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Polytope of Type {4,42,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42,4}*1344a
Also Known As : {{4,42|2},{42,4|2}}. if this polytope has another name.
Group : SmallGroup(1344,10970)
Rank : 4
Schlafli Type : {4,42,4}
Number of vertices, edges, etc : 4, 84, 84, 4
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,42,4}*672a, {4,42,2}*672a
   3-fold quotients : {4,14,4}*448
   4-fold quotients : {2,42,2}*336
   6-fold quotients : {2,14,4}*224, {4,14,2}*224
   7-fold quotients : {4,6,4}*192a
   8-fold quotients : {2,21,2}*168
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,6,4}*96a, {4,6,2}*96a
   21-fold quotients : {4,2,4}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,6,2}*48
   42-fold quotients : {2,2,4}*32, {4,2,2}*32
   56-fold quotients : {2,3,2}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,337)(  2,338)(  3,339)(  4,340)(  5,341)(  6,342)(  7,343)(  8,344)
(  9,345)( 10,346)( 11,347)( 12,348)( 13,349)( 14,350)( 15,351)( 16,352)
( 17,353)( 18,354)( 19,355)( 20,356)( 21,357)( 22,358)( 23,359)( 24,360)
( 25,361)( 26,362)( 27,363)( 28,364)( 29,365)( 30,366)( 31,367)( 32,368)
( 33,369)( 34,370)( 35,371)( 36,372)( 37,373)( 38,374)( 39,375)( 40,376)
( 41,377)( 42,378)( 43,379)( 44,380)( 45,381)( 46,382)( 47,383)( 48,384)
( 49,385)( 50,386)( 51,387)( 52,388)( 53,389)( 54,390)( 55,391)( 56,392)
( 57,393)( 58,394)( 59,395)( 60,396)( 61,397)( 62,398)( 63,399)( 64,400)
( 65,401)( 66,402)( 67,403)( 68,404)( 69,405)( 70,406)( 71,407)( 72,408)
( 73,409)( 74,410)( 75,411)( 76,412)( 77,413)( 78,414)( 79,415)( 80,416)
( 81,417)( 82,418)( 83,419)( 84,420)( 85,442)( 86,443)( 87,444)( 88,445)
( 89,446)( 90,447)( 91,448)( 92,449)( 93,450)( 94,451)( 95,452)( 96,453)
( 97,454)( 98,455)( 99,456)(100,457)(101,458)(102,459)(103,460)(104,461)
(105,462)(106,421)(107,422)(108,423)(109,424)(110,425)(111,426)(112,427)
(113,428)(114,429)(115,430)(116,431)(117,432)(118,433)(119,434)(120,435)
(121,436)(122,437)(123,438)(124,439)(125,440)(126,441)(127,484)(128,485)
(129,486)(130,487)(131,488)(132,489)(133,490)(134,491)(135,492)(136,493)
(137,494)(138,495)(139,496)(140,497)(141,498)(142,499)(143,500)(144,501)
(145,502)(146,503)(147,504)(148,463)(149,464)(150,465)(151,466)(152,467)
(153,468)(154,469)(155,470)(156,471)(157,472)(158,473)(159,474)(160,475)
(161,476)(162,477)(163,478)(164,479)(165,480)(166,481)(167,482)(168,483)
(169,505)(170,506)(171,507)(172,508)(173,509)(174,510)(175,511)(176,512)
(177,513)(178,514)(179,515)(180,516)(181,517)(182,518)(183,519)(184,520)
(185,521)(186,522)(187,523)(188,524)(189,525)(190,526)(191,527)(192,528)
(193,529)(194,530)(195,531)(196,532)(197,533)(198,534)(199,535)(200,536)
(201,537)(202,538)(203,539)(204,540)(205,541)(206,542)(207,543)(208,544)
(209,545)(210,546)(211,547)(212,548)(213,549)(214,550)(215,551)(216,552)
(217,553)(218,554)(219,555)(220,556)(221,557)(222,558)(223,559)(224,560)
(225,561)(226,562)(227,563)(228,564)(229,565)(230,566)(231,567)(232,568)
(233,569)(234,570)(235,571)(236,572)(237,573)(238,574)(239,575)(240,576)
(241,577)(242,578)(243,579)(244,580)(245,581)(246,582)(247,583)(248,584)
(249,585)(250,586)(251,587)(252,588)(253,610)(254,611)(255,612)(256,613)
(257,614)(258,615)(259,616)(260,617)(261,618)(262,619)(263,620)(264,621)
(265,622)(266,623)(267,624)(268,625)(269,626)(270,627)(271,628)(272,629)
(273,630)(274,589)(275,590)(276,591)(277,592)(278,593)(279,594)(280,595)
(281,596)(282,597)(283,598)(284,599)(285,600)(286,601)(287,602)(288,603)
(289,604)(290,605)(291,606)(292,607)(293,608)(294,609)(295,652)(296,653)
(297,654)(298,655)(299,656)(300,657)(301,658)(302,659)(303,660)(304,661)
(305,662)(306,663)(307,664)(308,665)(309,666)(310,667)(311,668)(312,669)
(313,670)(314,671)(315,672)(316,631)(317,632)(318,633)(319,634)(320,635)
(321,636)(322,637)(323,638)(324,639)(325,640)(326,641)(327,642)(328,643)
(329,644)(330,645)(331,646)(332,647)(333,648)(334,649)(335,650)(336,651);;
s1 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)
( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 44, 49)( 45, 48)( 46, 47)( 50, 57)
( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)( 66, 69)
( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)
( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)( 96,102)
( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)(115,125)
(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)(134,141)
(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)(150,153)
(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)
(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)
(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)(199,209)
(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)(218,225)
(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)(234,237)
(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)(245,247)
(254,259)(255,258)(256,257)(260,267)(261,273)(262,272)(263,271)(264,270)
(265,269)(266,268)(275,280)(276,279)(277,278)(281,288)(282,294)(283,293)
(284,292)(285,291)(286,290)(287,289)(296,301)(297,300)(298,299)(302,309)
(303,315)(304,314)(305,313)(306,312)(307,311)(308,310)(317,322)(318,321)
(319,320)(323,330)(324,336)(325,335)(326,334)(327,333)(328,332)(329,331)
(337,358)(338,364)(339,363)(340,362)(341,361)(342,360)(343,359)(344,372)
(345,378)(346,377)(347,376)(348,375)(349,374)(350,373)(351,365)(352,371)
(353,370)(354,369)(355,368)(356,367)(357,366)(379,400)(380,406)(381,405)
(382,404)(383,403)(384,402)(385,401)(386,414)(387,420)(388,419)(389,418)
(390,417)(391,416)(392,415)(393,407)(394,413)(395,412)(396,411)(397,410)
(398,409)(399,408)(421,442)(422,448)(423,447)(424,446)(425,445)(426,444)
(427,443)(428,456)(429,462)(430,461)(431,460)(432,459)(433,458)(434,457)
(435,449)(436,455)(437,454)(438,453)(439,452)(440,451)(441,450)(463,484)
(464,490)(465,489)(466,488)(467,487)(468,486)(469,485)(470,498)(471,504)
(472,503)(473,502)(474,501)(475,500)(476,499)(477,491)(478,497)(479,496)
(480,495)(481,494)(482,493)(483,492)(505,526)(506,532)(507,531)(508,530)
(509,529)(510,528)(511,527)(512,540)(513,546)(514,545)(515,544)(516,543)
(517,542)(518,541)(519,533)(520,539)(521,538)(522,537)(523,536)(524,535)
(525,534)(547,568)(548,574)(549,573)(550,572)(551,571)(552,570)(553,569)
(554,582)(555,588)(556,587)(557,586)(558,585)(559,584)(560,583)(561,575)
(562,581)(563,580)(564,579)(565,578)(566,577)(567,576)(589,610)(590,616)
(591,615)(592,614)(593,613)(594,612)(595,611)(596,624)(597,630)(598,629)
(599,628)(600,627)(601,626)(602,625)(603,617)(604,623)(605,622)(606,621)
(607,620)(608,619)(609,618)(631,652)(632,658)(633,657)(634,656)(635,655)
(636,654)(637,653)(638,666)(639,672)(640,671)(641,670)(642,669)(643,668)
(644,667)(645,659)(646,665)(647,664)(648,663)(649,662)(650,661)(651,660);;
s2 := (  1, 93)(  2, 92)(  3, 98)(  4, 97)(  5, 96)(  6, 95)(  7, 94)(  8, 86)
(  9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)( 16, 99)
( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)( 24,119)
( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)( 32,111)
( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43,135)( 44,134)( 45,140)( 46,139)( 47,138)( 48,137)
( 49,136)( 50,128)( 51,127)( 52,133)( 53,132)( 54,131)( 55,130)( 56,129)
( 57,142)( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)( 64,156)
( 65,155)( 66,161)( 67,160)( 68,159)( 69,158)( 70,157)( 71,149)( 72,148)
( 73,154)( 74,153)( 75,152)( 76,151)( 77,150)( 78,163)( 79,162)( 80,168)
( 81,167)( 82,166)( 83,165)( 84,164)(169,261)(170,260)(171,266)(172,265)
(173,264)(174,263)(175,262)(176,254)(177,253)(178,259)(179,258)(180,257)
(181,256)(182,255)(183,268)(184,267)(185,273)(186,272)(187,271)(188,270)
(189,269)(190,282)(191,281)(192,287)(193,286)(194,285)(195,284)(196,283)
(197,275)(198,274)(199,280)(200,279)(201,278)(202,277)(203,276)(204,289)
(205,288)(206,294)(207,293)(208,292)(209,291)(210,290)(211,303)(212,302)
(213,308)(214,307)(215,306)(216,305)(217,304)(218,296)(219,295)(220,301)
(221,300)(222,299)(223,298)(224,297)(225,310)(226,309)(227,315)(228,314)
(229,313)(230,312)(231,311)(232,324)(233,323)(234,329)(235,328)(236,327)
(237,326)(238,325)(239,317)(240,316)(241,322)(242,321)(243,320)(244,319)
(245,318)(246,331)(247,330)(248,336)(249,335)(250,334)(251,333)(252,332)
(337,450)(338,449)(339,455)(340,454)(341,453)(342,452)(343,451)(344,443)
(345,442)(346,448)(347,447)(348,446)(349,445)(350,444)(351,457)(352,456)
(353,462)(354,461)(355,460)(356,459)(357,458)(358,429)(359,428)(360,434)
(361,433)(362,432)(363,431)(364,430)(365,422)(366,421)(367,427)(368,426)
(369,425)(370,424)(371,423)(372,436)(373,435)(374,441)(375,440)(376,439)
(377,438)(378,437)(379,492)(380,491)(381,497)(382,496)(383,495)(384,494)
(385,493)(386,485)(387,484)(388,490)(389,489)(390,488)(391,487)(392,486)
(393,499)(394,498)(395,504)(396,503)(397,502)(398,501)(399,500)(400,471)
(401,470)(402,476)(403,475)(404,474)(405,473)(406,472)(407,464)(408,463)
(409,469)(410,468)(411,467)(412,466)(413,465)(414,478)(415,477)(416,483)
(417,482)(418,481)(419,480)(420,479)(505,618)(506,617)(507,623)(508,622)
(509,621)(510,620)(511,619)(512,611)(513,610)(514,616)(515,615)(516,614)
(517,613)(518,612)(519,625)(520,624)(521,630)(522,629)(523,628)(524,627)
(525,626)(526,597)(527,596)(528,602)(529,601)(530,600)(531,599)(532,598)
(533,590)(534,589)(535,595)(536,594)(537,593)(538,592)(539,591)(540,604)
(541,603)(542,609)(543,608)(544,607)(545,606)(546,605)(547,660)(548,659)
(549,665)(550,664)(551,663)(552,662)(553,661)(554,653)(555,652)(556,658)
(557,657)(558,656)(559,655)(560,654)(561,667)(562,666)(563,672)(564,671)
(565,670)(566,669)(567,668)(568,639)(569,638)(570,644)(571,643)(572,642)
(573,641)(574,640)(575,632)(576,631)(577,637)(578,636)(579,635)(580,634)
(581,633)(582,646)(583,645)(584,651)(585,650)(586,649)(587,648)(588,647);;
s3 := (  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)(  8,176)
(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)
( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)( 64,232)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,295)( 86,296)( 87,297)( 88,298)
( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)( 96,306)
( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)(104,314)
(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)(112,322)
(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)(120,330)
(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)(128,254)
(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)(136,262)
(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)(144,270)
(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)(152,278)
(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)
(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)
(337,505)(338,506)(339,507)(340,508)(341,509)(342,510)(343,511)(344,512)
(345,513)(346,514)(347,515)(348,516)(349,517)(350,518)(351,519)(352,520)
(353,521)(354,522)(355,523)(356,524)(357,525)(358,526)(359,527)(360,528)
(361,529)(362,530)(363,531)(364,532)(365,533)(366,534)(367,535)(368,536)
(369,537)(370,538)(371,539)(372,540)(373,541)(374,542)(375,543)(376,544)
(377,545)(378,546)(379,547)(380,548)(381,549)(382,550)(383,551)(384,552)
(385,553)(386,554)(387,555)(388,556)(389,557)(390,558)(391,559)(392,560)
(393,561)(394,562)(395,563)(396,564)(397,565)(398,566)(399,567)(400,568)
(401,569)(402,570)(403,571)(404,572)(405,573)(406,574)(407,575)(408,576)
(409,577)(410,578)(411,579)(412,580)(413,581)(414,582)(415,583)(416,584)
(417,585)(418,586)(419,587)(420,588)(421,631)(422,632)(423,633)(424,634)
(425,635)(426,636)(427,637)(428,638)(429,639)(430,640)(431,641)(432,642)
(433,643)(434,644)(435,645)(436,646)(437,647)(438,648)(439,649)(440,650)
(441,651)(442,652)(443,653)(444,654)(445,655)(446,656)(447,657)(448,658)
(449,659)(450,660)(451,661)(452,662)(453,663)(454,664)(455,665)(456,666)
(457,667)(458,668)(459,669)(460,670)(461,671)(462,672)(463,589)(464,590)
(465,591)(466,592)(467,593)(468,594)(469,595)(470,596)(471,597)(472,598)
(473,599)(474,600)(475,601)(476,602)(477,603)(478,604)(479,605)(480,606)
(481,607)(482,608)(483,609)(484,610)(485,611)(486,612)(487,613)(488,614)
(489,615)(490,616)(491,617)(492,618)(493,619)(494,620)(495,621)(496,622)
(497,623)(498,624)(499,625)(500,626)(501,627)(502,628)(503,629)(504,630);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(672)!(  1,337)(  2,338)(  3,339)(  4,340)(  5,341)(  6,342)(  7,343)
(  8,344)(  9,345)( 10,346)( 11,347)( 12,348)( 13,349)( 14,350)( 15,351)
( 16,352)( 17,353)( 18,354)( 19,355)( 20,356)( 21,357)( 22,358)( 23,359)
( 24,360)( 25,361)( 26,362)( 27,363)( 28,364)( 29,365)( 30,366)( 31,367)
( 32,368)( 33,369)( 34,370)( 35,371)( 36,372)( 37,373)( 38,374)( 39,375)
( 40,376)( 41,377)( 42,378)( 43,379)( 44,380)( 45,381)( 46,382)( 47,383)
( 48,384)( 49,385)( 50,386)( 51,387)( 52,388)( 53,389)( 54,390)( 55,391)
( 56,392)( 57,393)( 58,394)( 59,395)( 60,396)( 61,397)( 62,398)( 63,399)
( 64,400)( 65,401)( 66,402)( 67,403)( 68,404)( 69,405)( 70,406)( 71,407)
( 72,408)( 73,409)( 74,410)( 75,411)( 76,412)( 77,413)( 78,414)( 79,415)
( 80,416)( 81,417)( 82,418)( 83,419)( 84,420)( 85,442)( 86,443)( 87,444)
( 88,445)( 89,446)( 90,447)( 91,448)( 92,449)( 93,450)( 94,451)( 95,452)
( 96,453)( 97,454)( 98,455)( 99,456)(100,457)(101,458)(102,459)(103,460)
(104,461)(105,462)(106,421)(107,422)(108,423)(109,424)(110,425)(111,426)
(112,427)(113,428)(114,429)(115,430)(116,431)(117,432)(118,433)(119,434)
(120,435)(121,436)(122,437)(123,438)(124,439)(125,440)(126,441)(127,484)
(128,485)(129,486)(130,487)(131,488)(132,489)(133,490)(134,491)(135,492)
(136,493)(137,494)(138,495)(139,496)(140,497)(141,498)(142,499)(143,500)
(144,501)(145,502)(146,503)(147,504)(148,463)(149,464)(150,465)(151,466)
(152,467)(153,468)(154,469)(155,470)(156,471)(157,472)(158,473)(159,474)
(160,475)(161,476)(162,477)(163,478)(164,479)(165,480)(166,481)(167,482)
(168,483)(169,505)(170,506)(171,507)(172,508)(173,509)(174,510)(175,511)
(176,512)(177,513)(178,514)(179,515)(180,516)(181,517)(182,518)(183,519)
(184,520)(185,521)(186,522)(187,523)(188,524)(189,525)(190,526)(191,527)
(192,528)(193,529)(194,530)(195,531)(196,532)(197,533)(198,534)(199,535)
(200,536)(201,537)(202,538)(203,539)(204,540)(205,541)(206,542)(207,543)
(208,544)(209,545)(210,546)(211,547)(212,548)(213,549)(214,550)(215,551)
(216,552)(217,553)(218,554)(219,555)(220,556)(221,557)(222,558)(223,559)
(224,560)(225,561)(226,562)(227,563)(228,564)(229,565)(230,566)(231,567)
(232,568)(233,569)(234,570)(235,571)(236,572)(237,573)(238,574)(239,575)
(240,576)(241,577)(242,578)(243,579)(244,580)(245,581)(246,582)(247,583)
(248,584)(249,585)(250,586)(251,587)(252,588)(253,610)(254,611)(255,612)
(256,613)(257,614)(258,615)(259,616)(260,617)(261,618)(262,619)(263,620)
(264,621)(265,622)(266,623)(267,624)(268,625)(269,626)(270,627)(271,628)
(272,629)(273,630)(274,589)(275,590)(276,591)(277,592)(278,593)(279,594)
(280,595)(281,596)(282,597)(283,598)(284,599)(285,600)(286,601)(287,602)
(288,603)(289,604)(290,605)(291,606)(292,607)(293,608)(294,609)(295,652)
(296,653)(297,654)(298,655)(299,656)(300,657)(301,658)(302,659)(303,660)
(304,661)(305,662)(306,663)(307,664)(308,665)(309,666)(310,667)(311,668)
(312,669)(313,670)(314,671)(315,672)(316,631)(317,632)(318,633)(319,634)
(320,635)(321,636)(322,637)(323,638)(324,639)(325,640)(326,641)(327,642)
(328,643)(329,644)(330,645)(331,646)(332,647)(333,648)(334,649)(335,650)
(336,651);
s1 := Sym(672)!(  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)
( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)
( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 44, 49)( 45, 48)( 46, 47)
( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 86, 91)( 87, 90)( 88, 89)( 92, 99)( 93,105)( 94,104)( 95,103)
( 96,102)( 97,101)( 98,100)(107,112)(108,111)(109,110)(113,120)(114,126)
(115,125)(116,124)(117,123)(118,122)(119,121)(128,133)(129,132)(130,131)
(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)
(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)
(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)
(180,186)(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)
(199,209)(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)
(218,225)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)
(234,237)(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)
(245,247)(254,259)(255,258)(256,257)(260,267)(261,273)(262,272)(263,271)
(264,270)(265,269)(266,268)(275,280)(276,279)(277,278)(281,288)(282,294)
(283,293)(284,292)(285,291)(286,290)(287,289)(296,301)(297,300)(298,299)
(302,309)(303,315)(304,314)(305,313)(306,312)(307,311)(308,310)(317,322)
(318,321)(319,320)(323,330)(324,336)(325,335)(326,334)(327,333)(328,332)
(329,331)(337,358)(338,364)(339,363)(340,362)(341,361)(342,360)(343,359)
(344,372)(345,378)(346,377)(347,376)(348,375)(349,374)(350,373)(351,365)
(352,371)(353,370)(354,369)(355,368)(356,367)(357,366)(379,400)(380,406)
(381,405)(382,404)(383,403)(384,402)(385,401)(386,414)(387,420)(388,419)
(389,418)(390,417)(391,416)(392,415)(393,407)(394,413)(395,412)(396,411)
(397,410)(398,409)(399,408)(421,442)(422,448)(423,447)(424,446)(425,445)
(426,444)(427,443)(428,456)(429,462)(430,461)(431,460)(432,459)(433,458)
(434,457)(435,449)(436,455)(437,454)(438,453)(439,452)(440,451)(441,450)
(463,484)(464,490)(465,489)(466,488)(467,487)(468,486)(469,485)(470,498)
(471,504)(472,503)(473,502)(474,501)(475,500)(476,499)(477,491)(478,497)
(479,496)(480,495)(481,494)(482,493)(483,492)(505,526)(506,532)(507,531)
(508,530)(509,529)(510,528)(511,527)(512,540)(513,546)(514,545)(515,544)
(516,543)(517,542)(518,541)(519,533)(520,539)(521,538)(522,537)(523,536)
(524,535)(525,534)(547,568)(548,574)(549,573)(550,572)(551,571)(552,570)
(553,569)(554,582)(555,588)(556,587)(557,586)(558,585)(559,584)(560,583)
(561,575)(562,581)(563,580)(564,579)(565,578)(566,577)(567,576)(589,610)
(590,616)(591,615)(592,614)(593,613)(594,612)(595,611)(596,624)(597,630)
(598,629)(599,628)(600,627)(601,626)(602,625)(603,617)(604,623)(605,622)
(606,621)(607,620)(608,619)(609,618)(631,652)(632,658)(633,657)(634,656)
(635,655)(636,654)(637,653)(638,666)(639,672)(640,671)(641,670)(642,669)
(643,668)(644,667)(645,659)(646,665)(647,664)(648,663)(649,662)(650,661)
(651,660);
s2 := Sym(672)!(  1, 93)(  2, 92)(  3, 98)(  4, 97)(  5, 96)(  6, 95)(  7, 94)
(  8, 86)(  9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)
( 16, 99)( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)
( 24,119)( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)
( 32,111)( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43,135)( 44,134)( 45,140)( 46,139)( 47,138)
( 48,137)( 49,136)( 50,128)( 51,127)( 52,133)( 53,132)( 54,131)( 55,130)
( 56,129)( 57,142)( 58,141)( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)
( 64,156)( 65,155)( 66,161)( 67,160)( 68,159)( 69,158)( 70,157)( 71,149)
( 72,148)( 73,154)( 74,153)( 75,152)( 76,151)( 77,150)( 78,163)( 79,162)
( 80,168)( 81,167)( 82,166)( 83,165)( 84,164)(169,261)(170,260)(171,266)
(172,265)(173,264)(174,263)(175,262)(176,254)(177,253)(178,259)(179,258)
(180,257)(181,256)(182,255)(183,268)(184,267)(185,273)(186,272)(187,271)
(188,270)(189,269)(190,282)(191,281)(192,287)(193,286)(194,285)(195,284)
(196,283)(197,275)(198,274)(199,280)(200,279)(201,278)(202,277)(203,276)
(204,289)(205,288)(206,294)(207,293)(208,292)(209,291)(210,290)(211,303)
(212,302)(213,308)(214,307)(215,306)(216,305)(217,304)(218,296)(219,295)
(220,301)(221,300)(222,299)(223,298)(224,297)(225,310)(226,309)(227,315)
(228,314)(229,313)(230,312)(231,311)(232,324)(233,323)(234,329)(235,328)
(236,327)(237,326)(238,325)(239,317)(240,316)(241,322)(242,321)(243,320)
(244,319)(245,318)(246,331)(247,330)(248,336)(249,335)(250,334)(251,333)
(252,332)(337,450)(338,449)(339,455)(340,454)(341,453)(342,452)(343,451)
(344,443)(345,442)(346,448)(347,447)(348,446)(349,445)(350,444)(351,457)
(352,456)(353,462)(354,461)(355,460)(356,459)(357,458)(358,429)(359,428)
(360,434)(361,433)(362,432)(363,431)(364,430)(365,422)(366,421)(367,427)
(368,426)(369,425)(370,424)(371,423)(372,436)(373,435)(374,441)(375,440)
(376,439)(377,438)(378,437)(379,492)(380,491)(381,497)(382,496)(383,495)
(384,494)(385,493)(386,485)(387,484)(388,490)(389,489)(390,488)(391,487)
(392,486)(393,499)(394,498)(395,504)(396,503)(397,502)(398,501)(399,500)
(400,471)(401,470)(402,476)(403,475)(404,474)(405,473)(406,472)(407,464)
(408,463)(409,469)(410,468)(411,467)(412,466)(413,465)(414,478)(415,477)
(416,483)(417,482)(418,481)(419,480)(420,479)(505,618)(506,617)(507,623)
(508,622)(509,621)(510,620)(511,619)(512,611)(513,610)(514,616)(515,615)
(516,614)(517,613)(518,612)(519,625)(520,624)(521,630)(522,629)(523,628)
(524,627)(525,626)(526,597)(527,596)(528,602)(529,601)(530,600)(531,599)
(532,598)(533,590)(534,589)(535,595)(536,594)(537,593)(538,592)(539,591)
(540,604)(541,603)(542,609)(543,608)(544,607)(545,606)(546,605)(547,660)
(548,659)(549,665)(550,664)(551,663)(552,662)(553,661)(554,653)(555,652)
(556,658)(557,657)(558,656)(559,655)(560,654)(561,667)(562,666)(563,672)
(564,671)(565,670)(566,669)(567,668)(568,639)(569,638)(570,644)(571,643)
(572,642)(573,641)(574,640)(575,632)(576,631)(577,637)(578,636)(579,635)
(580,634)(581,633)(582,646)(583,645)(584,651)(585,650)(586,649)(587,648)
(588,647);
s3 := Sym(672)!(  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)
(  8,176)(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)
( 40,208)( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)
( 56,224)( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)
( 64,232)( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)
( 72,240)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,295)( 86,296)( 87,297)
( 88,298)( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)
( 96,306)( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)
(104,314)(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)
(112,322)(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)
(120,330)(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)
(128,254)(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)
(136,262)(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)
(144,270)(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)
(152,278)(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)
(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)
(168,294)(337,505)(338,506)(339,507)(340,508)(341,509)(342,510)(343,511)
(344,512)(345,513)(346,514)(347,515)(348,516)(349,517)(350,518)(351,519)
(352,520)(353,521)(354,522)(355,523)(356,524)(357,525)(358,526)(359,527)
(360,528)(361,529)(362,530)(363,531)(364,532)(365,533)(366,534)(367,535)
(368,536)(369,537)(370,538)(371,539)(372,540)(373,541)(374,542)(375,543)
(376,544)(377,545)(378,546)(379,547)(380,548)(381,549)(382,550)(383,551)
(384,552)(385,553)(386,554)(387,555)(388,556)(389,557)(390,558)(391,559)
(392,560)(393,561)(394,562)(395,563)(396,564)(397,565)(398,566)(399,567)
(400,568)(401,569)(402,570)(403,571)(404,572)(405,573)(406,574)(407,575)
(408,576)(409,577)(410,578)(411,579)(412,580)(413,581)(414,582)(415,583)
(416,584)(417,585)(418,586)(419,587)(420,588)(421,631)(422,632)(423,633)
(424,634)(425,635)(426,636)(427,637)(428,638)(429,639)(430,640)(431,641)
(432,642)(433,643)(434,644)(435,645)(436,646)(437,647)(438,648)(439,649)
(440,650)(441,651)(442,652)(443,653)(444,654)(445,655)(446,656)(447,657)
(448,658)(449,659)(450,660)(451,661)(452,662)(453,663)(454,664)(455,665)
(456,666)(457,667)(458,668)(459,669)(460,670)(461,671)(462,672)(463,589)
(464,590)(465,591)(466,592)(467,593)(468,594)(469,595)(470,596)(471,597)
(472,598)(473,599)(474,600)(475,601)(476,602)(477,603)(478,604)(479,605)
(480,606)(481,607)(482,608)(483,609)(484,610)(485,611)(486,612)(487,613)
(488,614)(489,615)(490,616)(491,617)(492,618)(493,619)(494,620)(495,621)
(496,622)(497,623)(498,624)(499,625)(500,626)(501,627)(502,628)(503,629)
(504,630);
poly := sub<Sym(672)|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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope