Mod-3-Cohomology of SymplecticGroup(4,8), a group of order 1056706560

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • SymplecticGroup(4,8) is a group of order 1056706560.
  • The group order factors as 212 · 34 · 5 · 72 · 13.
  • The group is defined by Group([(2,3,5,8,13,21,34)(6,10,17,28,47,74,112)(9,15,25,42,68,104,156)(11,19,16,24,40,64,37)(14,23,38,60,94,144,218)(18,30,26,39,62,46,59)(20,32,41,66,27,45,65)(22,36,58,91,139,213,317)(29,49,43,61,96,73,92)(31,52,83,128,193,291,221)(33,55,75,114,105,145,220)(35,57,90,138,212,316,200)(44,71,50,79,121,183,275)(48,76,69,95,146,102,140)(51,81,70,107,161,241,345)(53,85,132,201,198,299,409)(54,87,113,170,157,219,324)(56,89,137,211,315,302,249)(63,99,141,77,117,175,158)(67,103,155,234,235,323,303)(72,110,167,251,354,430,391)(78,119,179,268,115,93,142)(80,123,186,280,350,423,522)(82,126,190,286,351,455,539)(84,130,197,297,405,191,288)(86,134,205,294,131,199,301)(88,135,207,147,116,173,259)(97,106,159,237,171,98,149)(100,152,229,335,439,454,514)(108,163,245,308,419,519,548)(109,165,248,178,267,153,231)(111,169,255,360,180,270,124)(118,177,265,369,471,557,537)(120,181,272,377,478,561,563)(122,185,278,300,410,509,313)(125,188,284,349,452,544,523)(127,192,189,164,247,208,244)(129,195,295,210,314,262,367)(133,203,306,416,515,143,216)(136,209,312,421,446,540,547)(148,223,327,429,424,451,543)(150,226,330,432,496,567,254)(151,228,333,435,431,202,246)(154,232,296,376,309,368,250)(160,239,343,336,440,399,494)(162,243,348,450,366,467,433)(168,253,357,380,482,459,292)(172,257,362,462,551,464,331)(174,261,365,465,370,461,550)(176,217,322,182,274,258,320)(184,277,382,187,282,390,491)(194,293,403,504,542,483,560)(204,307,418,517,575,570,584)(214,318,406,393,443,266,371)(215,319,425,441,510,417,516)(222,326,334,437,533,536,427)(224,329,227,332,434,342,240)(225,285,394,495,566,529,562)(230,337,442,287,396,304,412)(236,339,444,305,414,512,355)(238,341,445,538,581,518,530)(242,347,383,438,535,479,545)(252,356,458,475,554,498,568)(256,361,344,447,283,392,493)(260,359,460,397,499,415,507)(263,364,276,373,340,402,346)(269,374,474,520,577,422,289)(273,379,480,513,574,485,555)(279,385,468,466,532,472,559)(281,388,489,521,481,549,582)(290,400,502,381,375,469,352)(298,407,506,572,527,526,310)(321,426,524,578,558,576,573)(325,428,411,395,497,389,490)(328,372,473,378,413,387,488)(358,386,487,564,449,492,565)(363,463,553,583,534,528,580)(384,486,398,501,511,569,500)(401,503,571,484,476,556,456)(408,508,505,436,531,453,546)(448,541,552,457,525,579,585),(1,2,4,7,12)(3,6,11,20,33)(5,9,16,27,46)(8,14,24,41,67)(10,18,31,53,86)(13,22,37,45,73)(15,26,44,72,111)(17,29,50,80,124)(19,32,54,34,56)(21,35,40,65,102)(23,39,63,100,153)(25,43,70,108,164)(28,48,77,118,178)(30,51,82,127,89)(36,59,93,143,217)(38,61,97,148,224)(42,69,106,160,240)(47,75,115,172,258)(49,78,120,182,137)(52,84,131,200,303)(55,88,136,210,139)(57,62,98,150,227)(58,92,141,214,231)(60,95,79,122,169)(64,101,154,233,66)(68,105,158,236,267)(71,109,166,250,353)(74,113,159,238,342)(76,116,174,262,211)(81,125,189,112,155)(83,129,196,296,404)(85,133,204,308,343)(87,121,184,255,213)(90,96,147,222,314)(91,140,128,194,294)(94,145,221,325,205)(99,151,165,249,234)(103,149,225,329,317)(104,157,173,260,195)(107,162,244,138,146)(110,168,254,359,461)(114,171,256,332,315)(117,176,264,368,470)(119,180,271,376,477)(123,187,283,393,357)(126,191,289,399,410)(130,198,300,411,510)(132,202,305,415,514)(134,206,309,420,345)(135,208,311,232,338)(142,215,320,156,235)(144,219,161,242,247)(152,230,288,398,293)(163,246,351,277,383)(167,252,251,355,457)(170,193,292,199,302)(175,263,248,352,456)(177,266,372,437,534)(179,269,322,316,324)(181,273,380,483,385)(183,276,270,375,476)(185,279,386,379,481)(186,281,389,390,492)(188,285,395,429,526)(190,287,397,500,203)(192,290,401,241,346)(197,298,408,506,573)(201,304,413,511,463)(207,310,295,218,323)(209,313,423,523,575)(212,220,275,307,360)(216,321,427,516,546)(223,328,431,528,581)(226,331,433,444,537)(228,334,438,509,495)(229,336,441,354,330)(237,340,434,502,571)(239,344,448,374,475)(243,349,453,517,426)(245,350,454,547,566)(253,358,459,549,535)(257,363,464,280,387)(259,364,367,469,556)(261,366,468,467,555)(265,370,312,422,326)(268,373,274,381,484)(272,378,479,553,284)(278,384,394,496,442)(282,391,412,455,501)(286,369,472,414,513)(291,402,301,400,503)(297,406,348,451,403)(299,341,446,339,377)(306,417,327,430,527)(318,424,478,562,491)(319,388,335,347,449)(333,436,532,578,574)(337,443,486,371,416)(356,425,462,552,490)(361,418,518,557,362)(365,466,382,485,563)(392,494,559,409,480)(396,498,569,540,579)(405,505,471,558,584)(407,507,564,439,489)(419,520,515,504,561)(421,521,543,538,487)(428,525,544,568,512)(432,529,508,519,576)(435,530,474,497,550)(440,536,567,450,542)(445,539,493,482,460)(447,452,545,551,572)(458,548,488,541,580)(465,554,499,570,585)(473,560,583,522,533)(524,577,565,531,582)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(648,252); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • 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. c_4_0, a Duflot element of degree 4
  3. a_7_1, 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 "obvious" relations:
   a_3_02, a_7_12

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.
  • However, the last relation was already found in degree 0 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_0, an element of degree 4
    2. c_8_1, an element of degree 8
  • The above filter regular HSOP forms a Duflot regular sequence.
  • 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*(SmallGroup(648,252); GF(3))

  1. a_3_0 → a_3_0
  2. c_4_0 → c_4_0
  3. a_7_1 → a_7_1
  4. c_8_1 → c_8_1

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_3_0 → 0, an element of degree 3
  2. c_4_0 →  − c_2_22 + c_2_1·c_2_2 + c_2_12, an element of degree 4
  3. a_7_1 → 0, an element of degree 7
  4. c_8_1 → c_2_24 + c_2_1·c_2_23 + c_2_12·c_2_22, 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