Mod-11-Cohomology of McL, a group of order 898128000

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General information on the group

  • McL is a group of order 898128000.
  • The group order factors as 27 · 36 · 53 · 7 · 11.
  • The group is defined by Group([(1,191)(2,182)(4,81)(5,55)(6,60)(8,66)(9,272)(10,177)(11,192)(12,163)(14,242)(15,133)(16,107)(18,267)(19,108)(20,218)(21,198)(22,185)(23,211)(24,82)(25,204)(26,195)(27,132)(28,253)(29,207)(30,59)(31,179)(32,154)(33,264)(34,152)(35,92)(36,189)(37,217)(38,197)(39,85)(40,156)(42,184)(43,102)(44,50)(45,216)(46,99)(47,181)(49,199)(51,111)(53,158)(54,236)(56,210)(58,103)(61,263)(63,119)(64,138)(65,127)(67,105)(68,137)(70,125)(71,144)(72,219)(73,261)(75,175)(77,269)(78,237)(79,268)(83,232)(84,256)(86,104)(87,95)(89,234)(90,233)(91,140)(94,149)(97,173)(98,160)(100,112)(101,123)(106,221)(109,131)(110,176)(113,262)(114,257)(115,201)(117,260)(118,238)(120,275)(121,214)(122,225)(124,246)(126,170)(129,141)(134,196)(135,167)(142,235)(143,224)(145,205)(146,249)(148,226)(150,243)(153,193)(155,228)(161,231)(164,215)(165,180)(168,222)(169,270)(171,241)(172,259)(174,212)(178,188)(183,266)(186,203)(190,250)(202,247)(208,255)(209,251)(213,252)(223,271)(227,274)(230,240)(239,254)(245,258)(265,273),(1,24,204,92,155)(2,28,272,165,78)(3,67,142,31,255)(4,168,77,17,100)(5,118,19,223,211)(6,274,137,79,245)(7,98,75,73,14)(8,209,43,139,193)(9,266,104,70,145)(10,271,12,13,71)(11,247,138,121,269)(15,218,133,164,196)(16,170,182,65,171)(18,26,198,124,185)(20,128,159,83,38)(21,264,27,64,162)(22,116,53,101,51)(23,179,244,80,203)(25,40,177,85,191)(29,56,135,68,195)(30,132,42,248,146)(32,148,114,49,134)(33,61,163,90,227)(34,241,233,95,181)(35,89,82,205,41)(36,239,275,257,183)(37,54,99,249,176)(39,126,189,136,230)(44,172,153,125,119)(45,234,222,232,212)(46,214,69,167,190)(47,140,268,174,62)(48,55,113,220,235)(50,213,224,202,130)(52,107,262,226,88)(57,221,261,129,58)(59,252,260,216,166)(60,84,188,208,201)(63,94,173,210,81)(66,192,93,169,110)(72,152,197,217,254)(74,206,154,186,219)(76,91,180,238,112)(86,215,231,131,225)(87,115,158,178,240)(96,265,161,120,144)(97,250,243,263,109)(102,207,246,122,127)(103,156,160,151,150)(105,267,199,111,117)(106,273,242,149,143)(108,256,157,147,184)(123,141,259,175,200)(187,251,258,194,236)(228,237,253,270,229)]).
  • It is non-abelian.
  • It has 11-Rank 1.
  • The centre of a Sylow 11-subgroup has rank 1.
  • Its Sylow 11-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(55,1); GF(11)).

General information

  • The cohomology ring is of dimension 1 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2) · (1  −  t3  +  t6))
    ( − 1  +  t) · (1  −  t  +  t2  −  t3  +  t4) · (1  +  t  +  t2  +  t3  +  t4)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 10:

  1. a_9_0, a nilpotent element of degree 9
  2. c_10_0, a Duflot element of degree 10

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Ring relations

There is one "obvious" relation:
   a_9_02

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 10 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_10_0, an element of degree 10
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 9].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(55,1); GF(11))

  1. a_9_0 → a_9_0
  2. c_10_0 → c_10_0

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

  1. a_9_0 → c_2_04·a_1_0, an element of degree 9
  2. c_10_0 → c_2_05, an element of degree 10

About the group Ring generators Ring relations Completion information Restriction maps


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