Cohomology of group number 91 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
    ( − 1) · (t4  −  t3  +  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 12 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. a_3_3, a nilpotent element of degree 3
  6. b_4_3, an element of degree 4
  7. a_5_4, a nilpotent element of degree 5
  8. b_5_6, an element of degree 5
  9. a_6_5, a nilpotent element of degree 6
  10. b_7_10, an element of degree 7
  11. b_8_11, an element of degree 8
  12. c_8_13, a Duflot regular element of degree 8

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Ring relations

There are 52 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. b_2_2·a_1_0 + a_2_1·b_1_1
  5. a_2_12
  6. a_1_0·a_3_3
  7. b_1_1·a_3_3 + a_2_1·b_2_2
  8. a_2_1·b_2_2·b_1_1
  9. a_2_1·a_3_3
  10. b_4_3·a_1_0
  11. a_2_1·b_2_22 + a_3_32
  12. a_2_1·b_4_3
  13. a_1_0·a_5_4
  14. b_1_1·a_5_4
  15. a_1_0·b_5_6
  16. b_4_3·a_3_3
  17. a_2_1·a_5_4
  18. b_2_2·a_5_4 + a_2_1·b_5_6
  19. a_6_5·a_1_0
  20. a_6_5·b_1_1 + b_2_2·a_5_4
  21. a_3_3·a_5_4
  22. b_1_13·b_5_6 + b_4_3·b_1_14 + b_4_32
  23. a_3_3·b_5_6 + b_2_2·a_6_5
  24. a_2_1·a_6_5
  25. a_1_0·b_7_10
  26. b_1_1·b_7_10 + b_2_2·b_1_1·b_5_6 + b_2_22·b_4_3 + a_3_3·b_5_6
  27. b_4_3·a_5_4
  28. a_2_1·b_2_2·b_5_6 + a_6_5·a_3_3
  29. a_2_1·b_7_10
  30. b_8_11·a_1_0
  31. b_8_11·b_1_1 + b_4_3·b_5_6 + b_4_3·b_1_15 + b_4_32·b_1_1 + b_2_2·b_1_12·b_5_6
       + b_2_22·b_4_3·b_1_1 + b_2_23·b_1_13
  32. a_5_42
  33. a_5_4·b_5_6
  34. b_4_3·a_6_5
  35. a_3_3·b_7_10
  36. b_5_62 + b_4_3·b_1_16 + b_4_32·b_1_12 + b_2_2·b_4_3·b_1_14
       + b_2_22·b_1_1·b_5_6 + b_2_22·b_1_16 + b_2_24·b_1_12 + c_8_13·b_1_12
  37. a_2_1·b_8_11
  38. a_6_5·a_5_4
  39. b_4_3·b_7_10 + b_2_2·b_4_3·b_5_6 + b_2_22·b_1_12·b_5_6 + b_2_22·b_4_3·b_1_13
       + b_2_2·a_6_5·a_3_3
  40. a_6_5·b_5_6 + b_2_2·a_6_5·a_3_3 + a_2_1·c_8_13·b_1_1
  41. b_8_11·a_3_3 + b_2_2·a_6_5·a_3_3
  42. a_6_52
  43. a_5_4·b_7_10
  44. b_5_6·b_7_10 + b_2_2·b_4_3·b_1_16 + b_2_2·b_4_32·b_1_12 + b_2_22·b_8_11
       + b_2_22·b_4_32 + b_2_23·b_1_16 + b_2_24·b_4_3 + b_2_23·a_6_5
       + b_2_2·c_8_13·b_1_12 + a_2_1·b_2_2·c_8_13
  45. b_4_3·b_1_18 + b_4_3·b_8_11 + b_4_32·b_1_14 + b_2_2·b_4_3·b_1_1·b_5_6
       + b_2_2·b_4_3·b_1_16 + b_2_22·b_1_18 + b_2_22·b_4_3·b_1_14
       + b_2_23·b_4_3·b_1_12 + b_2_24·b_1_14 + c_8_13·b_1_14
  46. a_6_5·b_7_10
  47. b_8_11·a_5_4
  48. b_8_11·b_5_6 + b_4_32·b_5_6 + b_2_2·b_4_3·b_1_17 + b_2_23·b_1_17
       + b_2_24·b_4_3·b_1_1 + b_2_25·b_1_13 + b_2_22·a_6_5·a_3_3 + b_4_3·c_8_13·b_1_1
       + b_2_2·c_8_13·b_1_13
  49. b_7_102 + b_2_22·b_4_3·b_1_16 + b_2_22·b_4_32·b_1_12 + b_2_23·b_4_3·b_1_14
       + b_2_24·b_1_16 + b_2_24·b_4_3·b_1_12 + b_2_26·b_1_12 + b_2_24·a_3_32
       + b_2_22·c_8_13·b_1_12 + c_8_13·a_3_32
  50. a_6_5·b_8_11
  51. b_8_11·b_7_10 + b_2_2·b_4_32·b_5_6 + b_2_22·b_4_32·b_1_13 + b_2_23·b_4_3·b_5_6
       + b_2_23·b_4_3·b_1_15 + b_2_24·b_4_3·b_1_13 + b_2_2·b_4_3·c_8_13·b_1_1
  52. b_8_112 + b_4_32·b_8_11 + b_2_2·b_4_32·b_1_1·b_5_6 + b_2_2·b_4_32·b_1_16
       + b_2_2·b_4_33·b_1_12 + b_2_22·b_4_3·b_8_11 + b_2_22·b_4_33
       + b_2_23·b_4_3·b_1_1·b_5_6 + b_2_23·b_4_3·b_1_16 + b_2_23·b_4_32·b_1_12
       + b_2_24·b_1_18 + b_2_25·b_4_3·b_1_12 + b_4_3·c_8_13·b_1_14 + b_4_32·c_8_13
       + b_2_22·c_8_13·b_1_14

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 16.
  • 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_13, a Duflot regular element of degree 8
    2. b_1_12 + b_2_2, an element of degree 2
    3. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 7, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. b_4_3 → 0, an element of degree 4
  7. a_5_4 → 0, an element of degree 5
  8. b_5_6 → 0, an element of degree 5
  9. a_6_5 → 0, an element of degree 6
  10. b_7_10 → 0, an element of degree 7
  11. b_8_11 → 0, an element of degree 8
  12. c_8_13 → 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. b_1_1 → c_1_1, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. a_3_3 → 0, an element of degree 3
  6. b_4_3 → c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_0·c_1_13 + c_1_02·c_1_12, an element of degree 4
  7. a_5_4 → 0, an element of degree 5
  8. b_5_6 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_04·c_1_1, an element of degree 5
  9. a_6_5 → 0, an element of degree 6
  10. b_7_10 → c_1_13·c_1_24 + c_1_15·c_1_22 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_15·c_1_2
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_13·c_1_22 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2, an element of degree 7
  11. b_8_11 → c_1_14·c_1_24 + c_1_17·c_1_2 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_0·c_1_17 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16
       + c_1_03·c_1_15 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12, an element of degree 8
  12. c_8_13 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_0·c_1_17
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16
       + 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

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