Cohomology of group number 1415 of order 128

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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 3.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

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

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. c_2_7, a Duflot regular element of degree 2
  6. c_2_8, a Duflot regular element of degree 2
  7. a_4_9, a nilpotent element of degree 4
  8. b_4_19, an element of degree 4
  9. b_4_20, an element of degree 4
  10. c_4_24, a Duflot regular element of degree 4

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Ring relations

There are 18 minimal relations of maximal degree 8:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. b_1_2·b_1_3 + a_1_1·b_1_2
  3. a_1_1·b_1_3 + a_1_0·b_1_3 + a_1_0·b_1_2 + a_1_12 + a_1_02
  4. a_1_03
  5. a_4_9·a_1_1 + c_2_8·a_1_02·b_1_2 + c_2_7·a_1_02·b_1_3 + c_2_7·a_1_02·b_1_2
       + c_2_8·a_1_02·a_1_1
  6. a_4_9·a_1_0 + c_2_8·a_1_02·b_1_3 + c_2_8·a_1_02·b_1_2 + c_2_7·a_1_02·b_1_3
       + c_2_8·a_1_02·a_1_1 + c_2_7·a_1_02·a_1_1
  7. b_4_19·b_1_2 + a_4_9·b_1_2 + c_2_7·a_1_1·b_1_22 + c_2_8·a_1_02·b_1_3
       + c_2_8·a_1_02·b_1_2 + c_2_8·a_1_02·a_1_1
  8. b_4_19·a_1_1 + a_4_9·b_1_3 + c_2_7·a_1_02·b_1_3 + c_2_7·a_1_02·b_1_2
       + c_2_7·a_1_02·a_1_1
  9. b_4_19·a_1_0 + a_4_9·b_1_3 + c_2_8·a_1_02·b_1_2 + c_2_8·a_1_02·a_1_1
  10. b_4_20·a_1_1 + a_4_9·b_1_2 + c_2_8·a_1_02·b_1_3 + c_2_8·a_1_02·b_1_2
       + c_2_7·a_1_02·b_1_2
  11. b_4_20·a_1_0 + c_2_7·a_1_02·b_1_3 + c_2_7·a_1_02·b_1_2 + c_2_7·a_1_02·a_1_1
  12. b_4_20·b_1_3 + a_1_0·b_1_34 + a_4_9·b_1_3 + a_4_9·b_1_2 + c_2_8·a_1_0·b_1_32
       + c_2_8·a_1_02·a_1_1 + c_2_7·a_1_02·a_1_1
  13. a_4_92
  14. a_1_0·b_1_37 + a_4_9·b_1_34 + a_4_9·b_4_19 + c_4_24·a_1_0·b_1_33
       + c_2_7·a_1_0·b_1_35 + c_2_82·a_1_0·b_1_33 + c_2_7·c_2_8·a_1_0·b_1_33
       + c_2_72·a_1_0·b_1_33
  15. b_1_38 + b_4_19·b_1_34 + b_4_192 + c_4_24·b_1_34 + c_2_7·b_1_36
       + c_2_8·a_1_0·b_1_35 + c_2_7·a_1_0·b_1_35 + c_2_82·b_1_34 + c_2_7·c_2_8·b_1_34
       + c_2_72·b_1_34
  16. a_4_9·b_4_20 + c_4_24·a_1_1·b_1_23 + c_2_7·a_1_1·b_1_25
       + c_2_7·c_2_8·a_1_1·b_1_23
  17. b_4_202 + a_4_9·b_1_24 + c_4_24·b_1_24 + c_2_7·b_1_26 + c_2_7·a_1_1·b_1_25
       + c_2_7·c_2_8·b_1_24
  18. b_4_19·b_4_20 + a_1_0·b_1_37 + c_4_24·a_1_1·b_1_23 + c_4_24·a_1_0·b_1_33
       + c_2_8·a_4_9·b_1_32 + c_2_7·a_1_1·b_1_25 + c_2_7·a_1_0·b_1_35
       + c_2_7·a_4_9·b_1_22 + c_2_82·a_1_0·b_1_33 + c_2_7·c_2_8·a_1_1·b_1_23
       + c_2_7·c_2_8·a_1_0·b_1_33 + c_2_72·a_1_0·b_1_33

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_2_7, a Duflot regular element of degree 2
    2. c_2_8, a Duflot regular element of degree 2
    3. c_4_24, a Duflot regular element of degree 4
    4. b_1_32 + b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3, 6].
  • 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 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_7 → c_1_12, an element of degree 2
  6. c_2_8 → c_1_22, an element of degree 2
  7. a_4_9 → 0, an element of degree 4
  8. b_4_19 → 0, an element of degree 4
  9. b_4_20 → 0, an element of degree 4
  10. c_4_24 → c_1_12·c_1_22 + c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_7 → c_1_1·c_1_3 + c_1_12, an element of degree 2
  6. c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  7. a_4_9 → 0, an element of degree 4
  8. b_4_19 → 0, an element of degree 4
  9. b_4_20 → c_1_02·c_1_32, an element of degree 4
  10. c_4_24 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. c_2_7 → c_1_1·c_1_3 + c_1_12, an element of degree 2
  6. c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  7. a_4_9 → 0, an element of degree 4
  8. b_4_19 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_02·c_1_32, an element of degree 4
  9. b_4_20 → 0, an element of degree 4
  10. c_4_24 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + 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_12·c_1_22 + c_1_02·c_1_32
       + c_1_04, an element of degree 4

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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