Polytope of Type {12,6,9} with group SmallGroup(1296,2020)

Overview

Group: SmallGroup(1296,2020)
Rank: 4
Schläfli Type: {12,6,9}
Vertices, edges, …: 12, 36, 27, 9
Order of s₀s₁s₂s₃: 36
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{6,6,9}*648b

3-fold

6-fold

9-fold

12-fold

{3,2,9}*108

18-fold

27-fold

{4,2,3}*48

36-fold

{3,2,3}*36

54-fold

{2,2,3}*24

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)(187,265)(188,266)(189,267)(190,271)(191,272)(192,273)(193,277)(194,278)(195,279)(196,274)(197,275)(198,276)(199,280)(200,281)(201,282)(202,286)(203,287)(204,288)(205,283)(206,284)(207,285)(208,289)(209,290)(210,291)(211,295)(212,296)(213,297)(214,292)(215,293)(216,294)(217,298)(218,299)(219,300)(220,304)(221,305)(222,306)(223,301)(224,302)(225,303)(226,307)(227,308)(228,309)(229,313)(230,314)(231,315)(232,310)(233,311)(234,312)(235,316)(236,317)(237,318)(238,322)(239,323)(240,324)(241,319)(242,320)(243,321)(325,406)(326,407)(327,408)(328,412)(329,413)(330,414)(331,409)(332,410)(333,411)(334,415)(335,416)(336,417)(337,421)(338,422)(339,423)(340,418)(341,419)(342,420)(343,424)(344,425)(345,426)(346,430)(347,431)(348,432)(349,427)(350,428)(351,429)(352,433)(353,434)(354,435)(355,439)(356,440)(357,441)(358,436)(359,437)(360,438)(361,442)(362,443)(363,444)(364,448)(365,449)(366,450)(367,445)(368,446)(369,447)(370,451)(371,452)(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,460)(380,461)(381,462)(382,466)(383,467)(384,468)(385,463)(386,464)(387,465)(388,469)(389,470)(390,471)(391,475)(392,476)(393,477)(394,472)(395,473)(396,474)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)(404,482)(405,483)(490,493)(491,494)(492,495)(499,502)(500,503)(501,504)(508,511)(509,512)(510,513)(517,520)(518,521)(519,522)(526,529)(527,530)(528,531)(535,538)(536,539)(537,540)(544,547)(545,548)(546,549)(553,556)(554,557)(555,558)(562,565)(563,566)(564,567)(571,574)(572,575)(573,576)(580,583)(581,584)(582,585)(589,592)(590,593)(591,594)(598,601)(599,602)(600,603)(607,610)(608,611)(609,612)(616,619)(617,620)(618,621)(625,628)(626,629)(627,630)(634,637)(635,638)(636,639)(643,646)(644,647)(645,648);;
s1 := (  1,166)(  2,167)(  3,168)(  4,163)(  5,164)(  6,165)(  7,169)(  8,170)(  9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)( 16,187)( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)( 24,174)( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)( 32,191)( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)( 40,208)( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)( 48,204)( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,220)( 56,221)( 57,222)( 58,217)( 59,218)( 60,219)( 61,223)( 62,224)( 63,225)( 64,238)( 65,239)( 66,240)( 67,235)( 68,236)( 69,237)( 70,241)( 71,242)( 72,243)( 73,229)( 74,230)( 75,231)( 76,226)( 77,227)( 78,228)( 79,232)( 80,233)( 81,234)( 82,247)( 83,248)( 84,249)( 85,244)( 86,245)( 87,246)( 88,250)( 89,251)( 90,252)( 91,265)( 92,266)( 93,267)( 94,262)( 95,263)( 96,264)( 97,268)( 98,269)( 99,270)(100,256)(101,257)(102,258)(103,253)(104,254)(105,255)(106,259)(107,260)(108,261)(109,274)(110,275)(111,276)(112,271)(113,272)(114,273)(115,277)(116,278)(117,279)(118,292)(119,293)(120,294)(121,289)(122,290)(123,291)(124,295)(125,296)(126,297)(127,283)(128,284)(129,285)(130,280)(131,281)(132,282)(133,286)(134,287)(135,288)(136,301)(137,302)(138,303)(139,298)(140,299)(141,300)(142,304)(143,305)(144,306)(145,319)(146,320)(147,321)(148,316)(149,317)(150,318)(151,322)(152,323)(153,324)(154,310)(155,311)(156,312)(157,307)(158,308)(159,309)(160,313)(161,314)(162,315)(325,571)(326,572)(327,573)(328,568)(329,569)(330,570)(331,574)(332,575)(333,576)(334,589)(335,590)(336,591)(337,586)(338,587)(339,588)(340,592)(341,593)(342,594)(343,580)(344,581)(345,582)(346,577)(347,578)(348,579)(349,583)(350,584)(351,585)(352,598)(353,599)(354,600)(355,595)(356,596)(357,597)(358,601)(359,602)(360,603)(361,616)(362,617)(363,618)(364,613)(365,614)(366,615)(367,619)(368,620)(369,621)(370,607)(371,608)(372,609)(373,604)(374,605)(375,606)(376,610)(377,611)(378,612)(379,625)(380,626)(381,627)(382,622)(383,623)(384,624)(385,628)(386,629)(387,630)(388,643)(389,644)(390,645)(391,640)(392,641)(393,642)(394,646)(395,647)(396,648)(397,634)(398,635)(399,636)(400,631)(401,632)(402,633)(403,637)(404,638)(405,639)(406,490)(407,491)(408,492)(409,487)(410,488)(411,489)(412,493)(413,494)(414,495)(415,508)(416,509)(417,510)(418,505)(419,506)(420,507)(421,511)(422,512)(423,513)(424,499)(425,500)(426,501)(427,496)(428,497)(429,498)(430,502)(431,503)(432,504)(433,517)(434,518)(435,519)(436,514)(437,515)(438,516)(439,520)(440,521)(441,522)(442,535)(443,536)(444,537)(445,532)(446,533)(447,534)(448,538)(449,539)(450,540)(451,526)(452,527)(453,528)(454,523)(455,524)(456,525)(457,529)(458,530)(459,531)(460,544)(461,545)(462,546)(463,541)(464,542)(465,543)(466,547)(467,548)(468,549)(469,562)(470,563)(471,564)(472,559)(473,560)(474,561)(475,565)(476,566)(477,567)(478,553)(479,554)(480,555)(481,550)(482,551)(483,552)(484,556)(485,557)(486,558);;
s2 := (  1,496)(  2,498)(  3,497)(  4,499)(  5,501)(  6,500)(  7,502)(  8,504)(  9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)( 16,493)( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)( 24,509)( 25,511)( 26,513)( 27,512)( 28,552)( 29,551)( 30,550)( 31,555)( 32,554)( 33,553)( 34,558)( 35,557)( 36,556)( 37,543)( 38,542)( 39,541)( 40,546)( 41,545)( 42,544)( 43,549)( 44,548)( 45,547)( 46,561)( 47,560)( 48,559)( 49,564)( 50,563)( 51,562)( 52,567)( 53,566)( 54,565)( 55,525)( 56,524)( 57,523)( 58,528)( 59,527)( 60,526)( 61,531)( 62,530)( 63,529)( 64,516)( 65,515)( 66,514)( 67,519)( 68,518)( 69,517)( 70,522)( 71,521)( 72,520)( 73,534)( 74,533)( 75,532)( 76,537)( 77,536)( 78,535)( 79,540)( 80,539)( 81,538)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)( 88,583)( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)( 96,572)( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)(104,591)(105,590)(106,592)(107,594)(108,593)(109,633)(110,632)(111,631)(112,636)(113,635)(114,634)(115,639)(116,638)(117,637)(118,624)(119,623)(120,622)(121,627)(122,626)(123,625)(124,630)(125,629)(126,628)(127,642)(128,641)(129,640)(130,645)(131,644)(132,643)(133,648)(134,647)(135,646)(136,606)(137,605)(138,604)(139,609)(140,608)(141,607)(142,612)(143,611)(144,610)(145,597)(146,596)(147,595)(148,600)(149,599)(150,598)(151,603)(152,602)(153,601)(154,615)(155,614)(156,613)(157,618)(158,617)(159,616)(160,621)(161,620)(162,619)(163,415)(164,417)(165,416)(166,418)(167,420)(168,419)(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)(176,411)(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)(184,427)(185,429)(186,428)(187,430)(188,432)(189,431)(190,471)(191,470)(192,469)(193,474)(194,473)(195,472)(196,477)(197,476)(198,475)(199,462)(200,461)(201,460)(202,465)(203,464)(204,463)(205,468)(206,467)(207,466)(208,480)(209,479)(210,478)(211,483)(212,482)(213,481)(214,486)(215,485)(216,484)(217,444)(218,443)(219,442)(220,447)(221,446)(222,445)(223,450)(224,449)(225,448)(226,435)(227,434)(228,433)(229,438)(230,437)(231,436)(232,441)(233,440)(234,439)(235,453)(236,452)(237,451)(238,456)(239,455)(240,454)(241,459)(242,458)(243,457)(244,334)(245,336)(246,335)(247,337)(248,339)(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)(256,328)(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)(264,344)(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,390)(272,389)(273,388)(274,393)(275,392)(276,391)(277,396)(278,395)(279,394)(280,381)(281,380)(282,379)(283,384)(284,383)(285,382)(286,387)(287,386)(288,385)(289,399)(290,398)(291,397)(292,402)(293,401)(294,400)(295,405)(296,404)(297,403)(298,363)(299,362)(300,361)(301,366)(302,365)(303,364)(304,369)(305,368)(306,367)(307,354)(308,353)(309,352)(310,357)(311,356)(312,355)(313,360)(314,359)(315,358)(316,372)(317,371)(318,370)(319,375)(320,374)(321,373)(322,378)(323,377)(324,376);;
s3 := (  1,514)(  2,516)(  3,515)(  4,517)(  5,519)(  6,518)(  7,520)(  8,522)(  9,521)( 10,532)( 11,534)( 12,533)( 13,535)( 14,537)( 15,536)( 16,538)( 17,540)( 18,539)( 19,523)( 20,525)( 21,524)( 22,526)( 23,528)( 24,527)( 25,529)( 26,531)( 27,530)( 28,487)( 29,489)( 30,488)( 31,490)( 32,492)( 33,491)( 34,493)( 35,495)( 36,494)( 37,505)( 38,507)( 39,506)( 40,508)( 41,510)( 42,509)( 43,511)( 44,513)( 45,512)( 46,496)( 47,498)( 48,497)( 49,499)( 50,501)( 51,500)( 52,502)( 53,504)( 54,503)( 55,543)( 56,542)( 57,541)( 58,546)( 59,545)( 60,544)( 61,549)( 62,548)( 63,547)( 64,561)( 65,560)( 66,559)( 67,564)( 68,563)( 69,562)( 70,567)( 71,566)( 72,565)( 73,552)( 74,551)( 75,550)( 76,555)( 77,554)( 78,553)( 79,558)( 80,557)( 81,556)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)( 88,601)( 89,603)( 90,602)( 91,613)( 92,615)( 93,614)( 94,616)( 95,618)( 96,617)( 97,619)( 98,621)( 99,620)(100,604)(101,606)(102,605)(103,607)(104,609)(105,608)(106,610)(107,612)(108,611)(109,568)(110,570)(111,569)(112,571)(113,573)(114,572)(115,574)(116,576)(117,575)(118,586)(119,588)(120,587)(121,589)(122,591)(123,590)(124,592)(125,594)(126,593)(127,577)(128,579)(129,578)(130,580)(131,582)(132,581)(133,583)(134,585)(135,584)(136,624)(137,623)(138,622)(139,627)(140,626)(141,625)(142,630)(143,629)(144,628)(145,642)(146,641)(147,640)(148,645)(149,644)(150,643)(151,648)(152,647)(153,646)(154,633)(155,632)(156,631)(157,636)(158,635)(159,634)(160,639)(161,638)(162,637)(163,433)(164,435)(165,434)(166,436)(167,438)(168,437)(169,439)(170,441)(171,440)(172,451)(173,453)(174,452)(175,454)(176,456)(177,455)(178,457)(179,459)(180,458)(181,442)(182,444)(183,443)(184,445)(185,447)(186,446)(187,448)(188,450)(189,449)(190,406)(191,408)(192,407)(193,409)(194,411)(195,410)(196,412)(197,414)(198,413)(199,424)(200,426)(201,425)(202,427)(203,429)(204,428)(205,430)(206,432)(207,431)(208,415)(209,417)(210,416)(211,418)(212,420)(213,419)(214,421)(215,423)(216,422)(217,462)(218,461)(219,460)(220,465)(221,464)(222,463)(223,468)(224,467)(225,466)(226,480)(227,479)(228,478)(229,483)(230,482)(231,481)(232,486)(233,485)(234,484)(235,471)(236,470)(237,469)(238,474)(239,473)(240,472)(241,477)(242,476)(243,475)(244,352)(245,354)(246,353)(247,355)(248,357)(249,356)(250,358)(251,360)(252,359)(253,370)(254,372)(255,371)(256,373)(257,375)(258,374)(259,376)(260,378)(261,377)(262,361)(263,363)(264,362)(265,364)(266,366)(267,365)(268,367)(269,369)(270,368)(271,325)(272,327)(273,326)(274,328)(275,330)(276,329)(277,331)(278,333)(279,332)(280,343)(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)(288,350)(289,334)(290,336)(291,335)(292,337)(293,339)(294,338)(295,340)(296,342)(297,341)(298,381)(299,380)(300,379)(301,384)(302,383)(303,382)(304,387)(305,386)(306,385)(307,399)(308,398)(309,397)(310,402)(311,401)(312,400)(313,405)(314,404)(315,403)(316,390)(317,389)(318,388)(319,393)(320,392)(321,391)(322,396)(323,395)(324,394);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(648)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,244)(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)(171,249)(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)(179,257)(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)(187,265)(188,266)(189,267)(190,271)(191,272)(192,273)(193,277)(194,278)(195,279)(196,274)(197,275)(198,276)(199,280)(200,281)(201,282)(202,286)(203,287)(204,288)(205,283)(206,284)(207,285)(208,289)(209,290)(210,291)(211,295)(212,296)(213,297)(214,292)(215,293)(216,294)(217,298)(218,299)(219,300)(220,304)(221,305)(222,306)(223,301)(224,302)(225,303)(226,307)(227,308)(228,309)(229,313)(230,314)(231,315)(232,310)(233,311)(234,312)(235,316)(236,317)(237,318)(238,322)(239,323)(240,324)(241,319)(242,320)(243,321)(325,406)(326,407)(327,408)(328,412)(329,413)(330,414)(331,409)(332,410)(333,411)(334,415)(335,416)(336,417)(337,421)(338,422)(339,423)(340,418)(341,419)(342,420)(343,424)(344,425)(345,426)(346,430)(347,431)(348,432)(349,427)(350,428)(351,429)(352,433)(353,434)(354,435)(355,439)(356,440)(357,441)(358,436)(359,437)(360,438)(361,442)(362,443)(363,444)(364,448)(365,449)(366,450)(367,445)(368,446)(369,447)(370,451)(371,452)(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,460)(380,461)(381,462)(382,466)(383,467)(384,468)(385,463)(386,464)(387,465)(388,469)(389,470)(390,471)(391,475)(392,476)(393,477)(394,472)(395,473)(396,474)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)(404,482)(405,483)(490,493)(491,494)(492,495)(499,502)(500,503)(501,504)(508,511)(509,512)(510,513)(517,520)(518,521)(519,522)(526,529)(527,530)(528,531)(535,538)(536,539)(537,540)(544,547)(545,548)(546,549)(553,556)(554,557)(555,558)(562,565)(563,566)(564,567)(571,574)(572,575)(573,576)(580,583)(581,584)(582,585)(589,592)(590,593)(591,594)(598,601)(599,602)(600,603)(607,610)(608,611)(609,612)(616,619)(617,620)(618,621)(625,628)(626,629)(627,630)(634,637)(635,638)(636,639)(643,646)(644,647)(645,648);
s1 := Sym(648)!(  1,166)(  2,167)(  3,168)(  4,163)(  5,164)(  6,165)(  7,169)(  8,170)(  9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)( 16,187)( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)( 24,174)( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)( 32,191)( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)( 40,208)( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)( 48,204)( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,220)( 56,221)( 57,222)( 58,217)( 59,218)( 60,219)( 61,223)( 62,224)( 63,225)( 64,238)( 65,239)( 66,240)( 67,235)( 68,236)( 69,237)( 70,241)( 71,242)( 72,243)( 73,229)( 74,230)( 75,231)( 76,226)( 77,227)( 78,228)( 79,232)( 80,233)( 81,234)( 82,247)( 83,248)( 84,249)( 85,244)( 86,245)( 87,246)( 88,250)( 89,251)( 90,252)( 91,265)( 92,266)( 93,267)( 94,262)( 95,263)( 96,264)( 97,268)( 98,269)( 99,270)(100,256)(101,257)(102,258)(103,253)(104,254)(105,255)(106,259)(107,260)(108,261)(109,274)(110,275)(111,276)(112,271)(113,272)(114,273)(115,277)(116,278)(117,279)(118,292)(119,293)(120,294)(121,289)(122,290)(123,291)(124,295)(125,296)(126,297)(127,283)(128,284)(129,285)(130,280)(131,281)(132,282)(133,286)(134,287)(135,288)(136,301)(137,302)(138,303)(139,298)(140,299)(141,300)(142,304)(143,305)(144,306)(145,319)(146,320)(147,321)(148,316)(149,317)(150,318)(151,322)(152,323)(153,324)(154,310)(155,311)(156,312)(157,307)(158,308)(159,309)(160,313)(161,314)(162,315)(325,571)(326,572)(327,573)(328,568)(329,569)(330,570)(331,574)(332,575)(333,576)(334,589)(335,590)(336,591)(337,586)(338,587)(339,588)(340,592)(341,593)(342,594)(343,580)(344,581)(345,582)(346,577)(347,578)(348,579)(349,583)(350,584)(351,585)(352,598)(353,599)(354,600)(355,595)(356,596)(357,597)(358,601)(359,602)(360,603)(361,616)(362,617)(363,618)(364,613)(365,614)(366,615)(367,619)(368,620)(369,621)(370,607)(371,608)(372,609)(373,604)(374,605)(375,606)(376,610)(377,611)(378,612)(379,625)(380,626)(381,627)(382,622)(383,623)(384,624)(385,628)(386,629)(387,630)(388,643)(389,644)(390,645)(391,640)(392,641)(393,642)(394,646)(395,647)(396,648)(397,634)(398,635)(399,636)(400,631)(401,632)(402,633)(403,637)(404,638)(405,639)(406,490)(407,491)(408,492)(409,487)(410,488)(411,489)(412,493)(413,494)(414,495)(415,508)(416,509)(417,510)(418,505)(419,506)(420,507)(421,511)(422,512)(423,513)(424,499)(425,500)(426,501)(427,496)(428,497)(429,498)(430,502)(431,503)(432,504)(433,517)(434,518)(435,519)(436,514)(437,515)(438,516)(439,520)(440,521)(441,522)(442,535)(443,536)(444,537)(445,532)(446,533)(447,534)(448,538)(449,539)(450,540)(451,526)(452,527)(453,528)(454,523)(455,524)(456,525)(457,529)(458,530)(459,531)(460,544)(461,545)(462,546)(463,541)(464,542)(465,543)(466,547)(467,548)(468,549)(469,562)(470,563)(471,564)(472,559)(473,560)(474,561)(475,565)(476,566)(477,567)(478,553)(479,554)(480,555)(481,550)(482,551)(483,552)(484,556)(485,557)(486,558);
s2 := Sym(648)!(  1,496)(  2,498)(  3,497)(  4,499)(  5,501)(  6,500)(  7,502)(  8,504)(  9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)( 16,493)( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)( 24,509)( 25,511)( 26,513)( 27,512)( 28,552)( 29,551)( 30,550)( 31,555)( 32,554)( 33,553)( 34,558)( 35,557)( 36,556)( 37,543)( 38,542)( 39,541)( 40,546)( 41,545)( 42,544)( 43,549)( 44,548)( 45,547)( 46,561)( 47,560)( 48,559)( 49,564)( 50,563)( 51,562)( 52,567)( 53,566)( 54,565)( 55,525)( 56,524)( 57,523)( 58,528)( 59,527)( 60,526)( 61,531)( 62,530)( 63,529)( 64,516)( 65,515)( 66,514)( 67,519)( 68,518)( 69,517)( 70,522)( 71,521)( 72,520)( 73,534)( 74,533)( 75,532)( 76,537)( 77,536)( 78,535)( 79,540)( 80,539)( 81,538)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)( 88,583)( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)( 96,572)( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)(104,591)(105,590)(106,592)(107,594)(108,593)(109,633)(110,632)(111,631)(112,636)(113,635)(114,634)(115,639)(116,638)(117,637)(118,624)(119,623)(120,622)(121,627)(122,626)(123,625)(124,630)(125,629)(126,628)(127,642)(128,641)(129,640)(130,645)(131,644)(132,643)(133,648)(134,647)(135,646)(136,606)(137,605)(138,604)(139,609)(140,608)(141,607)(142,612)(143,611)(144,610)(145,597)(146,596)(147,595)(148,600)(149,599)(150,598)(151,603)(152,602)(153,601)(154,615)(155,614)(156,613)(157,618)(158,617)(159,616)(160,621)(161,620)(162,619)(163,415)(164,417)(165,416)(166,418)(167,420)(168,419)(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)(176,411)(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)(184,427)(185,429)(186,428)(187,430)(188,432)(189,431)(190,471)(191,470)(192,469)(193,474)(194,473)(195,472)(196,477)(197,476)(198,475)(199,462)(200,461)(201,460)(202,465)(203,464)(204,463)(205,468)(206,467)(207,466)(208,480)(209,479)(210,478)(211,483)(212,482)(213,481)(214,486)(215,485)(216,484)(217,444)(218,443)(219,442)(220,447)(221,446)(222,445)(223,450)(224,449)(225,448)(226,435)(227,434)(228,433)(229,438)(230,437)(231,436)(232,441)(233,440)(234,439)(235,453)(236,452)(237,451)(238,456)(239,455)(240,454)(241,459)(242,458)(243,457)(244,334)(245,336)(246,335)(247,337)(248,339)(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)(256,328)(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)(264,344)(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,390)(272,389)(273,388)(274,393)(275,392)(276,391)(277,396)(278,395)(279,394)(280,381)(281,380)(282,379)(283,384)(284,383)(285,382)(286,387)(287,386)(288,385)(289,399)(290,398)(291,397)(292,402)(293,401)(294,400)(295,405)(296,404)(297,403)(298,363)(299,362)(300,361)(301,366)(302,365)(303,364)(304,369)(305,368)(306,367)(307,354)(308,353)(309,352)(310,357)(311,356)(312,355)(313,360)(314,359)(315,358)(316,372)(317,371)(318,370)(319,375)(320,374)(321,373)(322,378)(323,377)(324,376);
s3 := Sym(648)!(  1,514)(  2,516)(  3,515)(  4,517)(  5,519)(  6,518)(  7,520)(  8,522)(  9,521)( 10,532)( 11,534)( 12,533)( 13,535)( 14,537)( 15,536)( 16,538)( 17,540)( 18,539)( 19,523)( 20,525)( 21,524)( 22,526)( 23,528)( 24,527)( 25,529)( 26,531)( 27,530)( 28,487)( 29,489)( 30,488)( 31,490)( 32,492)( 33,491)( 34,493)( 35,495)( 36,494)( 37,505)( 38,507)( 39,506)( 40,508)( 41,510)( 42,509)( 43,511)( 44,513)( 45,512)( 46,496)( 47,498)( 48,497)( 49,499)( 50,501)( 51,500)( 52,502)( 53,504)( 54,503)( 55,543)( 56,542)( 57,541)( 58,546)( 59,545)( 60,544)( 61,549)( 62,548)( 63,547)( 64,561)( 65,560)( 66,559)( 67,564)( 68,563)( 69,562)( 70,567)( 71,566)( 72,565)( 73,552)( 74,551)( 75,550)( 76,555)( 77,554)( 78,553)( 79,558)( 80,557)( 81,556)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)( 88,601)( 89,603)( 90,602)( 91,613)( 92,615)( 93,614)( 94,616)( 95,618)( 96,617)( 97,619)( 98,621)( 99,620)(100,604)(101,606)(102,605)(103,607)(104,609)(105,608)(106,610)(107,612)(108,611)(109,568)(110,570)(111,569)(112,571)(113,573)(114,572)(115,574)(116,576)(117,575)(118,586)(119,588)(120,587)(121,589)(122,591)(123,590)(124,592)(125,594)(126,593)(127,577)(128,579)(129,578)(130,580)(131,582)(132,581)(133,583)(134,585)(135,584)(136,624)(137,623)(138,622)(139,627)(140,626)(141,625)(142,630)(143,629)(144,628)(145,642)(146,641)(147,640)(148,645)(149,644)(150,643)(151,648)(152,647)(153,646)(154,633)(155,632)(156,631)(157,636)(158,635)(159,634)(160,639)(161,638)(162,637)(163,433)(164,435)(165,434)(166,436)(167,438)(168,437)(169,439)(170,441)(171,440)(172,451)(173,453)(174,452)(175,454)(176,456)(177,455)(178,457)(179,459)(180,458)(181,442)(182,444)(183,443)(184,445)(185,447)(186,446)(187,448)(188,450)(189,449)(190,406)(191,408)(192,407)(193,409)(194,411)(195,410)(196,412)(197,414)(198,413)(199,424)(200,426)(201,425)(202,427)(203,429)(204,428)(205,430)(206,432)(207,431)(208,415)(209,417)(210,416)(211,418)(212,420)(213,419)(214,421)(215,423)(216,422)(217,462)(218,461)(219,460)(220,465)(221,464)(222,463)(223,468)(224,467)(225,466)(226,480)(227,479)(228,478)(229,483)(230,482)(231,481)(232,486)(233,485)(234,484)(235,471)(236,470)(237,469)(238,474)(239,473)(240,472)(241,477)(242,476)(243,475)(244,352)(245,354)(246,353)(247,355)(248,357)(249,356)(250,358)(251,360)(252,359)(253,370)(254,372)(255,371)(256,373)(257,375)(258,374)(259,376)(260,378)(261,377)(262,361)(263,363)(264,362)(265,364)(266,366)(267,365)(268,367)(269,369)(270,368)(271,325)(272,327)(273,326)(274,328)(275,330)(276,329)(277,331)(278,333)(279,332)(280,343)(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)(288,350)(289,334)(290,336)(291,335)(292,337)(293,339)(294,338)(295,340)(296,342)(297,341)(298,381)(299,380)(300,379)(301,384)(302,383)(303,382)(304,387)(305,386)(306,385)(307,399)(308,398)(309,397)(310,402)(311,401)(312,400)(313,405)(314,404)(315,403)(316,390)(317,389)(318,388)(319,393)(320,392)(321,391)(322,396)(323,395)(324,394);
poly := sub<Sym(648)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References