Cohomology of group number 1687 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 3 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 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t6  +  t5  +  t2  +  t  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-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 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_1_3, an element of degree 1
  5. b_2_8, an element of degree 2
  6. b_5_26, an element of degree 5
  7. b_5_27, an element of degree 5
  8. c_8_57, a Duflot regular element of degree 8

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Ring relations

There are 10 minimal relations of maximal degree 10:

  1. a_1_02
  2. b_1_1·b_1_2 + a_1_0·b_1_1
  3. b_1_2·b_1_32 + b_1_22·b_1_3 + b_2_8·b_1_2 + b_2_8·b_1_1 + a_1_0·b_1_32 + b_2_8·a_1_0
  4. b_2_8·b_1_1·b_1_32
  5. b_1_1·b_5_26 + b_2_82·b_1_1·b_1_3
  6. b_1_2·b_5_27 + b_2_8·b_1_23·b_1_3 + b_2_82·b_1_2·b_1_3 + b_2_82·b_1_22
       + a_1_0·b_5_27 + b_2_8·a_1_0·b_1_22·b_1_3 + b_2_82·a_1_0·b_1_3
  7. b_1_32·b_5_26 + b_1_2·b_1_3·b_5_26 + b_2_8·b_5_27 + b_2_8·b_5_26 + b_2_8·b_1_35
       + b_2_8·b_1_24·b_1_3 + b_2_82·b_1_23 + b_2_83·b_1_2 + a_1_0·b_1_3·b_5_26
       + b_2_82·a_1_0·b_1_22 + b_2_83·a_1_0
  8. b_5_26·b_5_27 + b_2_8·b_1_33·b_5_27 + b_2_8·b_1_22·b_1_3·b_5_26
       + b_2_82·b_1_3·b_5_27 + b_2_82·b_1_3·b_5_26 + b_2_82·b_1_36 + b_2_82·b_1_2·b_5_26
       + b_2_83·b_1_23·b_1_3 + b_2_84·b_1_32 + b_2_84·b_1_2·b_1_3 + b_2_84·b_1_22
       + b_2_82·a_1_0·b_5_26 + b_2_83·a_1_0·b_1_22·b_1_3 + b_2_84·a_1_0·b_1_3
  9. b_5_262 + b_2_8·b_1_22·b_1_3·b_5_26 + b_2_82·b_1_36 + b_2_82·b_1_2·b_5_26
       + b_2_82·b_1_26 + b_2_83·b_1_34 + b_2_83·b_1_24 + b_2_84·b_1_32
       + b_2_84·b_1_22 + b_2_84·b_1_1·b_1_3 + b_2_82·a_1_0·b_5_26 + c_8_57·b_1_22
  10. b_5_272 + b_1_1·b_1_34·b_5_27 + b_1_12·b_1_33·b_5_27 + b_1_14·b_1_3·b_5_27
       + b_2_8·b_1_38 + b_2_8·b_1_27·b_1_3 + b_2_82·b_1_25·b_1_3 + b_2_82·b_1_26
       + b_2_83·b_1_34 + b_2_83·b_1_23·b_1_3 + b_2_84·b_1_32
       + b_2_8·a_1_0·b_1_26·b_1_3 + b_2_82·a_1_0·b_1_24·b_1_3
       + b_2_83·a_1_0·b_1_22·b_1_3 + c_8_57·b_1_12

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • However, the last relation was already found in degree 10 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_57, a Duflot regular element of degree 8
    2. b_1_34 + b_1_23·b_1_3 + b_1_24 + b_1_12·b_1_32 + b_1_14 + b_2_8·b_1_32
         + b_2_8·b_1_2·b_1_3 + b_2_8·b_1_22 + b_2_82, an element of degree 4
    3. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 11].
  • 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 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_1_3 → 0, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_5_26 → 0, an element of degree 5
  7. b_5_27 → 0, an element of degree 5
  8. c_8_57 → 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 → 0, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. b_2_8 → 0, an element of degree 2
  6. b_5_26 → 0, an element of degree 5
  7. b_5_27 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  8. c_8_57 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_15·c_1_2
       + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_13·c_1_2 + 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 → 0, an element of degree 1
  4. b_1_3 → c_1_1, an element of degree 1
  5. b_2_8 → c_1_22, an element of degree 2
  6. b_5_26 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
  7. b_5_27 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  8. c_8_57 → c_1_16·c_1_22 + c_1_17·c_1_2 + 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_1, an element of degree 1
  4. b_1_3 → c_1_2 + c_1_1, an element of degree 1
  5. b_2_8 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_5_26 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_13
       + c_1_04·c_1_1, an element of degree 5
  7. b_5_27 → c_1_25 + c_1_14·c_1_2, an element of degree 5
  8. c_8_57 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_15·c_1_2 + 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_04·c_1_14 + c_1_08, an element of degree 8

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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