree 4
  8. b_4_8 → c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3, an element of degree 4
  9. c_4_9 → c_1_34 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + 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, an element of degree 4
  10. c_4_10 → c_1_04, an element of degree 4
  11. a_5_17 → 0, an element of degree 5
  12. b_5_16 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3
       + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23 + c_1_14·c_1_2
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_23, an element of degree 5
  13. b_7_25 → c_1_2·c_1_36 + c_1_22·c_1_35 + c_1_24·c_1_33 + c_1_25·c_1_32
       + c_1_0·c_1_24·c_1_32 + c_1_0·c_1_25·c_1_3 + c_1_02·c_1_2·c_1_34
       + c_1_02·c_1_23·c_1_32 + c_1_02·c_1_25 + c_1_03·c_1_22·c_1_32
       + c_1_03·c_1_23·c_1_3 + c_1_04·c_1_23, an element of degree 7

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