Polytope of Type {168,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {168,4}*1344c
if this polytope has a name.
Group : SmallGroup(1344,6459)
Rank : 3
Schlafli Type : {168,4}
Number of vertices, edges, etc : 168, 336, 4
Order of s₀s₁s₂ : 168
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {84,4}*672b
   4-fold quotients : {42,4}*336b
   7-fold quotients : {24,4}*192c
   8-fold quotients : {21,4}*168
   14-fold quotients : {12,4}*96b
   28-fold quotients : {6,4}*48c
   56-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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