Cohomology of group number 139 of order 64

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General information on the group

  • The group has 3 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 5 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) · (t7  +  2·t5  +  t3  +  t2  +  t  +  1)
    (t  +  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 10 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. 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. b_3_9, an element of degree 3
  7. b_5_16, an element of degree 5
  8. b_5_18, an element of degree 5
  9. b_5_19, an element of degree 5
  10. c_8_40, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 10:

  1. a_1_0·b_1_1
  2. a_1_0·b_1_2
  3. a_1_03
  4. b_2_5·b_1_1 + b_2_4·b_1_2
  5. b_2_4·b_1_22 + b_2_4·b_1_1·b_1_2
  6. b_1_2·b_3_9 + b_2_5·a_1_02
  7. b_1_1·b_3_9 + b_2_4·a_1_02
  8. b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2
  9. a_1_02·b_3_9
  10. a_1_0·b_5_16
  11. b_1_2·b_5_18 + b_2_52·b_1_22 + b_2_4·b_1_13·b_1_2 + b_2_42·b_1_1·b_1_2
       + b_2_4·b_2_5·a_1_02
  12. b_3_92 + b_2_4·b_2_52 + b_2_42·b_2_5 + a_1_0·b_5_18 + b_2_5·a_1_0·b_3_9
       + b_2_4·a_1_0·b_3_9 + b_2_4·b_2_5·a_1_02
  13. b_1_1·b_5_18 + b_2_4·b_1_13·b_1_2 + b_2_42·b_1_1·b_1_2 + b_2_42·b_1_12
       + b_2_4·b_2_5·a_1_02
  14. b_3_92 + b_2_4·b_2_52 + b_2_42·b_2_5 + a_1_0·b_5_19
  15. b_1_2·b_5_19 + b_1_2·b_5_16 + b_1_1·b_5_19 + b_2_52·b_1_22 + b_2_4·b_1_13·b_1_2
       + b_2_42·b_1_1·b_1_2 + b_2_42·b_1_12 + b_2_52·a_1_02 + b_2_42·a_1_02
  16. a_1_02·b_5_18
  17. b_2_5·b_5_19 + b_2_5·b_5_18 + b_2_5·b_5_16 + b_2_52·b_3_9 + b_2_4·b_5_19 + b_2_4·b_5_18
       + b_2_42·b_3_9 + b_2_42·b_1_12·b_1_2 + b_2_43·b_1_2
  18. b_2_4·b_1_2·b_5_16 + b_2_42·b_1_13·b_1_2 + b_2_43·b_1_1·b_1_2
  19. b_3_9·b_5_16 + b_2_5·a_1_0·b_5_18 + b_2_4·a_1_0·b_5_18 + b_2_42·b_2_5·a_1_02
  20. b_3_9·b_5_19 + b_3_9·b_5_18 + b_3_9·b_5_16 + b_2_4·b_2_53 + b_2_43·b_2_5
       + b_2_5·a_1_0·b_5_18 + b_2_52·a_1_0·b_3_9 + b_2_42·a_1_0·b_3_9
  21. b_2_4·b_2_5·b_5_16 + b_2_43·b_1_12·b_1_2 + b_2_44·b_1_2 + a_1_0·b_3_9·b_5_18
  22. b_5_16·b_5_18 + b_2_52·b_1_2·b_5_16 + b_2_42·b_1_1·b_5_16 + b_2_42·b_1_15·b_1_2
       + b_2_43·b_1_13·b_1_2 + b_2_4·b_2_52·a_1_0·b_3_9 + b_2_42·b_2_5·a_1_0·b_3_9
       + b_2_43·b_2_5·a_1_02
  23. b_5_18·b_5_19 + b_5_182 + b_2_5·b_3_9·b_5_18 + b_2_52·b_1_2·b_5_16
       + b_2_4·b_3_9·b_5_18 + b_2_4·b_1_13·b_5_19 + b_2_42·b_1_15·b_1_2 + b_2_43·b_1_14
       + b_2_44·b_1_1·b_1_2 + b_2_4·b_2_52·a_1_0·b_3_9 + b_2_43·b_2_5·a_1_02
  24. b_5_192 + b_5_182 + b_1_25·b_5_16 + b_1_12·b_1_23·b_5_16 + b_1_14·b_1_2·b_5_16
       + b_1_15·b_5_19 + b_2_52·b_1_26 + b_2_4·b_2_54 + b_2_42·b_1_16
       + b_2_42·b_2_53 + b_2_43·b_2_52 + b_2_44·b_2_5 + b_2_52·a_1_0·b_5_18
       + b_2_53·a_1_0·b_3_9 + b_2_4·b_2_52·a_1_0·b_3_9 + b_2_42·a_1_0·b_5_18
       + b_2_42·b_2_5·a_1_0·b_3_9 + b_2_43·a_1_0·b_3_9 + b_2_43·b_2_5·a_1_02
       + c_8_40·b_1_22
  25. b_5_192 + b_5_18·b_5_19 + b_5_16·b_5_19 + b_1_1·b_1_24·b_5_16
       + b_1_13·b_1_22·b_5_16 + b_1_15·b_5_19 + b_2_5·b_3_9·b_5_18 + b_2_4·b_3_9·b_5_18
       + b_2_4·b_1_17·b_1_2 + b_2_4·b_2_54 + b_2_42·b_1_1·b_5_19 + b_2_42·b_1_1·b_5_16
       + b_2_42·b_1_15·b_1_2 + b_2_42·b_1_16 + b_2_42·b_2_53 + b_2_43·b_1_13·b_1_2
       + b_2_43·b_2_52 + b_2_44·b_1_12 + b_2_44·b_2_5 + b_2_53·a_1_0·b_3_9
       + b_2_4·b_2_52·a_1_0·b_3_9 + b_2_43·a_1_0·b_3_9 + b_2_43·b_2_5·a_1_02
       + b_2_44·a_1_02 + c_8_40·b_1_1·b_1_2
  26. b_5_182 + b_2_54·b_1_22 + b_2_42·b_1_15·b_1_2 + b_2_42·b_2_53
       + b_2_43·b_2_52 + b_2_44·b_1_12 + b_2_4·b_2_5·a_1_0·b_5_18
       + b_2_43·b_2_5·a_1_02 + b_2_44·a_1_02 + c_8_40·a_1_02
  27. b_5_192 + b_5_182 + b_5_162 + b_1_12·b_1_23·b_5_16 + b_1_14·b_1_2·b_5_16
       + b_1_15·b_5_19 + b_1_15·b_5_16 + b_2_4·b_1_13·b_5_16 + b_2_4·b_2_54
       + b_2_42·b_2_53 + b_2_43·b_1_13·b_1_2 + b_2_43·b_2_52 + b_2_44·b_1_12
       + b_2_44·b_2_5 + b_2_52·a_1_0·b_5_18 + b_2_53·a_1_0·b_3_9
       + b_2_4·b_2_52·a_1_0·b_3_9 + b_2_42·a_1_0·b_5_18 + b_2_42·b_2_5·a_1_0·b_3_9
       + b_2_43·a_1_0·b_3_9 + b_2_43·b_2_5·a_1_02 + c_8_40·b_1_12

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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_8_40, a Duflot regular element of degree 8
    2. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_52 + b_2_4·b_2_5 + b_2_42, an element of degree 4
    3. b_3_9 + b_1_1·b_1_22 + b_1_12·b_1_2 + b_2_5·b_1_2 + b_2_4·b_1_2 + b_2_4·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 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. 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. b_3_9 → 0, an element of degree 3
  7. b_5_16 → 0, an element of degree 5
  8. b_5_18 → 0, an element of degree 5
  9. b_5_19 → 0, an element of degree 5
  10. c_8_40 → 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. b_1_2 → c_1_2, 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. b_3_9 → 0, an element of degree 3
  7. b_5_16 → c_1_0·c_1_1·c_1_23 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_23 + c_1_02·c_1_13
       + c_1_04·c_1_2 + c_1_04·c_1_1, an element of degree 5
  8. b_5_18 → 0, an element of degree 5
  9. b_5_19 → c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_23
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2, an element of degree 5
  10. c_8_40 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_26
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_16 + c_1_04·c_1_1·c_1_23
       + c_1_04·c_1_13·c_1_2 + 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. b_1_2 → 0, an element of degree 1
  4. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_9 → 0, an element of degree 3
  7. b_5_16 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  8. b_5_18 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_19 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. c_8_40 → c_1_28 + c_1_16·c_1_22 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + 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 → 0, an element of degree 1
  3. b_1_2 → c_1_1, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_9 → 0, an element of degree 3
  7. b_5_16 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  8. b_5_18 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_19 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. c_8_40 → c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_12·c_1_24 + 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_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 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_4 → c_1_12, an element of degree 2
  5. b_2_5 → c_1_22, an element of degree 2
  6. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_5_16 → 0, an element of degree 5
  8. b_5_18 → c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
  9. b_5_19 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  10. c_8_40 → c_1_12·c_1_26 + c_1_14·c_1_24 + 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

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. b_1_2 → c_1_1, an element of degree 1
  4. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_9 → 0, an element of degree 3
  7. b_5_16 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  8. b_5_18 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  9. b_5_19 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. c_8_40 → c_1_28 + 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_02·c_1_12·c_1_24 + 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_08, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64


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