Cohomology of group number 110 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 16.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  −  t5  −  t4  +  2·t3  −  2·t2  +  t  −  1
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-5,-3,-3. 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 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. b_3_4, an element of degree 3
  8. b_3_5, an element of degree 3
  9. a_5_6, a nilpotent element of degree 5
  10. b_5_9, an element of degree 5
  11. a_6_6, a nilpotent element of degree 6
  12. b_6_10, an element of degree 6
  13. c_8_17, a Duflot regular element of degree 8

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Ring relations

There are 53 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_0
  5. a_2_1·a_1_1
  6. a_2_1·a_1_0 + a_2_0·a_1_1
  7. b_2_3·a_1_0 + b_2_2·a_1_1 + a_2_0·a_1_1
  8. a_2_02
  9. a_2_0·a_2_1
  10. a_2_12
  11. a_1_1·b_3_4 + a_2_0·b_2_3
  12. a_1_0·b_3_4 + a_2_0·b_2_2
  13. a_1_1·b_3_5 + a_2_1·b_2_3
  14. a_1_0·b_3_5 + a_2_1·b_2_2
  15. b_2_2·b_2_3·a_1_1 + b_2_22·a_1_1
  16. b_2_22·a_1_0 + a_2_0·b_3_4
  17. a_2_1·b_3_4 + a_2_0·b_3_5
  18. a_2_1·b_3_5
  19. b_3_42 + b_2_23 + a_2_0·b_2_2·b_2_3 + a_2_0·b_2_22
  20. b_3_52 + b_2_2·b_2_32 + b_2_22·b_2_3 + a_2_1·b_2_32 + a_2_1·b_2_2·b_2_3
  21. a_2_0·b_2_32 + a_2_0·b_2_2·b_2_3 + a_1_1·a_5_6
  22. a_1_0·a_5_6
  23. a_1_1·b_5_9 + a_2_1·b_2_32 + a_2_0·b_2_32 + a_2_0·b_2_2·b_2_3
  24. a_1_0·b_5_9 + a_2_1·b_2_22
  25. a_2_0·a_5_6
  26. b_2_32·b_3_4 + b_2_2·b_5_9 + b_2_2·b_2_3·b_3_4 + b_2_22·b_3_5 + b_2_3·a_5_6
       + b_2_2·a_5_6 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_3·b_3_4 + a_2_0·b_2_2·b_3_4 + a_2_1·a_5_6
  27. a_2_0·b_5_9 + a_2_0·b_2_2·b_3_5 + a_2_1·a_5_6
  28. a_2_1·b_5_9 + a_2_1·a_5_6
  29. a_6_6·a_1_1 + a_2_1·a_5_6
  30. a_6_6·a_1_0
  31. b_6_10·a_1_1 + a_2_0·b_2_3·b_3_5 + a_2_1·a_5_6
  32. b_6_10·a_1_0 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_3·b_3_4 + a_2_0·b_2_2·b_3_4 + a_2_1·a_5_6
  33. b_3_4·b_5_9 + b_2_2·b_3_4·b_3_5 + b_2_22·b_2_32 + b_2_23·b_2_3 + b_3_4·a_5_6
       + b_2_3·a_6_6 + a_2_1·b_2_33 + a_2_0·b_2_23 + b_2_3·a_1_1·a_5_6
  34. b_3_4·a_5_6 + b_2_2·a_6_6 + a_2_0·b_2_23
  35. a_2_0·a_6_6
  36. a_2_1·a_6_6
  37. b_3_5·b_5_9 + b_3_4·b_5_9 + b_2_3·b_3_4·b_3_5 + b_2_3·b_6_10 + b_2_2·b_3_4·b_3_5
       + b_2_22·b_2_32 + b_2_23·b_2_3 + b_3_5·a_5_6 + b_3_4·a_5_6 + a_2_1·b_2_22·b_2_3
       + a_2_0·b_3_4·b_3_5 + a_2_0·b_2_22·b_2_3 + a_2_0·b_2_23 + b_2_3·a_1_1·a_5_6
  38. b_2_3·b_3_4·b_3_5 + b_2_2·b_6_10 + b_2_23·b_2_3 + b_2_24 + b_3_5·a_5_6 + b_3_4·a_5_6
       + a_2_0·b_3_4·b_3_5 + a_2_0·b_2_22·b_2_3 + a_2_0·b_2_23
  39. a_2_1·b_2_22·b_2_3 + a_2_0·b_6_10 + a_2_0·b_2_22·b_2_3 + a_2_0·b_2_23
  40. a_2_1·b_6_10 + a_2_1·b_2_22·b_2_3 + a_2_0·b_3_4·b_3_5
  41. a_6_6·b_3_4 + b_2_22·a_5_6 + a_2_0·b_2_22·b_3_4 + a_2_1·b_2_3·a_5_6
  42. b_6_10·b_3_4 + b_2_22·b_2_3·b_3_5 + b_2_22·b_2_3·b_3_4 + b_2_23·b_3_4 + a_6_6·b_3_5
       + b_2_22·a_5_6 + a_2_0·b_2_2·b_2_3·b_3_4 + a_2_0·b_2_22·b_3_4 + a_2_1·b_2_3·a_5_6
  43. b_6_10·b_3_5 + b_2_2·b_2_3·b_5_9 + b_2_23·b_3_5 + a_6_6·b_3_5 + a_2_1·b_2_3·a_5_6
  44. a_5_62
  45. b_5_92 + b_2_2·b_2_34 + b_2_24·b_2_3 + a_5_6·b_5_9 + b_2_32·a_6_6
       + b_2_2·b_3_5·a_5_6 + b_2_2·b_2_3·a_6_6 + b_2_32·a_1_1·a_5_6
  46. b_5_92 + b_2_2·b_2_34 + b_2_24·b_2_3 + a_2_1·b_2_34 + a_2_0·b_2_2·b_6_10
       + a_2_0·b_2_23·b_2_3 + a_2_0·b_2_24
  47. a_6_6·a_5_6 + a_2_1·b_2_32·a_5_6
  48. b_6_10·a_5_6 + a_6_6·b_5_9 + b_2_3·a_6_6·b_3_5 + b_2_2·a_6_6·b_3_5 + b_2_2·b_2_32·a_5_6
       + b_2_23·a_5_6 + a_2_0·b_2_22·b_2_3·b_3_5
  49. b_6_10·b_5_9 + b_2_2·b_2_33·b_3_5 + b_2_22·b_2_32·b_3_5 + b_2_23·b_5_9
       + b_2_23·b_2_3·b_3_5 + b_2_2·a_6_6·b_3_5 + b_2_2·b_2_32·a_5_6 + b_2_22·b_2_3·a_5_6
  50. a_6_6·b_5_9 + b_2_2·a_6_6·b_3_5 + b_2_2·b_2_32·a_5_6 + b_2_22·b_2_3·a_5_6
       + a_2_1·b_2_32·a_5_6 + a_2_0·c_8_17·a_1_1
  51. a_6_62
  52. b_6_102 + b_2_22·b_2_34 + b_2_23·b_2_33 + b_2_24·b_2_32 + b_2_26
  53. a_6_6·b_6_10 + b_2_2·b_2_3·b_3_5·a_5_6 + b_2_22·b_2_3·a_6_6 + b_2_23·a_6_6
       + a_2_0·b_2_22·b_6_10 + a_2_0·b_2_24·b_2_3 + a_2_0·b_2_25

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_8_17, a Duflot regular element of degree 8
    2. b_2_32 + b_2_2·b_2_3 + b_2_22, an element of degree 4
    3. b_3_5, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. b_3_4 → 0, an element of degree 3
  8. b_3_5 → 0, an element of degree 3
  9. a_5_6 → 0, an element of degree 5
  10. b_5_9 → 0, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_6_10 → 0, an element of degree 6
  13. c_8_17 → c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_12, an element of degree 2
  6. b_2_3 → c_1_22, an element of degree 2
  7. b_3_4 → c_1_13, an element of degree 3
  8. b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  9. a_5_6 → 0, an element of degree 5
  10. b_5_9 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_6_10 → c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_16, an element of degree 6
  13. c_8_17 → c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23 + c_1_17·c_1_2 + c_1_18
       + 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

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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