Cohomology of group number 170 of order 64

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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 3.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t4  −  t3  −  1
    (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 4:

  1. a_1_1, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_0, an element of degree 1
  4. a_3_3, a nilpotent element of degree 3
  5. b_3_2, an element of degree 3
  6. b_3_4, an element of degree 3
  7. c_4_6, a Duflot regular element of degree 4
  8. c_4_7, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

  1. a_1_1·b_1_0 + a_1_22
  2. a_1_2·b_1_0 + a_1_12
  3. a_1_13
  4. a_1_1·a_1_22 + a_1_12·a_1_2
  5. b_1_0·a_3_3 + a_1_2·a_3_3 + a_1_1·a_3_3
  6. a_1_2·b_3_2 + a_1_1·b_3_2
  7. b_1_0·a_3_3 + a_1_1·b_3_4 + a_1_1·b_3_2 + a_1_1·a_3_3
  8. b_1_0·a_3_3 + a_1_2·b_3_4 + a_1_1·b_3_2
  9. a_1_12·b_3_2 + a_1_1·a_1_2·a_3_3
  10. a_3_3·b_3_2
  11. b_3_22 + c_4_6·b_1_02 + c_4_6·a_1_22 + c_4_6·a_1_12
  12. b_3_42 + b_3_22 + b_1_03·b_3_4 + a_3_32 + c_4_7·b_1_02 + c_4_6·b_1_02
       + c_4_7·a_1_12 + c_4_6·a_1_12
  13. a_3_3·b_3_4 + c_4_7·a_1_1·a_1_2 + c_4_7·a_1_12 + c_4_6·a_1_1·a_1_2 + c_4_6·a_1_12
  14. b_3_22 + a_3_32 + c_4_6·b_1_02 + c_4_7·a_1_22 + c_4_7·a_1_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 7.
  • However, the last relation was already found in degree 6 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_6, a Duflot regular element of degree 4
    2. c_4_7, a Duflot regular element of degree 4
    3. b_1_0, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_3_2 → 0, an element of degree 3
  6. b_3_4 → 0, an element of degree 3
  7. c_4_6 → c_1_14, an element of degree 4
  8. c_4_7 → c_1_14 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → c_1_2, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_3_2 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. c_4_6 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  8. c_4_7 → c_1_1·c_1_23 + c_1_14 + c_1_0·c_1_23 + 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