Cohomology of group number 245 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 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-4,-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 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. c_4_14, a Duflot regular element of degree 4
  11. c_4_15, a Duflot regular element of degree 4
  12. a_5_12, a nilpotent element of degree 5

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Ring relations

There are 34 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·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 + b_2_4·a_1_0·b_1_2
  8. b_2_5·a_1_0·b_1_2 + 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_2_4·b_1_23 + b_1_22·a_3_5 + b_2_5·a_3_3
  14. b_1_22·b_3_9 + b_2_5·b_1_23 + b_2_5·a_3_3
  15. b_2_4·b_3_9 + b_2_4·b_2_5·b_1_2 + b_2_5·a_3_5 + b_2_4·a_3_3
  16. b_1_22·a_3_5 + b_4_13·a_1_0 + b_2_5·a_3_3 + b_2_4·a_3_3
  17. b_4_13·b_1_1
  18. a_3_32
  19. a_3_52 + b_2_4·a_1_0·a_3_5
  20. a_3_3·b_3_9 + b_2_5·b_1_2·a_3_3
  21. a_3_5·b_3_9 + b_2_5·b_1_2·a_3_5 + a_3_3·a_3_5
  22. b_3_92 + b_2_4·b_1_24 + c_4_14·b_1_12
  23. b_4_13·a_1_0·b_1_2 + a_3_3·a_3_5
  24. a_3_3·a_3_5 + a_1_0·a_5_12 + b_2_4·a_1_0·a_3_5
  25. b_1_1·a_5_12
  26. b_4_13·b_3_9 + b_2_5·b_4_13·b_1_2 + b_2_5·a_5_12 + b_2_4·b_1_22·a_3_3
       + b_2_4·b_2_5·a_3_5 + b_2_5·c_4_15·a_1_0 + b_2_4·c_4_15·a_1_0
  27. b_1_22·a_5_12 + b_4_13·a_3_3 + 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·a_3_3 + b_2_5·c_4_15·a_1_0 + b_2_4·c_4_15·a_1_0
  28. b_4_13·b_3_9 + b_2_5·b_4_13·b_1_2 + b_4_13·a_3_5 + b_2_4·a_5_12 + b_2_4·b_2_5·a_3_3
       + b_2_42·a_3_5 + b_2_5·c_4_15·a_1_0 + b_2_5·c_4_14·a_1_0 + b_2_4·c_4_15·a_1_0
  29. b_1_28 + b_4_13·b_1_24 + b_4_132 + b_2_5·b_4_13·b_1_22 + b_2_4·b_1_26
       + b_2_4·b_4_13·b_1_22 + b_1_25·a_3_3 + b_2_4·b_1_23·a_3_3 + b_2_42·b_1_2·a_3_3
       + a_3_3·a_5_12 + c_4_14·b_1_24 + b_2_4·c_4_15·b_1_22 + b_2_42·c_4_15
       + c_4_15·a_1_0·a_3_3
  30. b_1_28 + b_4_13·b_1_24 + b_4_132 + b_2_5·b_4_13·b_1_22 + b_2_4·b_1_26
       + b_2_4·b_4_13·b_1_22 + b_1_25·a_3_3 + b_2_4·b_1_23·a_3_3 + b_2_42·b_1_2·a_3_3
       + b_2_4·a_1_0·a_5_12 + b_2_42·a_1_0·a_3_5 + c_4_14·b_1_24 + b_2_4·c_4_15·b_1_22
       + b_2_42·c_4_15 + c_4_15·a_1_0·a_3_5 + c_4_15·a_1_0·a_3_3
  31. b_1_28 + b_4_13·b_1_24 + b_4_132 + b_2_5·b_4_13·b_1_22 + b_2_4·b_1_26
       + b_2_4·b_4_13·b_1_22 + b_1_25·a_3_3 + b_2_4·b_1_23·a_3_3 + b_2_42·b_1_2·a_3_3
       + a_3_5·a_5_12 + b_2_42·a_1_0·a_3_5 + c_4_14·b_1_24 + b_2_4·c_4_15·b_1_22
       + b_2_42·c_4_15 + c_4_15·a_1_0·a_3_5 + c_4_15·a_1_0·a_3_3 + c_4_14·a_1_0·a_3_3
  32. b_1_28 + b_4_13·b_1_24 + b_4_132 + b_2_5·b_4_13·b_1_22 + b_2_4·b_1_26
       + b_2_4·b_4_13·b_1_22 + b_3_9·a_5_12 + b_1_25·a_3_3 + b_2_5·b_1_2·a_5_12
       + b_2_4·b_1_23·a_3_3 + b_2_42·b_1_2·a_3_3 + c_4_14·b_1_24 + b_2_4·c_4_15·b_1_22
       + b_2_42·c_4_15 + c_4_15·a_1_0·a_3_3
  33. b_1_26·a_3_3 + b_4_13·a_5_12 + b_4_13·b_1_22·a_3_3 + b_2_4·b_1_24·a_3_3
       + b_2_4·b_4_13·a_3_3 + b_2_4·b_2_5·a_5_12 + b_2_42·a_5_12 + b_2_42·b_1_22·a_3_3
       + b_2_42·b_2_5·a_3_5 + b_2_42·b_2_5·a_3_3 + b_2_43·a_3_5 + c_4_14·b_1_22·a_3_3
       + b_2_5·c_4_15·a_3_5 + b_2_4·c_4_15·a_3_5 + b_2_42·c_4_15·a_1_0
  34. a_5_122 + b_2_43·a_1_0·a_3_5 + b_2_4·c_4_15·a_1_0·a_3_5

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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_4_14, a Duflot regular element of degree 4
    2. c_4_15, 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, 4, 6, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. c_4_14 → c_1_04, an element of degree 4
  11. c_4_15 → c_1_14 + c_1_04, an element of degree 4
  12. a_5_12 → 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. c_4_14 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  11. c_4_15 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  12. a_5_12 → 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 → c_1_2·c_1_32, an element of degree 3
  9. b_4_13 → c_1_34 + 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_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  10. c_4_14 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + 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_0·c_1_22·c_1_3
       + c_1_04, an element of degree 4
  11. c_4_15 → c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  12. a_5_12 → 0, an element of degree 5

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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