Mod-3-Cohomology of L3(3).2, a group of order 11232

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General information on the group

  • L3(3).2 is a group of order 11232.
  • The group order factors as 25 · 33 · 13.
  • The group is defined by Group([(1,2)(3,5)(4,6)(7,10)(8,9)(11,15)(12,16)(13,18)(14,20)(17,22)(19,24)(21,23)(25,26),(1,3)(2,4,7,11)(5,8,12,17)(6,9,13,19)(10,14)(15,21,25,22)(16,20)(18,23)(24,26)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.


Structure of the cohomology ring

The computation was based on 1 stability condition for H*(SmallGroup(216,87); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  −  2·t  +  3·t2  −  3·t3  +  4·t4  −  5·t5  +  6·t6  −  6·t7  +  7·t8  −  8·t9  +  10·t10  −  9·t11  +  9·t12  −  9·t13  +  10·t14  −  8·t15  +  7·t16  −  6·t17  +  6·t18  −  5·t19  +  4·t20  −  3·t21  +  3·t22  −  2·t23  +  t24

    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4) · (1  +  t4) · (1  +  t8)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 16:

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_10_0, a nilpotent element of degree 10
  4. a_11_1, a nilpotent element of degree 11
  5. a_11_0, a nilpotent element of degree 11
  6. c_12_1, a Duflot element of degree 12
  7. a_15_1, a nilpotent element of degree 15
  8. b_16_1, an element of degree 16

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Ring relations

There are 4 "obvious" relations:
   a_3_02, a_11_02, a_11_12, a_15_12

Apart from that, there are 16 minimal relations of maximal degree 27:

  1. a_10_0·a_3_0
  2. a_3_0·a_11_0
  3. b_4_0·a_10_0 − a_3_0·a_11_1
  4. b_4_0·a_11_0
  5. a_3_0·a_15_1 − b_4_0·a_3_0·a_11_1
  6. b_4_0·a_15_1 − b_4_02·a_11_1 + b_4_04·a_3_0 + b_4_0·c_12_1·a_3_0
  7. b_16_1·a_3_0 − b_4_04·a_3_0 − b_4_0·c_12_1·a_3_0
  8. a_10_02
  9. b_4_0·b_16_1 − b_4_05 − b_4_02·c_12_1
  10. a_10_0·a_11_0
  11. a_10_0·a_11_1
  12. a_11_0·a_11_1
  13. a_10_0·a_15_1
  14. a_11_1·a_15_1 − a_11_0·a_15_1 − b_4_03·a_3_0·a_11_1 − c_12_1·a_3_0·a_11_1
  15. a_10_0·b_16_1 − a_11_0·a_15_1 − b_4_03·a_3_0·a_11_1 − c_12_1·a_3_0·a_11_1
  16. b_16_1·a_11_1 − b_16_1·a_11_0 − b_4_04·a_11_1 − b_4_0·c_12_1·a_11_1


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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 27 using the Hilbert-Poincaré criterion.
  • 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_12_1, an element of degree 12
    2. b_16_1, an element of degree 16
  • A Duflot regular sequence is given by c_12_1.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 26].


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Restriction maps

Expressing the generators as elements of H*(SmallGroup(216,87); GF(3))

  1. a_3_0a_3_1 + a_3_0
  2. b_4_0b_4_1 + b_4_0
  3. a_10_0a_10_0
  4. a_11_1a_11_1 + b_4_02·a_3_0
  5. a_11_0a_11_3
  6. c_12_1c_12_2
  7. a_15_1b_4_0·a_11_1 − c_12_2·a_3_0
  8. b_16_1b_4_04 + b_4_0·c_12_2

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_00, an element of degree 3
  2. b_4_00, an element of degree 4
  3. a_10_00, an element of degree 10
  4. a_11_10, an element of degree 11
  5. a_11_00, an element of degree 11
  6. c_12_1c_2_06, an element of degree 12
  7. a_15_10, an element of degree 15
  8. b_16_10, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_00, an element of degree 3
  2. b_4_00, an element of degree 4
  3. a_10_0c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  4. a_11_1 − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  5. a_11_0 − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  6. c_12_1c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  7. a_15_1c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  8. b_16_1 − c_2_12·c_2_26 − c_2_14·c_2_24 − c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. b_4_0c_2_22, an element of degree 4
  3. a_10_0 − c_2_1·c_2_23·a_1_0·a_1_1 + c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  4. a_11_1c_2_25·a_1_1 + c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  5. a_11_00, an element of degree 11
  6. c_12_1c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  7. a_15_1c_2_1·c_2_26·a_1_0 + c_2_12·c_2_25·a_1_1 − c_2_13·c_2_24·a_1_0
       − c_2_16·c_2_2·a_1_1, an element of degree 15
  8. b_16_1c_2_28 + c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_00, an element of degree 3
  2. b_4_00, an element of degree 4
  3. a_10_0c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  4. a_11_1 − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  5. a_11_0 − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  6. c_12_1c_2_26 + c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  7. a_15_1c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  8. b_16_1 − c_2_12·c_2_26 − c_2_14·c_2_24 − c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0c_2_2·a_1_1, an element of degree 3
  2. b_4_0c_2_22, an element of degree 4
  3. a_10_0 − c_2_1·c_2_23·a_1_0·a_1_1 + c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  4. a_11_1c_2_25·a_1_1 + c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  5. a_11_00, an element of degree 11
  6. c_12_1c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12
  7. a_15_1c_2_1·c_2_26·a_1_0 + c_2_12·c_2_25·a_1_1 − c_2_13·c_2_24·a_1_0
       − c_2_16·c_2_2·a_1_1, an element of degree 15
  8. b_16_1c_2_28 + c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16


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Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2



Last change: 14.12.2010