Polytope of Type {172,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {172,4}*1376
Also Known As : {172,4|2}. if this polytope has another name.
Group : SmallGroup(1376,116)
Rank : 3
Schlafli Type : {172,4}
Number of vertices, edges, etc : 172, 344, 4
Order of s₀s₁s₂ : 172
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {172,2}*688, {86,4}*688
   4-fold quotients : {86,2}*344
   8-fold quotients : {43,2}*172
   43-fold quotients : {4,4}*32
   86-fold quotients : {2,4}*16, {4,2}*16
   172-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,173)(  2,215)(  3,214)(  4,213)(  5,212)(  6,211)(  7,210)(  8,209)
(  9,208)( 10,207)( 11,206)( 12,205)( 13,204)( 14,203)( 15,202)( 16,201)
( 17,200)( 18,199)( 19,198)( 20,197)( 21,196)( 22,195)( 23,194)( 24,193)
( 25,192)( 26,191)( 27,190)( 28,189)( 29,188)( 30,187)( 31,186)( 32,185)
( 33,184)( 34,183)( 35,182)( 36,181)( 37,180)( 38,179)( 39,178)( 40,177)
( 41,176)( 42,175)( 43,174)( 44,216)( 45,258)( 46,257)( 47,256)( 48,255)
( 49,254)( 50,253)( 51,252)( 52,251)( 53,250)( 54,249)( 55,248)( 56,247)
( 57,246)( 58,245)( 59,244)( 60,243)( 61,242)( 62,241)( 63,240)( 64,239)
( 65,238)( 66,237)( 67,236)( 68,235)( 69,234)( 70,233)( 71,232)( 72,231)
( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,225)( 79,224)( 80,223)
( 81,222)( 82,221)( 83,220)( 84,219)( 85,218)( 86,217)( 87,259)( 88,301)
( 89,300)( 90,299)( 91,298)( 92,297)( 93,296)( 94,295)( 95,294)( 96,293)
( 97,292)( 98,291)( 99,290)(100,289)(101,288)(102,287)(103,286)(104,285)
(105,284)(106,283)(107,282)(108,281)(109,280)(110,279)(111,278)(112,277)
(113,276)(114,275)(115,274)(116,273)(117,272)(118,271)(119,270)(120,269)
(121,268)(122,267)(123,266)(124,265)(125,264)(126,263)(127,262)(128,261)
(129,260)(130,302)(131,344)(132,343)(133,342)(134,341)(135,340)(136,339)
(137,338)(138,337)(139,336)(140,335)(141,334)(142,333)(143,332)(144,331)
(145,330)(146,329)(147,328)(148,327)(149,326)(150,325)(151,324)(152,323)
(153,322)(154,321)(155,320)(156,319)(157,318)(158,317)(159,316)(160,315)
(161,314)(162,313)(163,312)(164,311)(165,310)(166,309)(167,308)(168,307)
(169,306)(170,305)(171,304)(172,303)(345,517)(346,559)(347,558)(348,557)
(349,556)(350,555)(351,554)(352,553)(353,552)(354,551)(355,550)(356,549)
(357,548)(358,547)(359,546)(360,545)(361,544)(362,543)(363,542)(364,541)
(365,540)(366,539)(367,538)(368,537)(369,536)(370,535)(371,534)(372,533)
(373,532)(374,531)(375,530)(376,529)(377,528)(378,527)(379,526)(380,525)
(381,524)(382,523)(383,522)(384,521)(385,520)(386,519)(387,518)(388,560)
(389,602)(390,601)(391,600)(392,599)(393,598)(394,597)(395,596)(396,595)
(397,594)(398,593)(399,592)(400,591)(401,590)(402,589)(403,588)(404,587)
(405,586)(406,585)(407,584)(408,583)(409,582)(410,581)(411,580)(412,579)
(413,578)(414,577)(415,576)(416,575)(417,574)(418,573)(419,572)(420,571)
(421,570)(422,569)(423,568)(424,567)(425,566)(426,565)(427,564)(428,563)
(429,562)(430,561)(431,603)(432,645)(433,644)(434,643)(435,642)(436,641)
(437,640)(438,639)(439,638)(440,637)(441,636)(442,635)(443,634)(444,633)
(445,632)(446,631)(447,630)(448,629)(449,628)(450,627)(451,626)(452,625)
(453,624)(454,623)(455,622)(456,621)(457,620)(458,619)(459,618)(460,617)
(461,616)(462,615)(463,614)(464,613)(465,612)(466,611)(467,610)(468,609)
(469,608)(470,607)(471,606)(472,605)(473,604)(474,646)(475,688)(476,687)
(477,686)(478,685)(479,684)(480,683)(481,682)(482,681)(483,680)(484,679)
(485,678)(486,677)(487,676)(488,675)(489,674)(490,673)(491,672)(492,671)
(493,670)(494,669)(495,668)(496,667)(497,666)(498,665)(499,664)(500,663)
(501,662)(502,661)(503,660)(504,659)(505,658)(506,657)(507,656)(508,655)
(509,654)(510,653)(511,652)(512,651)(513,650)(514,649)(515,648)(516,647);;
s1 := (  1,  2)(  3, 43)(  4, 42)(  5, 41)(  6, 40)(  7, 39)(  8, 38)(  9, 37)
( 10, 36)( 11, 35)( 12, 34)( 13, 33)( 14, 32)( 15, 31)( 16, 30)( 17, 29)
( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 44, 45)( 46, 86)( 47, 85)
( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53, 79)( 54, 78)( 55, 77)
( 56, 76)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 87, 88)( 89,129)( 90,128)( 91,127)( 92,126)( 93,125)
( 94,124)( 95,123)( 96,122)( 97,121)( 98,120)( 99,119)(100,118)(101,117)
(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(130,131)
(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,165)
(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)
(148,156)(149,155)(150,154)(151,153)(173,217)(174,216)(175,258)(176,257)
(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)(184,249)
(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)(192,241)
(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)(200,233)
(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)(208,225)
(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(259,303)
(260,302)(261,344)(262,343)(263,342)(264,341)(265,340)(266,339)(267,338)
(268,337)(269,336)(270,335)(271,334)(272,333)(273,332)(274,331)(275,330)
(276,329)(277,328)(278,327)(279,326)(280,325)(281,324)(282,323)(283,322)
(284,321)(285,320)(286,319)(287,318)(288,317)(289,316)(290,315)(291,314)
(292,313)(293,312)(294,311)(295,310)(296,309)(297,308)(298,307)(299,306)
(300,305)(301,304)(345,432)(346,431)(347,473)(348,472)(349,471)(350,470)
(351,469)(352,468)(353,467)(354,466)(355,465)(356,464)(357,463)(358,462)
(359,461)(360,460)(361,459)(362,458)(363,457)(364,456)(365,455)(366,454)
(367,453)(368,452)(369,451)(370,450)(371,449)(372,448)(373,447)(374,446)
(375,445)(376,444)(377,443)(378,442)(379,441)(380,440)(381,439)(382,438)
(383,437)(384,436)(385,435)(386,434)(387,433)(388,475)(389,474)(390,516)
(391,515)(392,514)(393,513)(394,512)(395,511)(396,510)(397,509)(398,508)
(399,507)(400,506)(401,505)(402,504)(403,503)(404,502)(405,501)(406,500)
(407,499)(408,498)(409,497)(410,496)(411,495)(412,494)(413,493)(414,492)
(415,491)(416,490)(417,489)(418,488)(419,487)(420,486)(421,485)(422,484)
(423,483)(424,482)(425,481)(426,480)(427,479)(428,478)(429,477)(430,476)
(517,647)(518,646)(519,688)(520,687)(521,686)(522,685)(523,684)(524,683)
(525,682)(526,681)(527,680)(528,679)(529,678)(530,677)(531,676)(532,675)
(533,674)(534,673)(535,672)(536,671)(537,670)(538,669)(539,668)(540,667)
(541,666)(542,665)(543,664)(544,663)(545,662)(546,661)(547,660)(548,659)
(549,658)(550,657)(551,656)(552,655)(553,654)(554,653)(555,652)(556,651)
(557,650)(558,649)(559,648)(560,604)(561,603)(562,645)(563,644)(564,643)
(565,642)(566,641)(567,640)(568,639)(569,638)(570,637)(571,636)(572,635)
(573,634)(574,633)(575,632)(576,631)(577,630)(578,629)(579,628)(580,627)
(581,626)(582,625)(583,624)(584,623)(585,622)(586,621)(587,620)(588,619)
(589,618)(590,617)(591,616)(592,615)(593,614)(594,613)(595,612)(596,611)
(597,610)(598,609)(599,608)(600,607)(601,606)(602,605);;
s2 := (  1,345)(  2,346)(  3,347)(  4,348)(  5,349)(  6,350)(  7,351)(  8,352)
(  9,353)( 10,354)( 11,355)( 12,356)( 13,357)( 14,358)( 15,359)( 16,360)
( 17,361)( 18,362)( 19,363)( 20,364)( 21,365)( 22,366)( 23,367)( 24,368)
( 25,369)( 26,370)( 27,371)( 28,372)( 29,373)( 30,374)( 31,375)( 32,376)
( 33,377)( 34,378)( 35,379)( 36,380)( 37,381)( 38,382)( 39,383)( 40,384)
( 41,385)( 42,386)( 43,387)( 44,388)( 45,389)( 46,390)( 47,391)( 48,392)
( 49,393)( 50,394)( 51,395)( 52,396)( 53,397)( 54,398)( 55,399)( 56,400)
( 57,401)( 58,402)( 59,403)( 60,404)( 61,405)( 62,406)( 63,407)( 64,408)
( 65,409)( 66,410)( 67,411)( 68,412)( 69,413)( 70,414)( 71,415)( 72,416)
( 73,417)( 74,418)( 75,419)( 76,420)( 77,421)( 78,422)( 79,423)( 80,424)
( 81,425)( 82,426)( 83,427)( 84,428)( 85,429)( 86,430)( 87,431)( 88,432)
( 89,433)( 90,434)( 91,435)( 92,436)( 93,437)( 94,438)( 95,439)( 96,440)
( 97,441)( 98,442)( 99,443)(100,444)(101,445)(102,446)(103,447)(104,448)
(105,449)(106,450)(107,451)(108,452)(109,453)(110,454)(111,455)(112,456)
(113,457)(114,458)(115,459)(116,460)(117,461)(118,462)(119,463)(120,464)
(121,465)(122,466)(123,467)(124,468)(125,469)(126,470)(127,471)(128,472)
(129,473)(130,474)(131,475)(132,476)(133,477)(134,478)(135,479)(136,480)
(137,481)(138,482)(139,483)(140,484)(141,485)(142,486)(143,487)(144,488)
(145,489)(146,490)(147,491)(148,492)(149,493)(150,494)(151,495)(152,496)
(153,497)(154,498)(155,499)(156,500)(157,501)(158,502)(159,503)(160,504)
(161,505)(162,506)(163,507)(164,508)(165,509)(166,510)(167,511)(168,512)
(169,513)(170,514)(171,515)(172,516)(173,517)(174,518)(175,519)(176,520)
(177,521)(178,522)(179,523)(180,524)(181,525)(182,526)(183,527)(184,528)
(185,529)(186,530)(187,531)(188,532)(189,533)(190,534)(191,535)(192,536)
(193,537)(194,538)(195,539)(196,540)(197,541)(198,542)(199,543)(200,544)
(201,545)(202,546)(203,547)(204,548)(205,549)(206,550)(207,551)(208,552)
(209,553)(210,554)(211,555)(212,556)(213,557)(214,558)(215,559)(216,560)
(217,561)(218,562)(219,563)(220,564)(221,565)(222,566)(223,567)(224,568)
(225,569)(226,570)(227,571)(228,572)(229,573)(230,574)(231,575)(232,576)
(233,577)(234,578)(235,579)(236,580)(237,581)(238,582)(239,583)(240,584)
(241,585)(242,586)(243,587)(244,588)(245,589)(246,590)(247,591)(248,592)
(249,593)(250,594)(251,595)(252,596)(253,597)(254,598)(255,599)(256,600)
(257,601)(258,602)(259,603)(260,604)(261,605)(262,606)(263,607)(264,608)
(265,609)(266,610)(267,611)(268,612)(269,613)(270,614)(271,615)(272,616)
(273,617)(274,618)(275,619)(276,620)(277,621)(278,622)(279,623)(280,624)
(281,625)(282,626)(283,627)(284,628)(285,629)(286,630)(287,631)(288,632)
(289,633)(290,634)(291,635)(292,636)(293,637)(294,638)(295,639)(296,640)
(297,641)(298,642)(299,643)(300,644)(301,645)(302,646)(303,647)(304,648)
(305,649)(306,650)(307,651)(308,652)(309,653)(310,654)(311,655)(312,656)
(313,657)(314,658)(315,659)(316,660)(317,661)(318,662)(319,663)(320,664)
(321,665)(322,666)(323,667)(324,668)(325,669)(326,670)(327,671)(328,672)
(329,673)(330,674)(331,675)(332,676)(333,677)(334,678)(335,679)(336,680)
(337,681)(338,682)(339,683)(340,684)(341,685)(342,686)(343,687)(344,688);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(688)!(  1,173)(  2,215)(  3,214)(  4,213)(  5,212)(  6,211)(  7,210)
(  8,209)(  9,208)( 10,207)( 11,206)( 12,205)( 13,204)( 14,203)( 15,202)
( 16,201)( 17,200)( 18,199)( 19,198)( 20,197)( 21,196)( 22,195)( 23,194)
( 24,193)( 25,192)( 26,191)( 27,190)( 28,189)( 29,188)( 30,187)( 31,186)
( 32,185)( 33,184)( 34,183)( 35,182)( 36,181)( 37,180)( 38,179)( 39,178)
( 40,177)( 41,176)( 42,175)( 43,174)( 44,216)( 45,258)( 46,257)( 47,256)
( 48,255)( 49,254)( 50,253)( 51,252)( 52,251)( 53,250)( 54,249)( 55,248)
( 56,247)( 57,246)( 58,245)( 59,244)( 60,243)( 61,242)( 62,241)( 63,240)
( 64,239)( 65,238)( 66,237)( 67,236)( 68,235)( 69,234)( 70,233)( 71,232)
( 72,231)( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,225)( 79,224)
( 80,223)( 81,222)( 82,221)( 83,220)( 84,219)( 85,218)( 86,217)( 87,259)
( 88,301)( 89,300)( 90,299)( 91,298)( 92,297)( 93,296)( 94,295)( 95,294)
( 96,293)( 97,292)( 98,291)( 99,290)(100,289)(101,288)(102,287)(103,286)
(104,285)(105,284)(106,283)(107,282)(108,281)(109,280)(110,279)(111,278)
(112,277)(113,276)(114,275)(115,274)(116,273)(117,272)(118,271)(119,270)
(120,269)(121,268)(122,267)(123,266)(124,265)(125,264)(126,263)(127,262)
(128,261)(129,260)(130,302)(131,344)(132,343)(133,342)(134,341)(135,340)
(136,339)(137,338)(138,337)(139,336)(140,335)(141,334)(142,333)(143,332)
(144,331)(145,330)(146,329)(147,328)(148,327)(149,326)(150,325)(151,324)
(152,323)(153,322)(154,321)(155,320)(156,319)(157,318)(158,317)(159,316)
(160,315)(161,314)(162,313)(163,312)(164,311)(165,310)(166,309)(167,308)
(168,307)(169,306)(170,305)(171,304)(172,303)(345,517)(346,559)(347,558)
(348,557)(349,556)(350,555)(351,554)(352,553)(353,552)(354,551)(355,550)
(356,549)(357,548)(358,547)(359,546)(360,545)(361,544)(362,543)(363,542)
(364,541)(365,540)(366,539)(367,538)(368,537)(369,536)(370,535)(371,534)
(372,533)(373,532)(374,531)(375,530)(376,529)(377,528)(378,527)(379,526)
(380,525)(381,524)(382,523)(383,522)(384,521)(385,520)(386,519)(387,518)
(388,560)(389,602)(390,601)(391,600)(392,599)(393,598)(394,597)(395,596)
(396,595)(397,594)(398,593)(399,592)(400,591)(401,590)(402,589)(403,588)
(404,587)(405,586)(406,585)(407,584)(408,583)(409,582)(410,581)(411,580)
(412,579)(413,578)(414,577)(415,576)(416,575)(417,574)(418,573)(419,572)
(420,571)(421,570)(422,569)(423,568)(424,567)(425,566)(426,565)(427,564)
(428,563)(429,562)(430,561)(431,603)(432,645)(433,644)(434,643)(435,642)
(436,641)(437,640)(438,639)(439,638)(440,637)(441,636)(442,635)(443,634)
(444,633)(445,632)(446,631)(447,630)(448,629)(449,628)(450,627)(451,626)
(452,625)(453,624)(454,623)(455,622)(456,621)(457,620)(458,619)(459,618)
(460,617)(461,616)(462,615)(463,614)(464,613)(465,612)(466,611)(467,610)
(468,609)(469,608)(470,607)(471,606)(472,605)(473,604)(474,646)(475,688)
(476,687)(477,686)(478,685)(479,684)(480,683)(481,682)(482,681)(483,680)
(484,679)(485,678)(486,677)(487,676)(488,675)(489,674)(490,673)(491,672)
(492,671)(493,670)(494,669)(495,668)(496,667)(497,666)(498,665)(499,664)
(500,663)(501,662)(502,661)(503,660)(504,659)(505,658)(506,657)(507,656)
(508,655)(509,654)(510,653)(511,652)(512,651)(513,650)(514,649)(515,648)
(516,647);
s1 := Sym(688)!(  1,  2)(  3, 43)(  4, 42)(  5, 41)(  6, 40)(  7, 39)(  8, 38)
(  9, 37)( 10, 36)( 11, 35)( 12, 34)( 13, 33)( 14, 32)( 15, 31)( 16, 30)
( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 44, 45)( 46, 86)
( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53, 79)( 54, 78)
( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 61, 71)( 62, 70)
( 63, 69)( 64, 68)( 65, 67)( 87, 88)( 89,129)( 90,128)( 91,127)( 92,126)
( 93,125)( 94,124)( 95,123)( 96,122)( 97,121)( 98,120)( 99,119)(100,118)
(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)
(130,131)(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)
(139,165)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)
(147,157)(148,156)(149,155)(150,154)(151,153)(173,217)(174,216)(175,258)
(176,257)(177,256)(178,255)(179,254)(180,253)(181,252)(182,251)(183,250)
(184,249)(185,248)(186,247)(187,246)(188,245)(189,244)(190,243)(191,242)
(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)
(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)
(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)
(259,303)(260,302)(261,344)(262,343)(263,342)(264,341)(265,340)(266,339)
(267,338)(268,337)(269,336)(270,335)(271,334)(272,333)(273,332)(274,331)
(275,330)(276,329)(277,328)(278,327)(279,326)(280,325)(281,324)(282,323)
(283,322)(284,321)(285,320)(286,319)(287,318)(288,317)(289,316)(290,315)
(291,314)(292,313)(293,312)(294,311)(295,310)(296,309)(297,308)(298,307)
(299,306)(300,305)(301,304)(345,432)(346,431)(347,473)(348,472)(349,471)
(350,470)(351,469)(352,468)(353,467)(354,466)(355,465)(356,464)(357,463)
(358,462)(359,461)(360,460)(361,459)(362,458)(363,457)(364,456)(365,455)
(366,454)(367,453)(368,452)(369,451)(370,450)(371,449)(372,448)(373,447)
(374,446)(375,445)(376,444)(377,443)(378,442)(379,441)(380,440)(381,439)
(382,438)(383,437)(384,436)(385,435)(386,434)(387,433)(388,475)(389,474)
(390,516)(391,515)(392,514)(393,513)(394,512)(395,511)(396,510)(397,509)
(398,508)(399,507)(400,506)(401,505)(402,504)(403,503)(404,502)(405,501)
(406,500)(407,499)(408,498)(409,497)(410,496)(411,495)(412,494)(413,493)
(414,492)(415,491)(416,490)(417,489)(418,488)(419,487)(420,486)(421,485)
(422,484)(423,483)(424,482)(425,481)(426,480)(427,479)(428,478)(429,477)
(430,476)(517,647)(518,646)(519,688)(520,687)(521,686)(522,685)(523,684)
(524,683)(525,682)(526,681)(527,680)(528,679)(529,678)(530,677)(531,676)
(532,675)(533,674)(534,673)(535,672)(536,671)(537,670)(538,669)(539,668)
(540,667)(541,666)(542,665)(543,664)(544,663)(545,662)(546,661)(547,660)
(548,659)(549,658)(550,657)(551,656)(552,655)(553,654)(554,653)(555,652)
(556,651)(557,650)(558,649)(559,648)(560,604)(561,603)(562,645)(563,644)
(564,643)(565,642)(566,641)(567,640)(568,639)(569,638)(570,637)(571,636)
(572,635)(573,634)(574,633)(575,632)(576,631)(577,630)(578,629)(579,628)
(580,627)(581,626)(582,625)(583,624)(584,623)(585,622)(586,621)(587,620)
(588,619)(589,618)(590,617)(591,616)(592,615)(593,614)(594,613)(595,612)
(596,611)(597,610)(598,609)(599,608)(600,607)(601,606)(602,605);
s2 := Sym(688)!(  1,345)(  2,346)(  3,347)(  4,348)(  5,349)(  6,350)(  7,351)
(  8,352)(  9,353)( 10,354)( 11,355)( 12,356)( 13,357)( 14,358)( 15,359)
( 16,360)( 17,361)( 18,362)( 19,363)( 20,364)( 21,365)( 22,366)( 23,367)
( 24,368)( 25,369)( 26,370)( 27,371)( 28,372)( 29,373)( 30,374)( 31,375)
( 32,376)( 33,377)( 34,378)( 35,379)( 36,380)( 37,381)( 38,382)( 39,383)
( 40,384)( 41,385)( 42,386)( 43,387)( 44,388)( 45,389)( 46,390)( 47,391)
( 48,392)( 49,393)( 50,394)( 51,395)( 52,396)( 53,397)( 54,398)( 55,399)
( 56,400)( 57,401)( 58,402)( 59,403)( 60,404)( 61,405)( 62,406)( 63,407)
( 64,408)( 65,409)( 66,410)( 67,411)( 68,412)( 69,413)( 70,414)( 71,415)
( 72,416)( 73,417)( 74,418)( 75,419)( 76,420)( 77,421)( 78,422)( 79,423)
( 80,424)( 81,425)( 82,426)( 83,427)( 84,428)( 85,429)( 86,430)( 87,431)
( 88,432)( 89,433)( 90,434)( 91,435)( 92,436)( 93,437)( 94,438)( 95,439)
( 96,440)( 97,441)( 98,442)( 99,443)(100,444)(101,445)(102,446)(103,447)
(104,448)(105,449)(106,450)(107,451)(108,452)(109,453)(110,454)(111,455)
(112,456)(113,457)(114,458)(115,459)(116,460)(117,461)(118,462)(119,463)
(120,464)(121,465)(122,466)(123,467)(124,468)(125,469)(126,470)(127,471)
(128,472)(129,473)(130,474)(131,475)(132,476)(133,477)(134,478)(135,479)
(136,480)(137,481)(138,482)(139,483)(140,484)(141,485)(142,486)(143,487)
(144,488)(145,489)(146,490)(147,491)(148,492)(149,493)(150,494)(151,495)
(152,496)(153,497)(154,498)(155,499)(156,500)(157,501)(158,502)(159,503)
(160,504)(161,505)(162,506)(163,507)(164,508)(165,509)(166,510)(167,511)
(168,512)(169,513)(170,514)(171,515)(172,516)(173,517)(174,518)(175,519)
(176,520)(177,521)(178,522)(179,523)(180,524)(181,525)(182,526)(183,527)
(184,528)(185,529)(186,530)(187,531)(188,532)(189,533)(190,534)(191,535)
(192,536)(193,537)(194,538)(195,539)(196,540)(197,541)(198,542)(199,543)
(200,544)(201,545)(202,546)(203,547)(204,548)(205,549)(206,550)(207,551)
(208,552)(209,553)(210,554)(211,555)(212,556)(213,557)(214,558)(215,559)
(216,560)(217,561)(218,562)(219,563)(220,564)(221,565)(222,566)(223,567)
(224,568)(225,569)(226,570)(227,571)(228,572)(229,573)(230,574)(231,575)
(232,576)(233,577)(234,578)(235,579)(236,580)(237,581)(238,582)(239,583)
(240,584)(241,585)(242,586)(243,587)(244,588)(245,589)(246,590)(247,591)
(248,592)(249,593)(250,594)(251,595)(252,596)(253,597)(254,598)(255,599)
(256,600)(257,601)(258,602)(259,603)(260,604)(261,605)(262,606)(263,607)
(264,608)(265,609)(266,610)(267,611)(268,612)(269,613)(270,614)(271,615)
(272,616)(273,617)(274,618)(275,619)(276,620)(277,621)(278,622)(279,623)
(280,624)(281,625)(282,626)(283,627)(284,628)(285,629)(286,630)(287,631)
(288,632)(289,633)(290,634)(291,635)(292,636)(293,637)(294,638)(295,639)
(296,640)(297,641)(298,642)(299,643)(300,644)(301,645)(302,646)(303,647)
(304,648)(305,649)(306,650)(307,651)(308,652)(309,653)(310,654)(311,655)
(312,656)(313,657)(314,658)(315,659)(316,660)(317,661)(318,662)(319,663)
(320,664)(321,665)(322,666)(323,667)(324,668)(325,669)(326,670)(327,671)
(328,672)(329,673)(330,674)(331,675)(332,676)(333,677)(334,678)(335,679)
(336,680)(337,681)(338,682)(339,683)(340,684)(341,685)(342,686)(343,687)
(344,688);
poly := sub<Sym(688)|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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 : None.
to this polytope