Cohomology of group number 224 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 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t5  −  3·t4  +  2·t3  −  t  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-5,-4,-4. 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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. b_1_3, an element of degree 1
  5. a_4_8, a nilpotent element of degree 4
  6. b_4_9, an element of degree 4
  7. b_4_10, an element of degree 4
  8. b_4_11, an element of degree 4
  9. b_4_12, an element of degree 4
  10. c_4_13, a Duflot regular element of degree 4
  11. c_4_14, a Duflot regular element of degree 4
  12. b_6_37, an element of degree 6

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

Ring relations

There are 38 minimal relations of maximal degree 12:

  1. a_1_22 + a_1_0·a_1_2 + a_1_02
  2. a_1_2·b_1_1 + a_1_0·b_1_3 + a_1_0·b_1_1
  3. a_1_03
  4. a_1_0·b_1_32 + a_1_0·b_1_1·b_1_3 + a_1_02·b_1_1
  5. a_4_8·a_1_0
  6. a_4_8·a_1_2
  7. b_4_9·a_1_2 + b_4_9·a_1_0
  8. b_4_10·b_1_3 + b_4_10·b_1_1 + b_4_9·b_1_3 + a_4_8·b_1_3 + a_4_8·b_1_1
  9. b_4_10·a_1_0 + b_4_9·a_1_0 + a_4_8·b_1_1
  10. b_4_10·a_1_2
  11. b_4_11·b_1_1 + b_4_10·b_1_3 + b_4_9·b_1_1 + a_4_8·b_1_3
  12. b_4_11·a_1_0 + a_4_8·b_1_1
  13. b_4_11·a_1_2 + a_4_8·b_1_3 + a_4_8·b_1_1
  14. b_4_12·a_1_0 + b_4_9·a_1_0 + a_4_8·b_1_1
  15. b_4_12·a_1_2
  16. b_6_37·a_1_0 + a_4_8·b_1_12·b_1_3 + c_4_14·a_1_0·b_1_12 + c_4_13·a_1_0·b_1_12
       + c_4_14·a_1_02·b_1_1 + c_4_13·a_1_02·b_1_1
  17. b_6_37·a_1_2 + a_4_8·b_1_33 + a_4_8·b_1_12·b_1_3 + c_4_14·a_1_2·b_1_32
       + c_4_14·a_1_0·b_1_1·b_1_3 + c_4_14·a_1_0·b_1_12 + c_4_13·a_1_0·b_1_1·b_1_3
       + c_4_13·a_1_0·b_1_12 + c_4_14·a_1_02·b_1_1 + c_4_13·a_1_02·b_1_3
       + c_4_13·a_1_02·b_1_1 + c_4_13·a_1_02·a_1_2
  18. a_4_82
  19. a_4_8·b_4_9 + c_4_14·a_1_0·b_1_12·b_1_3 + c_4_14·a_1_0·b_1_13
  20. b_4_92 + c_4_14·b_1_12·b_1_32 + c_4_14·b_1_14
  21. a_4_8·b_4_10 + c_4_14·a_1_0·b_1_12·b_1_3
  22. b_4_102 + c_4_14·b_1_12·b_1_32
  23. b_4_9·b_4_10 + c_4_14·b_1_12·b_1_32 + c_4_14·b_1_13·b_1_3
       + c_4_14·a_1_0·b_1_12·b_1_3 + c_4_14·a_1_0·b_1_13
  24. a_4_8·b_4_11 + c_4_14·a_1_2·b_1_33 + c_4_14·a_1_0·b_1_13
  25. b_4_112 + c_4_14·b_1_34 + c_4_14·b_1_12·b_1_32 + c_4_14·b_1_14
  26. b_4_10·b_4_11 + c_4_14·b_1_1·b_1_33 + c_4_14·b_1_12·b_1_32 + c_4_14·b_1_13·b_1_3
       + c_4_14·a_1_2·b_1_33 + c_4_14·a_1_0·b_1_13
  27. b_4_9·b_4_11 + c_4_14·b_1_1·b_1_33 + c_4_14·b_1_14
  28. a_4_8·b_4_12 + c_4_14·a_1_0·b_1_12·b_1_3
  29. b_4_122 + b_4_9·b_1_1·b_1_33 + b_4_9·b_1_12·b_1_32 + c_4_14·b_1_12·b_1_32
  30. b_6_37·b_1_32 + b_4_11·b_1_34 + b_4_11·b_4_12 + b_4_9·b_1_34 + b_4_9·b_4_12
       + a_4_8·b_1_13·b_1_3 + c_4_14·b_1_34 + c_4_14·b_1_12·b_1_32
       + c_4_13·b_1_12·b_1_32 + c_4_13·a_1_2·b_1_33
  31. b_6_37·b_1_1·b_1_3 + b_4_10·b_1_14 + b_4_10·b_4_12 + b_4_9·b_1_34
       + b_4_9·b_1_1·b_1_33 + b_4_9·b_1_12·b_1_32 + b_4_9·b_1_13·b_1_3
       + a_4_8·b_1_13·b_1_3 + a_4_8·b_1_14 + c_4_14·b_1_1·b_1_33 + c_4_14·b_1_13·b_1_3
       + c_4_13·b_1_13·b_1_3 + c_4_14·a_1_0·b_1_12·b_1_3
  32. b_6_37·b_1_12 + b_4_10·b_1_14 + b_4_10·b_4_12 + b_4_9·b_1_1·b_1_33
       + b_4_9·b_1_12·b_1_32 + b_4_9·b_1_13·b_1_3 + b_4_9·b_4_12 + a_4_8·b_1_13·b_1_3
       + a_4_8·b_1_14 + c_4_14·b_1_12·b_1_32 + c_4_14·b_1_14 + c_4_13·b_1_14
       + c_4_14·a_1_0·b_1_12·b_1_3
  33. a_4_8·b_6_37 + c_4_14·a_1_2·b_1_35 + c_4_14·a_1_0·b_1_14·b_1_3
       + a_4_8·c_4_14·b_1_32 + a_4_8·c_4_14·b_1_1·b_1_3 + a_4_8·c_4_14·b_1_12
       + a_4_8·c_4_13·b_1_12
  34. b_4_12·b_6_37 + b_4_11·b_4_12·b_1_32 + b_4_9·b_4_12·b_1_32
       + c_4_14·b_1_12·b_1_34 + c_4_14·b_1_14·b_1_32 + b_4_12·c_4_14·b_1_32
       + b_4_12·c_4_14·b_1_12 + b_4_12·c_4_13·b_1_12 + b_4_10·c_4_14·b_1_12
       + b_4_9·c_4_14·b_1_1·b_1_3 + c_4_14·a_1_0·b_1_14·b_1_3 + a_4_8·c_4_14·b_1_12
  35. b_4_11·b_6_37 + c_4_14·b_1_36 + c_4_14·b_1_1·b_1_35 + c_4_14·b_1_12·b_1_34
       + b_4_12·c_4_14·b_1_32 + b_4_12·c_4_14·b_1_1·b_1_3 + b_4_12·c_4_14·b_1_12
       + b_4_11·c_4_14·b_1_32 + b_4_10·c_4_14·b_1_12 + b_4_10·c_4_13·b_1_12
       + b_4_9·c_4_14·b_1_1·b_1_3 + b_4_9·c_4_14·b_1_12 + b_4_9·c_4_13·b_1_1·b_1_3
       + b_4_9·c_4_13·b_1_12 + c_4_14·a_1_0·b_1_14·b_1_3 + a_4_8·c_4_14·b_1_12
       + a_4_8·c_4_13·b_1_32 + a_4_8·c_4_13·b_1_1·b_1_3 + a_4_8·c_4_13·b_1_12
  36. b_4_10·b_6_37 + c_4_14·b_1_1·b_1_35 + b_4_12·c_4_14·b_1_1·b_1_3
       + b_4_10·c_4_13·b_1_12 + b_4_9·c_4_14·b_1_32 + b_4_9·c_4_14·b_1_1·b_1_3
       + c_4_14·a_1_2·b_1_35 + a_4_8·c_4_14·b_1_32 + a_4_8·c_4_14·b_1_1·b_1_3
       + a_4_8·c_4_14·b_1_12
  37. b_4_9·b_6_37 + c_4_14·b_1_1·b_1_35 + c_4_14·b_1_12·b_1_34
       + b_4_12·c_4_14·b_1_1·b_1_3 + b_4_12·c_4_14·b_1_12 + b_4_9·c_4_14·b_1_32
       + b_4_9·c_4_14·b_1_12 + b_4_9·c_4_13·b_1_12
  38. b_6_372 + c_4_14·b_1_38 + b_4_9·c_4_14·b_1_1·b_1_33
       + b_4_9·c_4_14·b_1_12·b_1_32 + c_4_142·b_1_34 + c_4_142·b_1_12·b_1_32
       + c_4_142·b_1_14 + c_4_132·b_1_14

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_4_13, a Duflot regular element of degree 4
    2. c_4_14, a Duflot regular element of degree 4
    3. b_1_32 + b_1_1·b_1_3 + b_1_12, an element of degree 2
    4. b_1_3, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 6, 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 64

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_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_4_8 → 0, an element of degree 4
  6. b_4_9 → 0, an element of degree 4
  7. b_4_10 → 0, an element of degree 4
  8. b_4_11 → 0, an element of degree 4
  9. b_4_12 → 0, an element of degree 4
  10. c_4_13 → c_1_04, an element of degree 4
  11. c_4_14 → c_1_14, an element of degree 4
  12. b_6_37 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. a_4_8 → 0, an element of degree 4
  6. b_4_9 → c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22, an element of degree 4
  7. b_4_10 → c_1_12·c_1_2·c_1_3, an element of degree 4
  8. b_4_11 → 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
  9. b_4_12 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3, an element of degree 4
  10. c_4_13 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3
       + 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
  11. c_4_14 → c_1_14, an element of degree 4
  12. b_6_37 → c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3 + c_1_12·c_1_34
       + c_1_12·c_1_23·c_1_3 + c_1_13·c_1_2·c_1_32 + c_1_13·c_1_22·c_1_3
       + c_1_14·c_1_32 + c_1_14·c_1_2·c_1_3 + c_1_14·c_1_22 + c_1_0·c_1_23·c_1_32
       + c_1_0·c_1_24·c_1_3 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_23·c_1_3
       + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6

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