Cohomology of group number 782 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 3.
  • Its center has rank 1.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t6  −  t5  +  2·t4  −  t3  +  t2  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-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_2, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_1, an element of degree 1
  4. a_2_3, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. a_4_7, a nilpotent element of degree 4
  8. b_5_15, an element of degree 5
  9. b_5_16, an element of degree 5
  10. b_5_17, an element of degree 5
  11. a_8_19, a nilpotent element of degree 8
  12. c_8_35, a Duflot regular element of degree 8

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Ring relations

There are 38 minimal relations of maximal degree 16:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_1 + a_1_22
  3. a_1_2·b_1_1
  4. a_2_3·a_1_2
  5. b_2_5·b_1_0 + b_2_4·b_1_1 + b_2_5·a_1_2 + b_2_4·a_1_2
  6. a_2_32
  7. b_2_4·b_2_5·b_1_1 + b_2_42·b_1_1 + b_2_4·b_2_5·a_1_2 + b_2_42·a_1_2
  8. a_4_7·b_1_1 + a_2_3·b_2_4·b_1_1
  9. a_4_7·b_1_0 + a_2_3·b_1_03 + a_2_3·b_2_4·b_1_1
  10. a_2_3·b_2_4·b_1_1 + a_4_7·a_1_2
  11. a_2_3·a_4_7
  12. b_1_1·b_5_15 + b_2_52·b_1_12 + a_2_3·b_1_14 + a_2_3·b_2_4·b_2_5
  13. b_1_0·b_5_15 + a_2_3·b_2_4·b_1_02 + a_2_3·b_2_4·b_2_5 + b_2_42·a_1_22
  14. b_1_1·b_5_16 + b_2_52·b_1_12 + a_1_2·b_5_15 + b_2_4·a_4_7 + a_2_3·b_2_5·b_1_12
       + a_2_3·b_2_4·b_1_02 + a_2_3·b_2_4·b_2_5 + b_2_42·a_1_22
  15. a_1_2·b_5_16 + a_1_2·b_5_15 + b_2_4·a_4_7 + a_2_3·b_2_4·b_1_02 + a_2_3·b_2_4·b_2_5
       + b_2_42·a_1_22
  16. b_1_0·b_5_17 + a_1_2·b_5_15 + b_2_5·a_4_7
  17. a_1_2·b_5_17 + b_2_5·a_4_7 + a_2_3·b_2_4·b_2_5 + b_2_42·a_1_22
  18. b_2_4·b_2_52·a_1_2 + b_2_42·b_2_5·a_1_2 + a_2_3·b_5_15 + a_2_3·b_2_52·b_1_1
       + b_2_4·a_4_7·a_1_2
  19. b_2_5·b_5_16 + b_2_53·b_1_1 + b_2_4·b_5_17 + b_2_4·b_5_15 + b_2_53·a_1_2
       + b_2_4·b_2_52·a_1_2 + b_2_42·b_2_5·a_1_2 + b_2_43·a_1_2 + a_2_3·b_2_52·b_1_1
       + a_2_3·b_2_42·b_1_0
  20. a_4_72
  21. a_4_7·b_5_16 + b_2_43·b_2_5·a_1_2 + b_2_44·a_1_2 + a_2_3·b_1_02·b_5_16
       + a_2_3·b_2_5·b_5_15 + a_2_3·b_2_53·b_1_1
  22. a_4_7·b_5_15 + a_2_3·b_2_4·b_5_17 + a_2_3·b_2_4·b_5_15
  23. a_4_7·b_5_17 + a_4_7·b_5_15 + b_2_54·a_1_2 + b_2_43·b_2_5·a_1_2 + a_2_3·b_2_5·b_5_15
       + a_2_3·b_2_53·b_1_1 + b_2_42·a_4_7·a_1_2
  24. a_8_19·b_1_1 + a_4_7·b_5_15 + a_2_3·b_1_17 + a_2_3·b_2_5·b_5_15 + a_2_3·b_2_53·b_1_1
       + a_2_3·b_2_4·b_5_15 + b_2_42·a_4_7·a_1_2
  25. a_8_19·b_1_0 + a_4_7·b_5_16 + a_4_7·b_5_15 + b_2_43·b_2_5·a_1_2 + b_2_44·a_1_2
       + a_2_3·b_1_07 + a_2_3·b_2_4·b_5_15 + a_2_3·b_2_4·b_1_05 + a_2_3·b_2_43·b_1_0
  26. a_4_7·b_5_15 + a_2_3·b_2_5·b_5_15 + a_2_3·b_2_53·b_1_1 + a_2_3·b_2_4·b_5_15
       + a_8_19·a_1_2 + b_2_42·a_4_7·a_1_2
  27. b_5_16·b_5_17 + b_5_15·b_5_16 + b_2_52·b_1_1·b_5_17 + b_2_54·b_1_12 + b_2_4·b_2_54
       + b_2_42·b_2_53 + b_2_43·b_2_52 + b_2_44·b_2_5 + b_2_53·a_4_7
       + b_2_4·b_2_52·a_4_7 + b_2_42·b_2_5·a_4_7 + b_2_43·a_4_7 + a_2_3·b_2_5·b_1_1·b_5_17
       + a_2_3·b_2_52·b_1_14 + a_2_3·b_2_53·b_1_12 + a_2_3·b_2_4·b_1_0·b_5_16
       + a_2_3·b_2_4·b_2_53 + a_2_3·b_2_43·b_1_02
  28. b_5_162 + b_5_152 + b_2_4·b_1_03·b_5_16 + b_2_42·b_1_0·b_5_16 + b_2_42·b_1_06
       + b_2_43·b_1_04 + b_2_44·b_1_02 + b_2_44·b_2_5 + b_2_45 + b_2_43·a_4_7
       + a_2_3·b_1_08 + c_8_35·b_1_02
  29. b_5_172 + b_2_52·b_1_1·b_5_17 + b_2_52·b_1_16 + b_2_53·b_1_14 + b_2_55
       + b_2_4·b_2_54 + b_2_53·a_4_7 + a_2_3·b_1_13·b_5_17 + a_2_3·b_1_18
       + a_2_3·b_2_5·b_1_16 + a_2_3·b_2_53·b_1_12 + a_2_3·b_2_4·b_2_53
       + c_8_35·b_1_12
  30. b_5_152 + b_2_54·b_1_12 + b_2_42·b_2_53 + b_2_43·b_2_52
       + b_2_42·a_1_2·b_5_15 + a_2_3·b_2_43·b_2_5 + b_2_44·a_1_22 + c_8_35·a_1_22
  31. b_5_15·b_5_17 + b_2_52·b_1_1·b_5_17 + b_2_4·b_2_54 + b_2_42·b_2_53 + b_2_5·a_8_19
       + b_2_4·b_2_52·a_4_7 + b_2_42·a_1_2·b_5_15 + b_2_42·b_2_5·a_4_7
       + a_2_3·b_1_13·b_5_17 + a_2_3·b_2_5·b_1_16 + a_2_3·b_2_42·b_2_52
       + a_2_3·b_2_43·b_2_5 + b_2_44·a_1_22
  32. b_5_15·b_5_16 + b_5_152 + b_2_43·b_2_52 + b_2_44·b_2_5 + b_2_4·a_8_19
       + b_2_4·b_2_52·a_4_7 + b_2_42·b_2_5·a_4_7 + b_2_43·a_4_7 + a_2_3·b_2_52·b_1_14
       + a_2_3·b_2_53·b_1_12 + a_2_3·b_2_4·b_1_06 + a_2_3·b_2_42·b_1_04
       + a_2_3·b_2_43·b_1_02 + a_2_3·b_2_44
  33. b_2_4·b_2_52·a_4_7 + b_2_42·b_2_5·a_4_7 + a_2_3·b_2_4·b_2_53 + a_2_3·b_2_43·b_2_5
       + a_2_3·a_8_19
  34. a_4_7·a_8_19 + a_2_3·b_2_5·a_8_19 + a_2_3·b_2_4·a_8_19
  35. b_2_4·b_2_53·b_5_15 + b_2_42·b_2_52·b_5_17 + b_2_42·b_2_52·b_5_15
       + b_2_43·b_2_5·b_5_17 + a_8_19·b_5_15 + a_2_3·b_2_52·b_1_17
       + a_2_3·b_2_4·b_2_52·b_5_17 + a_2_3·b_2_42·b_2_5·b_5_17 + a_2_3·b_2_43·b_5_17
       + a_2_3·b_2_43·b_5_15 + a_4_7·c_8_35·a_1_2
  36. b_2_4·b_2_53·b_5_15 + b_2_42·b_2_52·b_5_17 + b_2_43·b_2_5·b_5_15 + b_2_44·b_5_17
       + a_8_19·b_5_16 + a_2_3·b_1_06·b_5_16 + a_2_3·b_2_52·b_1_17
       + a_2_3·b_2_4·b_2_52·b_5_17 + a_2_3·b_2_42·b_1_02·b_5_16 + a_2_3·b_2_42·b_1_07
       + a_2_3·b_2_43·b_5_16 + a_2_3·b_2_43·b_5_15 + a_2_3·b_2_43·b_1_05
       + a_2_3·b_2_44·b_1_03 + a_2_3·b_2_45·b_1_0 + b_2_52·a_8_19·a_1_2
       + a_2_3·c_8_35·b_1_03
  37. b_2_54·b_5_15 + b_2_56·b_1_1 + b_2_4·b_2_53·b_5_17 + b_2_4·b_2_53·b_5_15
       + b_2_42·b_2_52·b_5_17 + b_2_46·b_1_1 + a_8_19·b_5_17 + b_2_45·b_2_5·a_1_2
       + b_2_46·a_1_2 + a_2_3·b_1_16·b_5_17 + a_2_3·b_2_54·b_1_13 + a_2_3·b_2_43·b_5_15
       + b_2_42·a_8_19·a_1_2 + a_4_7·c_8_35·a_1_2
  38. a_8_192

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_35, a Duflot regular element of degree 8
    2. b_1_14 + b_1_04 + b_2_52 + b_2_4·b_2_5 + b_2_42, an element of degree 4
    3. b_2_52·b_1_12 + b_2_4·b_2_52 + b_2_42·b_1_02 + b_2_42·b_2_5, an element of degree 6
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 15].
  • 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_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. b_5_15 → 0, an element of degree 5
  9. b_5_16 → 0, an element of degree 5
  10. b_5_17 → 0, an element of degree 5
  11. a_8_19 → 0, an element of degree 8
  12. c_8_35 → c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_1, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. b_5_15 → 0, an element of degree 5
  9. b_5_16 → c_1_25 + c_1_12·c_1_23 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_17 → 0, an element of degree 5
  11. a_8_19 → 0, an element of degree 8
  12. c_8_35 → c_1_28 + c_1_1·c_1_27 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → c_1_1, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. b_5_15 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_16 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. b_5_17 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_02·c_1_13
       + c_1_04·c_1_1, an element of degree 5
  11. a_8_19 → 0, an element of degree 8
  12. c_8_35 → 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_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

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_12, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. b_5_15 → c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
  9. b_5_16 → c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  10. b_5_17 → c_1_25 + c_1_1·c_1_24, an element of degree 5
  11. a_8_19 → 0, an element of degree 8
  12. c_8_35 → c_1_12·c_1_26 + c_1_16·c_1_22 + 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

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