Mod-2-Cohomology of SymplecticGroup(4,5), a group of order 9360000

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • SymplecticGroup(4,5) is a group of order 9360000.
  • The group order factors as 27 · 32 · 54 · 13.
  • The group is defined by Group([(1,2,4,7)(6,10,13,9)(11,16,23,37)(14,19,29,49)(15,21,33,57)(17,25,41,72)(18,27,45,80)(20,31,53,95)(22,35,61,108)(24,39,68,122)(26,43,76,136)(28,47,84,150)(30,51,91,163)(32,55,71,128)(34,59,105,188)(36,63,112,199)(38,66,118,209)(40,70,126,221)(42,74,133,233)(44,78,140,243)(46,82,147,256)(48,86,94,169)(50,89,159,276)(52,93,167,288)(54,97,173,297)(56,100,178,303)(58,103,184,312)(60,107,192,228)(62,110,129,226)(64,114,202,336)(65,116,206,343)(67,120,211,351)(69,124,210,349)(73,131,229,373)(75,135,201,334)(77,138,239,160)(79,142,246,262)(81,145,252,393)(83,149,260,325)(85,152,137,237)(87,155,269,413)(88,157,273,418)(90,161,278,424)(92,165,277,422)(96,171,293,442)(98,175,139,241)(99,176,222,253)(101,180,306,194)(102,182,291,438)(104,186,314,458)(106,190,313,456)(109,195,326,472)(111,198,268,411)(113,200,332,119)(115,204,339,170)(117,208,347,174)(121,213,255,397)(123,216,358,379)(125,219,361,500)(127,223,365,337)(130,156,271,416)(132,231,375,494)(134,183,311,363)(141,235,378,430)(143,248,390,318)(144,250,224,367)(146,254,395,528)(148,258,394,526)(151,263,405,537)(153,266,177,301)(154,267,289,185)(158,275,420,265)(162,280,187,316)(164,283,431,447)(166,286,321,466)(168,290,436,244)(172,295,444,550)(179,299,348,489)(181,308,342,215)(189,319,465,477)(191,322,285,432)(193,324,470,414)(196,328,474,570)(197,251,346,469)(203,330,476,357)(205,341,309,399)(207,345,488,583)(212,353,492,327)(214,356,496,475)(217,292,440,559)(218,359,259,402)(220,360,499,587)(225,369,507,320)(227,247,388,486)(230,279,426,459)(234,307,452,546)(236,380,515,401)(238,340,483,557)(240,368,506,391)(242,383,517,437)(245,385,451,565)(249,282,272,371)(257,400,533,540)(261,404,535,304)(264,407,510,593)(270,409,421,464)(274,364,503,591)(281,429,551,376)(284,331,478,577)(287,433,554,601)(294,396,529,352)(298,417,544,505)(300,439,558,454)(302,381,481,536)(305,449,543,585)(310,455,552,607)(315,460,425,406)(317,463,572,539)(323,467,574,590)(333,419,547,485)(335,441,450,366)(338,480,387,521)(344,487,525,410)(350,453,567,581)(355,495,443,493)(370,386,471,412)(372,508,446,561)(374,511,462,571)(377,513,382,490)(384,518,594,504)(389,523,600,457)(392,435,497,584)(398,532,603,445)(403,501,589,612)(415,542,482,580)(423,545,502,522)(428,512,538,549)(448,520,597,556)(473,576,531,560)(479,578,614,575)(484,555,566,527)(498,586,568,514)(509,573,599,524)(516,588,615,582)(519,596,618,609)(541,564,610,604)(553,608,606,562)(563,595,617,620)(579,611,619,592)(598,616,613,605),(1,3,6,11,17,26,44,79,143,249,391,413,169,292,441)(2,5,9,14,20,32,56,101,181,309,454,336,199,331,383)(4,8,13,18,28,48,87,156,272,390,525,243,136,236,381)(7,12,10,15,22,36,64,115,205,342,485,303,128,225,370)(16,24,40,71,129,227,372,509,465,63,113,201,335,479,579)(19,30,52,94,133,234,377,514,358,43,77,139,242,384,519)(21,34,60,76,137,238,382,516,533,86,154,268,412,541,605)(23,38,67,121,214,357,497,585,159,155,270,415,517,578,596)(25,42,75)(27,46,83,112,173,298,446,562,431,55,99,177,302,448,563)(29,50,90,162,281,430,552,580,252,114,203,338,481,518,595)(31,54,98)(33,58,104,187,317,464,503,565,118,100,179,305,450,564,611)(35,62,111)(37,65,117,202,337,182,310,275,229,316,462,324,471,575,613)(39,69,125,220,363,470,72,130,228,186,315,461,570,603,507)(41,73,132,232,353,493,200,333,478,209,348,190,321,467,175)(45,81,146,255,398,489,488,521,184,78,141,245,386,520,598)(47,85,153)(49,88,158,140,244,116,207,346,293,213,355,223,366,504,592)(51,92,166,287,347,365,95,170,221,120,212,354,494,572,559)(53,96,172,296,426,549,138,240,380,276,421,124,218,360,266)(57,102,183,269,414,250,392,208,326,397,531,404,536,604,620)(59,106,191,323,469,535,108,194,325,254,396,530,593,551,515)(61,109,196,329,460,511,267,410,440,312,378,258,361,501,135)(66,119,210,350,491,322,468,567,612,70,127,224,368,246,387)(68,123,217)(74,134,235,379,278,425,429,271,417,545,606,487,263,406,538)(80,144,251,178,304,157,274,311,405,280,428,290,437,556,609)(82,148,259,403,420,436,150,262,288,161,279,427,550,496,577)(84,151,264,408,529,576,176,300,369,393,476,165,285,433,198)(89,160,277,423,548,219,362,502,590,93,168,291,439,306,451)(91,164,284)(97,174,299,447,395,492,532,204,340,484,582,547,195,327,473)(103,185,313,457,569,402,534,523,601,107,193,273,419,339,482)(105,189,320)(110,197,330,477,211,352,356,180,307,453,568,558,171,294,443)(122,215,301)(126,222,364)(131,230,374,152,265,409,540,314,459,463,142,247,389,524,506)(145,253,394,527,602,286,434,555,587,149,261,206,344,416,543)(147,257,401)(163,282,411)(167,289,435)(188,318,241)(192,239,345)(216,237,308)(226,371,283)(231,376,424)(233,341,400)(248,319,297)(256,399,334)(260,332,455)(295,445,528)(328,475,351)(343,486,466,573,449)(349,490,499)(359,498,542,418,546)(367,505,432,553,480)(373,510,571)(375,512,537)(385,438,557,500,588)(388,522,599)(407,539,458)(422,508,554)(442,474,495)(444,560,472)(452,566,586)(456,561,574)(483,581,615)(513,589,526)(544,600,608)(594,616,621)(597,619,624)(610,618,623)(614,617,622)]).
  • It is non-abelian.
  • It has 2-Rank 2.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

The computation was based on 50 stability conditions for H*(Syl2(Sp4(3)); GF(2)).

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  −  t  +  t2) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6)
    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)
  • 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].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

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

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_7_0, a nilpotent element of degree 7
  4. c_8_1, a Duflot element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 2 minimal relations of maximal degree 14:

  1. a_3_02
  2. a_7_02

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 14 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_8_1, an element of degree 8
    2. b_4_0, an element of degree 4
  • A Duflot regular sequence is given by c_8_1.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(Syl2(Sp4(3)); GF(2))

  1. a_3_0 → a_2_4·a_1_2 + a_1_23 + a_1_13
  2. b_4_0 → b_1_04 + b_4_8
  3. a_7_0 → a_2_4·a_5_10 + a_2_4·a_2_5·b_1_03 + a_1_22·a_5_10 + a_1_12·a_5_10 + a_1_12·a_5_9
       + b_4_8·a_1_23 + b_4_8·a_1_1·a_1_22
  4. c_8_1 → b_4_82 + a_2_4·a_2_5·b_1_04 + a_2_42·b_1_04 + a_2_42·a_2_5·b_1_02
       + a_1_23·a_5_10 + a_1_13·a_5_10 + a_1_13·a_5_9 + b_4_8·a_1_12·a_1_22
       + b_4_8·a_1_13·a_1_2 + c_8_15

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

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → 0, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. c_8_1 → c_1_08, an element of degree 8

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

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → c_1_14, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. c_8_1 → c_1_04·c_1_14 + c_1_08, an element of degree 8

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

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → c_1_14, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. c_8_1 → c_1_04·c_1_14 + c_1_08, an element of degree 8

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