Cohomology of group number 233 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

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 of rank 3 and 4, respectively.

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) · (t5  −  t4  −  t2  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • 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 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_5, a nilpotent element of degree 3
  8. b_3_9, an element of degree 3
  9. b_4_13, an element of degree 4
  10. b_4_14, an element of degree 4
  11. c_4_15, a Duflot regular element of degree 4
  12. c_4_16, a Duflot regular element of degree 4
  13. a_5_14, a nilpotent element of degree 5

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Ring relations

There are 44 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_22 + a_1_0·b_1_22
  4. b_2_4·b_1_1 + a_1_0·b_1_22
  5. a_1_0·b_1_22 + b_2_5·a_1_0
  6. b_2_5·b_1_1 + a_1_0·b_1_22
  7. b_2_52 + b_2_4·b_1_22
  8. a_1_0·a_3_3
  9. b_1_1·a_3_3
  10. b_2_5·a_1_0·b_1_2 + b_2_4·a_1_0·b_1_2 + a_1_0·a_3_5
  11. b_1_1·a_3_5
  12. a_1_0·b_3_9
  13. b_2_5·b_3_9 + b_1_22·a_3_5 + b_2_5·a_3_3 + b_2_4·b_2_5·a_1_0
  14. b_1_22·b_3_9 + b_1_22·a_3_5 + b_2_5·a_3_3 + b_2_4·b_2_5·a_1_0
  15. b_2_4·b_3_9 + b_2_5·a_3_5 + b_2_4·a_3_3 + b_2_4·b_2_5·a_1_0
  16. b_1_22·a_3_5 + b_4_13·a_1_0 + b_2_5·a_3_3 + b_2_4·b_2_5·a_1_0
  17. b_4_13·b_1_1 + b_1_22·a_3_5 + b_2_5·a_3_3 + b_2_4·b_2_5·a_1_0
  18. b_4_14·a_1_0 + b_2_5·a_3_5 + b_2_4·a_3_3 + b_2_4·b_2_5·a_1_0
  19. b_4_14·b_1_1 + b_1_22·a_3_5 + b_2_5·a_3_3 + b_2_4·b_2_5·a_1_0
  20. a_3_32
  21. a_3_52
  22. a_3_3·b_3_9
  23. a_3_5·b_3_9 + a_3_3·a_3_5
  24. b_3_92 + c_4_15·b_1_12
  25. b_4_13·a_1_0·b_1_2 + b_2_5·b_1_2·a_3_5 + b_2_4·b_1_2·a_3_3 + b_2_42·a_1_0·b_1_2
       + a_3_3·a_3_5 + b_2_4·a_1_0·a_3_5
  26. b_2_5·b_4_14 + b_2_4·b_1_24 + b_2_4·b_4_13 + b_2_42·b_1_22
  27. b_4_14·b_1_22 + b_2_5·b_1_24 + b_2_5·b_4_13 + b_2_4·b_2_5·b_1_22
  28. a_3_3·a_3_5 + a_1_0·a_5_14
  29. b_1_1·a_5_14
  30. b_4_13·b_3_9 + b_2_42·b_2_5·a_1_0 + b_2_5·c_4_16·a_1_0 + b_2_5·c_4_15·a_1_0
  31. b_4_14·a_3_3 + b_4_13·a_3_5 + b_2_5·b_1_22·a_3_3 + b_2_4·b_4_13·a_1_0
       + b_2_4·b_2_5·a_3_3 + b_2_42·b_2_5·a_1_0 + b_2_5·c_4_15·a_1_0 + b_2_4·c_4_16·a_1_0
  32. b_4_14·b_3_9 + b_2_42·b_2_5·a_1_0 + b_2_5·c_4_15·a_1_0 + b_2_4·c_4_16·a_1_0
  33. b_4_13·a_3_5 + b_2_5·a_5_14 + b_2_4·b_1_22·a_3_3 + b_2_4·b_2_5·a_3_3
       + b_2_5·c_4_16·a_1_0 + b_2_5·c_4_15·a_1_0
  34. b_1_22·a_5_14 + b_4_13·a_3_3 + b_2_5·b_1_22·a_3_3 + b_2_4·b_1_22·a_3_3
       + b_2_4·b_4_13·a_1_0 + b_2_42·b_2_5·a_1_0
  35. b_4_14·a_3_5 + b_2_4·a_5_14 + b_2_4·b_1_22·a_3_3 + b_2_4·b_4_13·a_1_0
       + b_2_4·b_2_5·a_3_5 + b_2_4·b_2_5·a_3_3 + b_2_42·a_3_3 + b_2_42·b_2_5·a_1_0
       + b_2_5·c_4_15·a_1_0 + b_2_4·c_4_16·a_1_0
  36. b_4_13·b_1_24 + b_4_132 + b_2_5·b_1_26 + b_2_4·b_1_26 + b_2_4·b_4_13·b_1_22
       + b_2_4·b_2_5·b_1_24 + c_4_15·b_1_24 + b_2_4·c_4_16·b_1_22
  37. b_4_13·b_4_14 + b_2_4·b_1_26 + b_2_4·b_2_5·b_1_24 + b_2_42·b_1_24
       + b_2_5·c_4_15·b_1_22 + b_2_4·b_2_5·c_4_16
  38. b_4_142 + b_2_4·b_1_26 + b_2_4·b_4_13·b_1_22 + b_2_4·b_2_5·b_1_24
       + b_2_42·b_1_24 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_43·b_1_22
       + b_2_4·c_4_15·b_1_22 + b_2_42·c_4_16
  39. a_3_3·a_5_14 + c_4_16·a_1_0·a_3_5
  40. a_3_5·a_5_14 + b_2_4·a_1_0·a_5_14 + c_4_16·a_1_0·a_3_5
  41. b_3_9·a_5_14 + c_4_16·a_1_0·a_3_5
  42. b_4_13·a_5_14 + b_4_13·b_1_22·a_3_3 + b_2_5·b_1_24·a_3_3 + b_2_5·b_4_13·a_3_3
       + b_2_4·b_1_24·a_3_3 + b_2_4·b_2_5·b_1_22·a_3_3 + b_2_42·b_4_13·a_1_0
       + b_2_43·b_2_5·a_1_0 + c_4_15·b_1_22·a_3_3 + b_4_13·c_4_16·a_1_0 + b_2_5·c_4_16·a_3_5
       + b_2_4·b_2_5·c_4_15·a_1_0
  43. b_4_14·a_5_14 + b_2_4·b_4_13·a_3_3 + b_2_4·b_2_5·a_5_14 + b_2_42·b_1_22·a_3_3
       + b_2_5·c_4_16·a_3_5 + b_2_5·c_4_15·a_3_3 + b_2_4·c_4_16·a_3_5 + b_2_4·c_4_16·a_3_3
       + b_2_4·b_2_5·c_4_15·a_1_0 + b_2_42·c_4_16·a_1_0
  44. a_5_142

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_15, a Duflot regular element of degree 4
    2. c_4_16, 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_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 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. 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_5 → 0, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. b_4_13 → 0, an element of degree 4
  10. b_4_14 → 0, an element of degree 4
  11. c_4_15 → c_1_04, an element of degree 4
  12. c_4_16 → c_1_14 + c_1_04, an element of degree 4
  13. a_5_14 → 0, an element of degree 5

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_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 → 0, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_5 → 0, an element of degree 3
  8. b_3_9 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. b_4_13 → 0, an element of degree 4
  10. b_4_14 → 0, an element of degree 4
  11. c_4_15 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  12. c_4_16 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. a_5_14 → 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 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → c_1_2·c_1_3, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. a_3_5 → 0, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. b_4_13 → c_1_12·c_1_2·c_1_3 + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3, an element of degree 4
  10. b_4_14 → c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  11. c_4_15 → 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_0·c_1_2·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  12. c_4_16 → 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_33
       + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  13. a_5_14 → 0, 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