Polytope of Type {4,171}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,171}*1368
if this polytope has a name.
Group : SmallGroup(1368,55)
Rank : 3
Schlafli Type : {4,171}
Number of vertices, edges, etc : 4, 342, 171
Order of s₀s₁s₂ : 171
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,57}*456
   19-fold quotients : {4,9}*72
   57-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)(382,384)
(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)(414,416)
(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)(430,432)
(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)(446,448)
(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)(462,464)
(465,467)(466,468)(469,471)(470,472)(473,475)(474,476)(477,479)(478,480)
(481,483)(482,484)(485,487)(486,488)(489,491)(490,492)(493,495)(494,496)
(497,499)(498,500)(501,503)(502,504)(505,507)(506,508)(509,511)(510,512)
(513,515)(514,516)(517,519)(518,520)(521,523)(522,524)(525,527)(526,528)
(529,531)(530,532)(533,535)(534,536)(537,539)(538,540)(541,543)(542,544)
(545,547)(546,548)(549,551)(550,552)(553,555)(554,556)(557,559)(558,560)
(561,563)(562,564)(565,567)(566,568)(569,571)(570,572)(573,575)(574,576)
(577,579)(578,580)(581,583)(582,584)(585,587)(586,588)(589,591)(590,592)
(593,595)(594,596)(597,599)(598,600)(601,603)(602,604)(605,607)(606,608)
(609,611)(610,612)(613,615)(614,616)(617,619)(618,620)(621,623)(622,624)
(625,627)(626,628)(629,631)(630,632)(633,635)(634,636)(637,639)(638,640)
(641,643)(642,644)(645,647)(646,648)(649,651)(650,652)(653,655)(654,656)
(657,659)(658,660)(661,663)(662,664)(665,667)(666,668)(669,671)(670,672)
(673,675)(674,676)(677,679)(678,680)(681,683)(682,684);;
s1 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,217)( 14,218)( 15,220)
( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)
( 24,223)( 25,205)( 26,206)( 27,208)( 28,207)( 29,213)( 30,214)( 31,216)
( 32,215)( 33,209)( 34,210)( 35,212)( 36,211)( 37,193)( 38,194)( 39,196)
( 40,195)( 41,201)( 42,202)( 43,204)( 44,203)( 45,197)( 46,198)( 47,200)
( 48,199)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)
( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)
( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)
( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)
( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)
( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)
( 96,151)( 97,133)( 98,134)( 99,136)(100,135)(101,141)(102,142)(103,144)
(104,143)(105,137)(106,138)(107,140)(108,139)(109,121)(110,122)(111,124)
(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)
(120,127)(229,465)(230,466)(231,468)(232,467)(233,461)(234,462)(235,464)
(236,463)(237,457)(238,458)(239,460)(240,459)(241,681)(242,682)(243,684)
(244,683)(245,677)(246,678)(247,680)(248,679)(249,673)(250,674)(251,676)
(252,675)(253,669)(254,670)(255,672)(256,671)(257,665)(258,666)(259,668)
(260,667)(261,661)(262,662)(263,664)(264,663)(265,657)(266,658)(267,660)
(268,659)(269,653)(270,654)(271,656)(272,655)(273,649)(274,650)(275,652)
(276,651)(277,645)(278,646)(279,648)(280,647)(281,641)(282,642)(283,644)
(284,643)(285,637)(286,638)(287,640)(288,639)(289,633)(290,634)(291,636)
(292,635)(293,629)(294,630)(295,632)(296,631)(297,625)(298,626)(299,628)
(300,627)(301,621)(302,622)(303,624)(304,623)(305,617)(306,618)(307,620)
(308,619)(309,613)(310,614)(311,616)(312,615)(313,609)(314,610)(315,612)
(316,611)(317,605)(318,606)(319,608)(320,607)(321,601)(322,602)(323,604)
(324,603)(325,597)(326,598)(327,600)(328,599)(329,593)(330,594)(331,596)
(332,595)(333,589)(334,590)(335,592)(336,591)(337,585)(338,586)(339,588)
(340,587)(341,581)(342,582)(343,584)(344,583)(345,577)(346,578)(347,580)
(348,579)(349,573)(350,574)(351,576)(352,575)(353,569)(354,570)(355,572)
(356,571)(357,565)(358,566)(359,568)(360,567)(361,561)(362,562)(363,564)
(364,563)(365,557)(366,558)(367,560)(368,559)(369,553)(370,554)(371,556)
(372,555)(373,549)(374,550)(375,552)(376,551)(377,545)(378,546)(379,548)
(380,547)(381,541)(382,542)(383,544)(384,543)(385,537)(386,538)(387,540)
(388,539)(389,533)(390,534)(391,536)(392,535)(393,529)(394,530)(395,532)
(396,531)(397,525)(398,526)(399,528)(400,527)(401,521)(402,522)(403,524)
(404,523)(405,517)(406,518)(407,520)(408,519)(409,513)(410,514)(411,516)
(412,515)(413,509)(414,510)(415,512)(416,511)(417,505)(418,506)(419,508)
(420,507)(421,501)(422,502)(423,504)(424,503)(425,497)(426,498)(427,500)
(428,499)(429,493)(430,494)(431,496)(432,495)(433,489)(434,490)(435,492)
(436,491)(437,485)(438,486)(439,488)(440,487)(441,481)(442,482)(443,484)
(444,483)(445,477)(446,478)(447,480)(448,479)(449,473)(450,474)(451,476)
(452,475)(453,469)(454,470)(455,472)(456,471);;
s2 := (  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)(  8,250)
(  9,245)( 10,248)( 11,247)( 12,246)( 13,229)( 14,232)( 15,231)( 16,230)
( 17,237)( 18,240)( 19,239)( 20,238)( 21,233)( 22,236)( 23,235)( 24,234)
( 25,445)( 26,448)( 27,447)( 28,446)( 29,453)( 30,456)( 31,455)( 32,454)
( 33,449)( 34,452)( 35,451)( 36,450)( 37,433)( 38,436)( 39,435)( 40,434)
( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)( 48,438)
( 49,421)( 50,424)( 51,423)( 52,422)( 53,429)( 54,432)( 55,431)( 56,430)
( 57,425)( 58,428)( 59,427)( 60,426)( 61,409)( 62,412)( 63,411)( 64,410)
( 65,417)( 66,420)( 67,419)( 68,418)( 69,413)( 70,416)( 71,415)( 72,414)
( 73,397)( 74,400)( 75,399)( 76,398)( 77,405)( 78,408)( 79,407)( 80,406)
( 81,401)( 82,404)( 83,403)( 84,402)( 85,385)( 86,388)( 87,387)( 88,386)
( 89,393)( 90,396)( 91,395)( 92,394)( 93,389)( 94,392)( 95,391)( 96,390)
( 97,373)( 98,376)( 99,375)(100,374)(101,381)(102,384)(103,383)(104,382)
(105,377)(106,380)(107,379)(108,378)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,349)(122,352)(123,351)(124,350)(125,357)(126,360)(127,359)(128,358)
(129,353)(130,356)(131,355)(132,354)(133,337)(134,340)(135,339)(136,338)
(137,345)(138,348)(139,347)(140,346)(141,341)(142,344)(143,343)(144,342)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,313)(158,316)(159,315)(160,314)
(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)(168,318)
(169,301)(170,304)(171,303)(172,302)(173,309)(174,312)(175,311)(176,310)
(177,305)(178,308)(179,307)(180,306)(181,289)(182,292)(183,291)(184,290)
(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)(191,295)(192,294)
(193,277)(194,280)(195,279)(196,278)(197,285)(198,288)(199,287)(200,286)
(201,281)(202,284)(203,283)(204,282)(205,265)(206,268)(207,267)(208,266)
(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)(215,271)(216,270)
(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)(223,263)(224,262)
(225,257)(226,260)(227,259)(228,258)(457,477)(458,480)(459,479)(460,478)
(461,473)(462,476)(463,475)(464,474)(465,469)(466,472)(467,471)(468,470)
(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)(488,678)
(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)(496,670)
(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)(504,662)
(505,657)(506,660)(507,659)(508,658)(509,653)(510,656)(511,655)(512,654)
(513,649)(514,652)(515,651)(516,650)(517,645)(518,648)(519,647)(520,646)
(521,641)(522,644)(523,643)(524,642)(525,637)(526,640)(527,639)(528,638)
(529,633)(530,636)(531,635)(532,634)(533,629)(534,632)(535,631)(536,630)
(537,625)(538,628)(539,627)(540,626)(541,621)(542,624)(543,623)(544,622)
(545,617)(546,620)(547,619)(548,618)(549,613)(550,616)(551,615)(552,614)
(553,609)(554,612)(555,611)(556,610)(557,605)(558,608)(559,607)(560,606)
(561,601)(562,604)(563,603)(564,602)(565,597)(566,600)(567,599)(568,598)
(569,593)(570,596)(571,595)(572,594)(573,589)(574,592)(575,591)(576,590)
(577,585)(578,588)(579,587)(580,586)(582,584);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

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

Permutation Representation (Magma) :

s0 := Sym(684)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)
(382,384)(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)
(414,416)(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)
(430,432)(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)
(446,448)(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)
(462,464)(465,467)(466,468)(469,471)(470,472)(473,475)(474,476)(477,479)
(478,480)(481,483)(482,484)(485,487)(486,488)(489,491)(490,492)(493,495)
(494,496)(497,499)(498,500)(501,503)(502,504)(505,507)(506,508)(509,511)
(510,512)(513,515)(514,516)(517,519)(518,520)(521,523)(522,524)(525,527)
(526,528)(529,531)(530,532)(533,535)(534,536)(537,539)(538,540)(541,543)
(542,544)(545,547)(546,548)(549,551)(550,552)(553,555)(554,556)(557,559)
(558,560)(561,563)(562,564)(565,567)(566,568)(569,571)(570,572)(573,575)
(574,576)(577,579)(578,580)(581,583)(582,584)(585,587)(586,588)(589,591)
(590,592)(593,595)(594,596)(597,599)(598,600)(601,603)(602,604)(605,607)
(606,608)(609,611)(610,612)(613,615)(614,616)(617,619)(618,620)(621,623)
(622,624)(625,627)(626,628)(629,631)(630,632)(633,635)(634,636)(637,639)
(638,640)(641,643)(642,644)(645,647)(646,648)(649,651)(650,652)(653,655)
(654,656)(657,659)(658,660)(661,663)(662,664)(665,667)(666,668)(669,671)
(670,672)(673,675)(674,676)(677,679)(678,680)(681,683)(682,684);
s1 := Sym(684)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,217)( 14,218)
( 15,220)( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)
( 23,224)( 24,223)( 25,205)( 26,206)( 27,208)( 28,207)( 29,213)( 30,214)
( 31,216)( 32,215)( 33,209)( 34,210)( 35,212)( 36,211)( 37,193)( 38,194)
( 39,196)( 40,195)( 41,201)( 42,202)( 43,204)( 44,203)( 45,197)( 46,198)
( 47,200)( 48,199)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)
( 55,192)( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)
( 63,172)( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)
( 71,176)( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)
( 79,168)( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)
( 87,148)( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)
( 95,152)( 96,151)( 97,133)( 98,134)( 99,136)(100,135)(101,141)(102,142)
(103,144)(104,143)(105,137)(106,138)(107,140)(108,139)(109,121)(110,122)
(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)
(119,128)(120,127)(229,465)(230,466)(231,468)(232,467)(233,461)(234,462)
(235,464)(236,463)(237,457)(238,458)(239,460)(240,459)(241,681)(242,682)
(243,684)(244,683)(245,677)(246,678)(247,680)(248,679)(249,673)(250,674)
(251,676)(252,675)(253,669)(254,670)(255,672)(256,671)(257,665)(258,666)
(259,668)(260,667)(261,661)(262,662)(263,664)(264,663)(265,657)(266,658)
(267,660)(268,659)(269,653)(270,654)(271,656)(272,655)(273,649)(274,650)
(275,652)(276,651)(277,645)(278,646)(279,648)(280,647)(281,641)(282,642)
(283,644)(284,643)(285,637)(286,638)(287,640)(288,639)(289,633)(290,634)
(291,636)(292,635)(293,629)(294,630)(295,632)(296,631)(297,625)(298,626)
(299,628)(300,627)(301,621)(302,622)(303,624)(304,623)(305,617)(306,618)
(307,620)(308,619)(309,613)(310,614)(311,616)(312,615)(313,609)(314,610)
(315,612)(316,611)(317,605)(318,606)(319,608)(320,607)(321,601)(322,602)
(323,604)(324,603)(325,597)(326,598)(327,600)(328,599)(329,593)(330,594)
(331,596)(332,595)(333,589)(334,590)(335,592)(336,591)(337,585)(338,586)
(339,588)(340,587)(341,581)(342,582)(343,584)(344,583)(345,577)(346,578)
(347,580)(348,579)(349,573)(350,574)(351,576)(352,575)(353,569)(354,570)
(355,572)(356,571)(357,565)(358,566)(359,568)(360,567)(361,561)(362,562)
(363,564)(364,563)(365,557)(366,558)(367,560)(368,559)(369,553)(370,554)
(371,556)(372,555)(373,549)(374,550)(375,552)(376,551)(377,545)(378,546)
(379,548)(380,547)(381,541)(382,542)(383,544)(384,543)(385,537)(386,538)
(387,540)(388,539)(389,533)(390,534)(391,536)(392,535)(393,529)(394,530)
(395,532)(396,531)(397,525)(398,526)(399,528)(400,527)(401,521)(402,522)
(403,524)(404,523)(405,517)(406,518)(407,520)(408,519)(409,513)(410,514)
(411,516)(412,515)(413,509)(414,510)(415,512)(416,511)(417,505)(418,506)
(419,508)(420,507)(421,501)(422,502)(423,504)(424,503)(425,497)(426,498)
(427,500)(428,499)(429,493)(430,494)(431,496)(432,495)(433,489)(434,490)
(435,492)(436,491)(437,485)(438,486)(439,488)(440,487)(441,481)(442,482)
(443,484)(444,483)(445,477)(446,478)(447,480)(448,479)(449,473)(450,474)
(451,476)(452,475)(453,469)(454,470)(455,472)(456,471);
s2 := Sym(684)!(  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)
(  8,250)(  9,245)( 10,248)( 11,247)( 12,246)( 13,229)( 14,232)( 15,231)
( 16,230)( 17,237)( 18,240)( 19,239)( 20,238)( 21,233)( 22,236)( 23,235)
( 24,234)( 25,445)( 26,448)( 27,447)( 28,446)( 29,453)( 30,456)( 31,455)
( 32,454)( 33,449)( 34,452)( 35,451)( 36,450)( 37,433)( 38,436)( 39,435)
( 40,434)( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)
( 48,438)( 49,421)( 50,424)( 51,423)( 52,422)( 53,429)( 54,432)( 55,431)
( 56,430)( 57,425)( 58,428)( 59,427)( 60,426)( 61,409)( 62,412)( 63,411)
( 64,410)( 65,417)( 66,420)( 67,419)( 68,418)( 69,413)( 70,416)( 71,415)
( 72,414)( 73,397)( 74,400)( 75,399)( 76,398)( 77,405)( 78,408)( 79,407)
( 80,406)( 81,401)( 82,404)( 83,403)( 84,402)( 85,385)( 86,388)( 87,387)
( 88,386)( 89,393)( 90,396)( 91,395)( 92,394)( 93,389)( 94,392)( 95,391)
( 96,390)( 97,373)( 98,376)( 99,375)(100,374)(101,381)(102,384)(103,383)
(104,382)(105,377)(106,380)(107,379)(108,378)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,349)(122,352)(123,351)(124,350)(125,357)(126,360)(127,359)
(128,358)(129,353)(130,356)(131,355)(132,354)(133,337)(134,340)(135,339)
(136,338)(137,345)(138,348)(139,347)(140,346)(141,341)(142,344)(143,343)
(144,342)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,313)(158,316)(159,315)
(160,314)(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)
(168,318)(169,301)(170,304)(171,303)(172,302)(173,309)(174,312)(175,311)
(176,310)(177,305)(178,308)(179,307)(180,306)(181,289)(182,292)(183,291)
(184,290)(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)(191,295)
(192,294)(193,277)(194,280)(195,279)(196,278)(197,285)(198,288)(199,287)
(200,286)(201,281)(202,284)(203,283)(204,282)(205,265)(206,268)(207,267)
(208,266)(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)(215,271)
(216,270)(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)(223,263)
(224,262)(225,257)(226,260)(227,259)(228,258)(457,477)(458,480)(459,479)
(460,478)(461,473)(462,476)(463,475)(464,474)(465,469)(466,472)(467,471)
(468,470)(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)
(488,678)(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)
(496,670)(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)
(504,662)(505,657)(506,660)(507,659)(508,658)(509,653)(510,656)(511,655)
(512,654)(513,649)(514,652)(515,651)(516,650)(517,645)(518,648)(519,647)
(520,646)(521,641)(522,644)(523,643)(524,642)(525,637)(526,640)(527,639)
(528,638)(529,633)(530,636)(531,635)(532,634)(533,629)(534,632)(535,631)
(536,630)(537,625)(538,628)(539,627)(540,626)(541,621)(542,624)(543,623)
(544,622)(545,617)(546,620)(547,619)(548,618)(549,613)(550,616)(551,615)
(552,614)(553,609)(554,612)(555,611)(556,610)(557,605)(558,608)(559,607)
(560,606)(561,601)(562,604)(563,603)(564,602)(565,597)(566,600)(567,599)
(568,598)(569,593)(570,596)(571,595)(572,594)(573,589)(574,592)(575,591)
(576,590)(577,585)(578,588)(579,587)(580,586)(582,584);
poly := sub<Sym(684)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

References : None.
to this polytope