Cohomology of group number 125 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 4.
  • 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 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t3  −  2·t2  +  t  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-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 4:

  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. b_2_1, an element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. c_2_4, a Duflot regular element of degree 2
  8. b_3_5, an element of degree 3
  9. b_3_6, an element of degree 3
  10. b_3_7, an element of degree 3
  11. b_3_8, an element of degree 3
  12. b_4_11, an element of degree 4
  13. c_4_14, a Duflot regular element of degree 4

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Ring relations

There are 44 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. b_2_2·a_1_1 + b_2_1·a_1_1
  7. b_2_2·a_1_0 + b_2_1·a_1_1
  8. b_2_3·a_1_0
  9. a_2_02
  10. b_2_22 + b_2_1·b_2_2 + a_2_0·b_2_1
  11. a_1_1·b_3_5 + a_2_0·b_2_2
  12. a_1_0·b_3_5 + a_2_0·b_2_1
  13. a_1_1·b_3_6 + a_2_0·b_2_2
  14. a_1_0·b_3_6 + a_2_0·b_2_2
  15. a_1_1·b_3_7 + a_2_0·b_2_2
  16. a_1_0·b_3_7 + a_2_0·b_2_1
  17. a_1_1·b_3_8 + a_2_0·b_2_3 + a_2_0·b_2_2
  18. a_1_0·b_3_8 + a_2_0·b_2_1
  19. b_2_12·a_1_1 + a_2_0·b_3_5 + b_2_1·c_2_4·a_1_0
  20. b_2_2·b_3_5 + b_2_1·b_3_6
  21. b_2_2·b_3_6 + b_2_2·b_3_5 + b_2_12·a_1_1 + b_2_1·c_2_4·a_1_0
  22. b_2_12·a_1_1 + a_2_0·b_3_6 + b_2_1·c_2_4·a_1_1
  23. b_2_12·a_1_1 + a_2_0·b_3_7 + b_2_1·c_2_4·a_1_0
  24. b_2_2·b_3_7 + b_2_2·b_3_5 + b_2_1·b_3_8 + b_2_1·b_3_7
  25. b_2_2·b_3_8 + b_2_2·b_3_5
  26. b_2_12·a_1_1 + a_2_0·b_3_8 + b_2_1·c_2_4·a_1_0
  27. b_4_11·a_1_1 + b_2_32·a_1_1 + b_2_12·a_1_1 + b_2_3·c_2_4·a_1_1 + b_2_1·c_2_4·a_1_1
  28. b_4_11·a_1_0 + b_2_12·a_1_1 + b_2_1·c_2_4·a_1_1
  29. b_3_52 + b_2_12·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_12 + b_2_12·c_2_4
  30. b_3_5·b_3_6 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_1·b_2_2 + a_2_0·b_2_12
       + b_2_1·b_2_2·c_2_4
  31. b_3_62 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_12 + b_2_1·b_2_2·c_2_4
       + a_2_0·b_2_1·c_2_4
  32. b_3_72 + b_2_1·b_2_32 + b_2_12·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_12 + b_2_12·c_2_4
  33. b_3_6·b_3_7 + b_3_5·b_3_8 + b_3_5·b_3_7 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2
       + a_2_0·b_2_1·b_2_2 + a_2_0·b_2_12 + b_2_1·b_2_2·c_2_4
  34. b_3_6·b_3_8 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_1·b_2_2 + a_2_0·b_2_12
       + b_2_1·b_2_2·c_2_4
  35. b_3_7·b_3_8 + b_3_6·b_3_7 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3
       + b_2_12·b_2_3 + a_2_0·b_2_1·b_2_2 + b_2_1·b_2_2·c_2_4 + b_2_12·c_2_4
  36. b_3_82 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_12·b_2_3 + b_2_12·b_2_2
       + a_2_0·b_2_12 + b_2_12·c_2_4
  37. b_3_5·b_3_7 + b_2_1·b_4_11 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3
       + a_2_0·b_2_1·b_2_2 + b_2_1·b_2_3·c_2_4 + b_2_1·b_2_2·c_2_4 + b_2_12·c_2_4
       + a_2_0·b_2_1·c_2_4
  38. b_3_6·b_3_7 + b_2_2·b_4_11 + b_2_2·b_2_32 + a_2_0·b_2_1·b_2_2 + b_2_2·b_2_3·c_2_4
       + a_2_0·b_2_2·c_2_4 + a_2_0·b_2_1·c_2_4
  39. a_2_0·b_4_11 + a_2_0·b_2_32 + a_2_0·b_2_1·b_2_2 + a_2_0·b_2_3·c_2_4
       + a_2_0·b_2_2·c_2_4
  40. b_4_11·b_3_5 + b_2_32·b_3_5 + b_2_1·b_2_3·b_3_7 + b_2_1·b_2_3·b_3_6 + b_2_1·b_2_3·b_3_5
       + b_2_12·b_3_8 + b_2_12·b_3_7 + b_2_12·b_3_6 + b_2_3·c_2_4·b_3_5 + b_2_1·c_2_4·b_3_7
       + b_2_1·c_2_4·b_3_6 + b_2_1·c_2_4·b_3_5 + b_2_12·c_2_4·a_1_0 + b_2_1·c_2_42·a_1_0
  41. b_4_11·b_3_6 + b_2_32·b_3_6 + b_2_1·b_2_3·b_3_8 + b_2_1·b_2_3·b_3_7 + b_2_1·b_2_3·b_3_6
       + b_2_12·b_3_8 + b_2_12·b_3_7 + b_2_12·b_3_6 + a_2_0·b_2_1·b_3_5 + b_2_3·c_2_4·b_3_6
       + b_2_1·c_2_4·b_3_8 + b_2_1·c_2_4·b_3_7 + b_2_1·c_2_4·b_3_6 + b_2_1·c_2_42·a_1_1
       + b_2_1·c_2_42·a_1_0
  42. b_4_11·b_3_7 + b_2_32·b_3_7 + b_2_32·b_3_5 + b_2_1·b_2_3·b_3_8 + b_2_1·b_2_3·b_3_6
       + b_2_1·b_2_3·b_3_5 + b_2_12·b_3_6 + b_2_3·c_2_4·b_3_7 + b_2_1·c_2_4·b_3_8
       + b_2_1·c_2_4·b_3_6 + b_2_1·c_2_4·b_3_5 + b_2_12·c_2_4·a_1_0 + b_2_1·c_2_42·a_1_0
  43. b_4_11·b_3_8 + b_2_32·b_3_8 + b_2_32·b_3_6 + b_2_32·b_3_5 + b_2_1·b_2_3·b_3_7
       + b_2_1·b_2_3·b_3_6 + b_2_1·b_2_3·b_3_5 + b_2_12·b_3_8 + b_2_12·b_3_7 + b_2_12·b_3_6
       + b_2_3·c_2_4·b_3_8 + b_2_1·c_2_4·b_3_7 + b_2_1·c_2_4·b_3_6 + b_2_1·c_2_4·b_3_5
       + b_2_12·c_2_4·a_1_0 + b_2_1·c_2_42·a_1_0
  44. b_4_112 + b_2_34 + b_2_1·b_2_33 + b_2_13·b_2_2 + a_2_0·b_2_13
       + b_2_1·b_2_32·c_2_4 + b_2_32·c_2_42 + b_2_1·b_2_2·c_2_42 + a_2_0·b_2_1·c_2_42

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 8.
  • 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_2_4, a Duflot regular element of degree 2
    2. c_4_14, a Duflot regular element of degree 4
    3. b_2_3 + b_2_1, an element of degree 2
    4. b_3_7 + b_3_5, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 7].
  • 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. a_2_0 → 0, an element of degree 2
  4. b_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. c_2_4 → c_1_12, an element of degree 2
  8. b_3_5 → 0, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_3_7 → 0, an element of degree 3
  11. b_3_8 → 0, an element of degree 3
  12. b_4_11 → 0, an element of degree 4
  13. c_4_14 → c_1_14 + c_1_04, an element of degree 4

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. a_2_0 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  7. c_2_4 → c_1_2·c_1_3 + c_1_12, an element of degree 2
  8. b_3_5 → c_1_22·c_1_3 + c_1_1·c_1_22, an element of degree 3
  9. b_3_6 → 0, an element of degree 3
  10. b_3_7 → c_1_2·c_1_32 + c_1_1·c_1_22, an element of degree 3
  11. b_3_8 → c_1_2·c_1_32 + c_1_1·c_1_22, an element of degree 3
  12. b_4_11 → 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, an element of degree 4
  13. c_4_14 → c_1_2·c_1_33 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_14
       + 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

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. a_2_0 → 0, an element of degree 2
  4. b_2_1 → c_1_32, an element of degree 2
  5. b_2_2 → c_1_32, an element of degree 2
  6. b_2_3 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  7. c_2_4 → c_1_2·c_1_3 + c_1_12, an element of degree 2
  8. b_3_5 → c_1_33 + c_1_2·c_1_32 + c_1_1·c_1_32, an element of degree 3
  9. b_3_6 → c_1_33 + c_1_2·c_1_32 + c_1_1·c_1_32, an element of degree 3
  10. b_3_7 → c_1_33 + c_1_22·c_1_3 + c_1_1·c_1_32, an element of degree 3
  11. b_3_8 → c_1_33 + c_1_2·c_1_32 + c_1_1·c_1_32, an element of degree 3
  12. b_4_11 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_24 + 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, an element of degree 4
  13. c_4_14 → c_1_34 + c_1_22·c_1_32 + c_1_23·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
       + 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

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