Cohomology of group number 136 of order 64

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

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 4 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t5  +  t4  −  t3  +  t2  +  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 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_5_14, an element of degree 5
  7. b_5_15, an element of degree 5
  8. b_5_16, an element of degree 5
  9. b_6_21, an element of degree 6
  10. c_8_35, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

  1. a_1_0·b_1_1
  2. a_1_0·b_1_2
  3. b_2_4·a_1_0
  4. b_2_5·b_1_1 + a_1_03
  5. b_2_4·b_1_22 + b_2_4·b_1_1·b_1_2 + b_2_42
  6. b_2_5·a_1_03
  7. a_1_0·b_5_14
  8. a_1_0·b_5_15
  9. b_1_2·b_5_14 + b_1_1·b_5_15
  10. b_1_2·b_5_16 + b_2_4·b_2_52
  11. b_1_1·b_5_16
  12. b_2_5·b_5_14 + a_1_02·b_5_16
  13. b_2_4·b_5_16 + b_2_4·b_2_52·b_1_2
  14. b_1_1·b_1_2·b_5_15 + b_1_12·b_5_15 + b_6_21·b_1_2 + b_2_4·b_5_15 + b_2_42·b_2_5·b_1_2
  15. b_2_5·b_5_14 + b_6_21·a_1_0
  16. b_1_12·b_5_15 + b_1_12·b_5_14 + b_6_21·b_1_1 + b_2_4·b_5_14
  17. b_2_4·b_1_2·b_5_15 + b_2_4·b_1_1·b_5_14 + b_2_4·b_6_21 + b_2_43·b_2_5
  18. b_5_14·b_5_16
  19. b_5_15·b_5_16 + b_2_52·b_6_21 + b_2_42·b_2_53 + b_2_52·a_1_0·b_5_16
  20. b_5_152 + b_1_15·b_5_14 + b_6_21·b_1_24 + b_6_21·b_1_1·b_1_23
       + b_6_21·b_1_13·b_1_2 + b_6_21·b_1_14 + b_2_52·b_1_26 + b_2_53·b_1_24
       + b_2_4·b_1_13·b_5_14 + b_2_4·b_2_5·b_6_21 + b_2_42·b_1_1·b_5_15
       + b_2_42·b_1_1·b_5_14 + b_2_43·b_2_52 + b_2_44·b_2_5 + c_8_35·b_1_22
  21. b_5_14·b_5_15 + b_1_15·b_5_14 + b_6_21·b_1_1·b_1_23 + b_6_21·b_1_12·b_1_22
       + b_6_21·b_1_14 + b_2_4·b_6_21·b_1_12 + c_8_35·b_1_1·b_1_2
  22. b_5_162 + b_2_4·b_2_54 + b_2_52·a_1_0·b_5_16 + b_2_54·a_1_02 + c_8_35·a_1_02
  23. b_5_142 + b_1_15·b_5_14 + b_6_21·b_1_12·b_1_22 + b_6_21·b_1_13·b_1_2
       + b_2_4·b_1_13·b_5_14 + b_2_4·b_6_21·b_1_12 + b_2_42·b_1_1·b_5_14 + c_8_35·b_1_12
  24. b_6_21·b_5_16 + b_2_4·b_2_52·b_5_15 + b_2_42·b_2_53·b_1_2 + b_2_52·a_1_02·b_5_16
       + c_8_35·a_1_03
  25. b_6_21·b_5_15 + b_6_21·b_1_1·b_1_24 + b_6_21·b_1_13·b_1_22 + b_6_21·b_1_14·b_1_2
       + b_2_42·b_2_53·b_1_2 + b_2_43·b_5_15 + b_2_43·b_2_52·b_1_2
       + c_8_35·b_1_1·b_1_22 + c_8_35·b_1_12·b_1_2 + b_2_4·c_8_35·b_1_2
  26. b_6_21·b_5_14 + b_6_21·b_1_12·b_1_23 + b_6_21·b_1_14·b_1_2 + b_6_21·b_1_15
       + b_2_43·b_5_14 + c_8_35·b_1_12·b_1_2 + c_8_35·b_1_13 + b_2_4·c_8_35·b_1_1
  27. b_6_21·b_1_12·b_1_24 + b_6_21·b_1_13·b_1_23 + b_6_21·b_1_14·b_1_22
       + b_6_21·b_1_16 + b_6_212 + b_2_4·b_6_21·b_1_14 + b_2_42·b_6_21·b_1_12
       + b_2_42·b_2_5·b_6_21 + b_2_43·b_1_1·b_5_15 + b_2_43·b_1_1·b_5_14 + b_2_43·b_6_21
       + b_2_43·b_2_53 + b_2_44·b_2_52 + b_2_45·b_2_5 + c_8_35·b_1_12·b_1_22
       + c_8_35·b_1_14 + b_2_42·c_8_35

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

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_22 + b_1_1·b_1_2 + b_1_12 + b_2_5, an element of degree 2
    3. b_1_2, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 7, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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

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_5_14 → 0, an element of degree 5
  7. b_5_15 → 0, an element of degree 5
  8. b_5_16 → 0, an element of degree 5
  9. b_6_21 → 0, an element of degree 6
  10. 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_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_5_14 → 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
  7. b_5_15 → 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
  8. b_5_16 → 0, an element of degree 5
  9. b_6_21 → c_1_0·c_1_12·c_1_23 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_1·c_1_23
       + c_1_02·c_1_14 + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  10. c_8_35 → c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_14·c_1_23
       + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23
       + c_1_04·c_1_12·c_1_22 + 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 → c_1_2, 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_5_14 → c_1_1·c_1_24 + c_1_12·c_1_23 + 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
  7. b_5_15 → c_1_25 + c_1_1·c_1_24 + 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
  8. b_5_16 → 0, an element of degree 5
  9. b_6_21 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_23 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_22 + c_1_04·c_1_12, an element of degree 6
  10. c_8_35 → c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23
       + c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16
       + c_1_04·c_1_12·c_1_22 + 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_5_14 → 0, an element of degree 5
  7. b_5_15 → 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_16 → 0, an element of degree 5
  9. b_6_21 → 0, an element of degree 6
  10. 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_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. b_5_14 → 0, an element of degree 5
  7. b_5_15 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + 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
  8. b_5_16 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  9. b_6_21 → c_1_26 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24
       + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
  10. 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_0·c_1_1·c_1_26 + c_1_0·c_1_14·c_1_23 + c_1_02·c_1_26
       + c_1_04·c_1_1·c_1_23 + 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 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