Cohomology of group number 338 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) · (t5  −  t4  +  t3  −  t2  −  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 10 minimal generators of maximal degree 4:

  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. a_2_3, a nilpotent element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_3, a nilpotent element of degree 3
  7. b_3_8, an element of degree 3
  8. b_4_11, an element of degree 4
  9. c_4_12, a Duflot regular element of degree 4
  10. c_4_13, a Duflot regular element of degree 4

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Ring relations

There are 19 minimal relations of maximal degree 8:

  1. a_1_0·b_1_1
  2. a_1_0·b_1_2
  3. b_1_1·b_1_22 + a_1_03
  4. a_2_3·b_1_2 + a_1_03
  5. a_2_3·b_1_1 + a_1_03
  6. b_2_5·b_1_1 + a_1_03
  7. b_1_2·a_3_3
  8. a_1_0·a_3_3 + a_2_32
  9. b_1_1·a_3_3
  10. a_1_0·b_3_8
  11. a_2_3·b_3_8 + a_2_32·a_1_0
  12. b_4_11·a_1_0 + a_2_3·a_3_3 + a_2_3·b_2_5·a_1_0
  13. b_4_11·b_1_1
  14. a_3_3·b_3_8 + a_2_3·b_2_5·a_1_02
  15. a_3_32 + b_2_52·a_1_02 + a_2_32·b_2_5 + a_2_3·b_2_5·a_1_02 + a_2_33
       + c_4_12·a_1_02
  16. b_3_82 + b_1_26 + b_1_12·b_1_2·b_3_8 + b_2_52·b_1_22 + c_4_13·b_1_22
       + c_4_13·b_1_12 + c_4_12·b_1_12
  17. a_2_3·b_4_11 + a_3_32 + a_2_32·b_2_5
  18. b_4_11·a_3_3 + a_2_3·b_2_52·a_1_0 + a_2_3·c_4_12·a_1_0 + c_4_12·a_1_03
  19. b_1_25·b_3_8 + b_1_28 + b_4_112 + b_2_5·b_4_11·b_1_22 + b_2_52·b_1_2·b_3_8
       + b_2_53·b_1_22 + b_2_53·a_1_02 + a_2_32·b_2_52 + a_2_33·b_2_5
       + c_4_12·b_1_24 + b_2_5·c_4_12·a_1_02 + a_2_32·c_4_12 + a_2_3·c_4_12·a_1_02

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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_4_12, a Duflot regular element of degree 4
    2. c_4_13, a Duflot regular element of degree 4
    3. b_1_22 + b_1_12 + b_2_5, 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, 3, 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. a_2_3 → 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. b_3_8 → 0, an element of degree 3
  8. b_4_11 → 0, an element of degree 4
  9. c_4_12 → c_1_14, an element of degree 4
  10. c_4_13 → c_1_04, an element of degree 4

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. a_2_3 → 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. b_3_8 → 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
  8. b_4_11 → 0, an element of degree 4
  9. c_4_12 → c_1_1·c_1_23 + c_1_14 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  10. c_4_13 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + 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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_5 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. a_3_3 → 0, an element of degree 3
  7. b_3_8 → c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  8. b_4_11 → c_1_2·c_1_33 + c_1_23·c_1_3 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
  9. c_4_12 → c_1_34 + 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_14 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
  10. c_4_13 → c_1_34 + c_1_22·c_1_32 + c_1_24 + 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