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Polytope of Type {4,21,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,21,4}*1344a
if this polytope has a name.
Group : SmallGroup(1344,11703)
Rank : 4
Schlafli Type : {4,21,4}
Number of vertices, edges, etc : 4, 84, 84, 8
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   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) :
   2-fold quotients : {4,21,4}*672
   4-fold quotients : {4,21,2}*336
   7-fold quotients : {4,3,4}*192a
   14-fold quotients : {4,3,4}*96
   28-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)
(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)
(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)
(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)
(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)
(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)
(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)(264,272)
(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)(280,288)
(289,297)(290,298)(291,299)(292,300)(293,301)(294,302)(295,303)(296,304)
(305,313)(306,314)(307,315)(308,316)(309,317)(310,318)(311,319)(312,320)
(321,329)(322,330)(323,331)(324,332)(325,333)(326,334)(327,335)(328,336)
(337,345)(338,346)(339,347)(340,348)(341,349)(342,350)(343,351)(344,352)
(353,361)(354,362)(355,363)(356,364)(357,365)(358,366)(359,367)(360,368)
(369,377)(370,378)(371,379)(372,380)(373,381)(374,382)(375,383)(376,384)
(385,393)(386,394)(387,395)(388,396)(389,397)(390,398)(391,399)(392,400)
(401,409)(402,410)(403,411)(404,412)(405,413)(406,414)(407,415)(408,416)
(417,425)(418,426)(419,427)(420,428)(421,429)(422,430)(423,431)(424,432)
(433,441)(434,442)(435,443)(436,444)(437,445)(438,446)(439,447)(440,448)
(449,457)(450,458)(451,459)(452,460)(453,461)(454,462)(455,463)(456,464)
(465,473)(466,474)(467,475)(468,476)(469,477)(470,478)(471,479)(472,480)
(481,489)(482,490)(483,491)(484,492)(485,493)(486,494)(487,495)(488,496)
(497,505)(498,506)(499,507)(500,508)(501,509)(502,510)(503,511)(504,512)
(513,521)(514,522)(515,523)(516,524)(517,525)(518,526)(519,527)(520,528)
(529,537)(530,538)(531,539)(532,540)(533,541)(534,542)(535,543)(536,544)
(545,553)(546,554)(547,555)(548,556)(549,557)(550,558)(551,559)(552,560)
(561,569)(562,570)(563,571)(564,572)(565,573)(566,574)(567,575)(568,576)
(577,585)(578,586)(579,587)(580,588)(581,589)(582,590)(583,591)(584,592)
(593,601)(594,602)(595,603)(596,604)(597,605)(598,606)(599,607)(600,608)
(609,617)(610,618)(611,619)(612,620)(613,621)(614,622)(615,623)(616,624)
(625,633)(626,634)(627,635)(628,636)(629,637)(630,638)(631,639)(632,640)
(641,649)(642,650)(643,651)(644,652)(645,653)(646,654)(647,655)(648,656)
(657,665)(658,666)(659,667)(660,668)(661,669)(662,670)(663,671)(664,672);;
s1 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)( 18, 98)
( 19,100)( 20, 99)( 21,101)( 22,102)( 23,104)( 24,103)( 25,109)( 26,110)
( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)( 34, 82)
( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)
( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)( 50, 66)
( 51, 68)( 52, 67)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 77)( 58, 78)
( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75)(113,225)(114,226)
(115,228)(116,227)(117,229)(118,230)(119,232)(120,231)(121,237)(122,238)
(123,240)(124,239)(125,233)(126,234)(127,236)(128,235)(129,321)(130,322)
(131,324)(132,323)(133,325)(134,326)(135,328)(136,327)(137,333)(138,334)
(139,336)(140,335)(141,329)(142,330)(143,332)(144,331)(145,305)(146,306)
(147,308)(148,307)(149,309)(150,310)(151,312)(152,311)(153,317)(154,318)
(155,320)(156,319)(157,313)(158,314)(159,316)(160,315)(161,289)(162,290)
(163,292)(164,291)(165,293)(166,294)(167,296)(168,295)(169,301)(170,302)
(171,304)(172,303)(173,297)(174,298)(175,300)(176,299)(177,273)(178,274)
(179,276)(180,275)(181,277)(182,278)(183,280)(184,279)(185,285)(186,286)
(187,288)(188,287)(189,281)(190,282)(191,284)(192,283)(193,257)(194,258)
(195,260)(196,259)(197,261)(198,262)(199,264)(200,263)(201,269)(202,270)
(203,272)(204,271)(205,265)(206,266)(207,268)(208,267)(209,241)(210,242)
(211,244)(212,243)(213,245)(214,246)(215,248)(216,247)(217,253)(218,254)
(219,256)(220,255)(221,249)(222,250)(223,252)(224,251)(339,340)(343,344)
(345,349)(346,350)(347,352)(348,351)(353,433)(354,434)(355,436)(356,435)
(357,437)(358,438)(359,440)(360,439)(361,445)(362,446)(363,448)(364,447)
(365,441)(366,442)(367,444)(368,443)(369,417)(370,418)(371,420)(372,419)
(373,421)(374,422)(375,424)(376,423)(377,429)(378,430)(379,432)(380,431)
(381,425)(382,426)(383,428)(384,427)(385,401)(386,402)(387,404)(388,403)
(389,405)(390,406)(391,408)(392,407)(393,413)(394,414)(395,416)(396,415)
(397,409)(398,410)(399,412)(400,411)(449,561)(450,562)(451,564)(452,563)
(453,565)(454,566)(455,568)(456,567)(457,573)(458,574)(459,576)(460,575)
(461,569)(462,570)(463,572)(464,571)(465,657)(466,658)(467,660)(468,659)
(469,661)(470,662)(471,664)(472,663)(473,669)(474,670)(475,672)(476,671)
(477,665)(478,666)(479,668)(480,667)(481,641)(482,642)(483,644)(484,643)
(485,645)(486,646)(487,648)(488,647)(489,653)(490,654)(491,656)(492,655)
(493,649)(494,650)(495,652)(496,651)(497,625)(498,626)(499,628)(500,627)
(501,629)(502,630)(503,632)(504,631)(505,637)(506,638)(507,640)(508,639)
(509,633)(510,634)(511,636)(512,635)(513,609)(514,610)(515,612)(516,611)
(517,613)(518,614)(519,616)(520,615)(521,621)(522,622)(523,624)(524,623)
(525,617)(526,618)(527,620)(528,619)(529,593)(530,594)(531,596)(532,595)
(533,597)(534,598)(535,600)(536,599)(537,605)(538,606)(539,608)(540,607)
(541,601)(542,602)(543,604)(544,603)(545,577)(546,578)(547,580)(548,579)
(549,581)(550,582)(551,584)(552,583)(553,589)(554,590)(555,592)(556,591)
(557,585)(558,586)(559,588)(560,587);;
s2 := (  1,129)(  2,132)(  3,131)(  4,130)(  5,141)(  6,144)(  7,143)(  8,142)
(  9,137)( 10,140)( 11,139)( 12,138)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,113)( 18,116)( 19,115)( 20,114)( 21,125)( 22,128)( 23,127)( 24,126)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,117)( 30,120)( 31,119)( 32,118)
( 33,209)( 34,212)( 35,211)( 36,210)( 37,221)( 38,224)( 39,223)( 40,222)
( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)( 48,214)
( 49,193)( 50,196)( 51,195)( 52,194)( 53,205)( 54,208)( 55,207)( 56,206)
( 57,201)( 58,204)( 59,203)( 60,202)( 61,197)( 62,200)( 63,199)( 64,198)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,189)( 70,192)( 71,191)( 72,190)
( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,161)( 82,164)( 83,163)( 84,162)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,169)( 90,172)( 91,171)( 92,170)( 93,165)( 94,168)( 95,167)( 96,166)
( 97,145)( 98,148)( 99,147)(100,146)(101,157)(102,160)(103,159)(104,158)
(105,153)(106,156)(107,155)(108,154)(109,149)(110,152)(111,151)(112,150)
(225,241)(226,244)(227,243)(228,242)(229,253)(230,256)(231,255)(232,254)
(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)(240,246)
(257,321)(258,324)(259,323)(260,322)(261,333)(262,336)(263,335)(264,334)
(265,329)(266,332)(267,331)(268,330)(269,325)(270,328)(271,327)(272,326)
(273,305)(274,308)(275,307)(276,306)(277,317)(278,320)(279,319)(280,318)
(281,313)(282,316)(283,315)(284,314)(285,309)(286,312)(287,311)(288,310)
(290,292)(293,301)(294,304)(295,303)(296,302)(298,300)(337,465)(338,468)
(339,467)(340,466)(341,477)(342,480)(343,479)(344,478)(345,473)(346,476)
(347,475)(348,474)(349,469)(350,472)(351,471)(352,470)(353,449)(354,452)
(355,451)(356,450)(357,461)(358,464)(359,463)(360,462)(361,457)(362,460)
(363,459)(364,458)(365,453)(366,456)(367,455)(368,454)(369,545)(370,548)
(371,547)(372,546)(373,557)(374,560)(375,559)(376,558)(377,553)(378,556)
(379,555)(380,554)(381,549)(382,552)(383,551)(384,550)(385,529)(386,532)
(387,531)(388,530)(389,541)(390,544)(391,543)(392,542)(393,537)(394,540)
(395,539)(396,538)(397,533)(398,536)(399,535)(400,534)(401,513)(402,516)
(403,515)(404,514)(405,525)(406,528)(407,527)(408,526)(409,521)(410,524)
(411,523)(412,522)(413,517)(414,520)(415,519)(416,518)(417,497)(418,500)
(419,499)(420,498)(421,509)(422,512)(423,511)(424,510)(425,505)(426,508)
(427,507)(428,506)(429,501)(430,504)(431,503)(432,502)(433,481)(434,484)
(435,483)(436,482)(437,493)(438,496)(439,495)(440,494)(441,489)(442,492)
(443,491)(444,490)(445,485)(446,488)(447,487)(448,486)(561,577)(562,580)
(563,579)(564,578)(565,589)(566,592)(567,591)(568,590)(569,585)(570,588)
(571,587)(572,586)(573,581)(574,584)(575,583)(576,582)(593,657)(594,660)
(595,659)(596,658)(597,669)(598,672)(599,671)(600,670)(601,665)(602,668)
(603,667)(604,666)(605,661)(606,664)(607,663)(608,662)(609,641)(610,644)
(611,643)(612,642)(613,653)(614,656)(615,655)(616,654)(617,649)(618,652)
(619,651)(620,650)(621,645)(622,648)(623,647)(624,646)(626,628)(629,637)
(630,640)(631,639)(632,638)(634,636);;
s3 := (  1,338)(  2,337)(  3,340)(  4,339)(  5,342)(  6,341)(  7,344)(  8,343)
(  9,346)( 10,345)( 11,348)( 12,347)( 13,350)( 14,349)( 15,352)( 16,351)
( 17,354)( 18,353)( 19,356)( 20,355)( 21,358)( 22,357)( 23,360)( 24,359)
( 25,362)( 26,361)( 27,364)( 28,363)( 29,366)( 30,365)( 31,368)( 32,367)
( 33,370)( 34,369)( 35,372)( 36,371)( 37,374)( 38,373)( 39,376)( 40,375)
( 41,378)( 42,377)( 43,380)( 44,379)( 45,382)( 46,381)( 47,384)( 48,383)
( 49,386)( 50,385)( 51,388)( 52,387)( 53,390)( 54,389)( 55,392)( 56,391)
( 57,394)( 58,393)( 59,396)( 60,395)( 61,398)( 62,397)( 63,400)( 64,399)
( 65,402)( 66,401)( 67,404)( 68,403)( 69,406)( 70,405)( 71,408)( 72,407)
( 73,410)( 74,409)( 75,412)( 76,411)( 77,414)( 78,413)( 79,416)( 80,415)
( 81,418)( 82,417)( 83,420)( 84,419)( 85,422)( 86,421)( 87,424)( 88,423)
( 89,426)( 90,425)( 91,428)( 92,427)( 93,430)( 94,429)( 95,432)( 96,431)
( 97,434)( 98,433)( 99,436)(100,435)(101,438)(102,437)(103,440)(104,439)
(105,442)(106,441)(107,444)(108,443)(109,446)(110,445)(111,448)(112,447)
(113,450)(114,449)(115,452)(116,451)(117,454)(118,453)(119,456)(120,455)
(121,458)(122,457)(123,460)(124,459)(125,462)(126,461)(127,464)(128,463)
(129,466)(130,465)(131,468)(132,467)(133,470)(134,469)(135,472)(136,471)
(137,474)(138,473)(139,476)(140,475)(141,478)(142,477)(143,480)(144,479)
(145,482)(146,481)(147,484)(148,483)(149,486)(150,485)(151,488)(152,487)
(153,490)(154,489)(155,492)(156,491)(157,494)(158,493)(159,496)(160,495)
(161,498)(162,497)(163,500)(164,499)(165,502)(166,501)(167,504)(168,503)
(169,506)(170,505)(171,508)(172,507)(173,510)(174,509)(175,512)(176,511)
(177,514)(178,513)(179,516)(180,515)(181,518)(182,517)(183,520)(184,519)
(185,522)(186,521)(187,524)(188,523)(189,526)(190,525)(191,528)(192,527)
(193,530)(194,529)(195,532)(196,531)(197,534)(198,533)(199,536)(200,535)
(201,538)(202,537)(203,540)(204,539)(205,542)(206,541)(207,544)(208,543)
(209,546)(210,545)(211,548)(212,547)(213,550)(214,549)(215,552)(216,551)
(217,554)(218,553)(219,556)(220,555)(221,558)(222,557)(223,560)(224,559)
(225,562)(226,561)(227,564)(228,563)(229,566)(230,565)(231,568)(232,567)
(233,570)(234,569)(235,572)(236,571)(237,574)(238,573)(239,576)(240,575)
(241,578)(242,577)(243,580)(244,579)(245,582)(246,581)(247,584)(248,583)
(249,586)(250,585)(251,588)(252,587)(253,590)(254,589)(255,592)(256,591)
(257,594)(258,593)(259,596)(260,595)(261,598)(262,597)(263,600)(264,599)
(265,602)(266,601)(267,604)(268,603)(269,606)(270,605)(271,608)(272,607)
(273,610)(274,609)(275,612)(276,611)(277,614)(278,613)(279,616)(280,615)
(281,618)(282,617)(283,620)(284,619)(285,622)(286,621)(287,624)(288,623)
(289,626)(290,625)(291,628)(292,627)(293,630)(294,629)(295,632)(296,631)
(297,634)(298,633)(299,636)(300,635)(301,638)(302,637)(303,640)(304,639)
(305,642)(306,641)(307,644)(308,643)(309,646)(310,645)(311,648)(312,647)
(313,650)(314,649)(315,652)(316,651)(317,654)(318,653)(319,656)(320,655)
(321,658)(322,657)(323,660)(324,659)(325,662)(326,661)(327,664)(328,663)
(329,666)(330,665)(331,668)(332,667)(333,670)(334,669)(335,672)(336,671);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)
(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)
(184,192)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)
(200,208)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)
(216,224)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)
(232,240)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)
(248,256)(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)
(264,272)(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)
(280,288)(289,297)(290,298)(291,299)(292,300)(293,301)(294,302)(295,303)
(296,304)(305,313)(306,314)(307,315)(308,316)(309,317)(310,318)(311,319)
(312,320)(321,329)(322,330)(323,331)(324,332)(325,333)(326,334)(327,335)
(328,336)(337,345)(338,346)(339,347)(340,348)(341,349)(342,350)(343,351)
(344,352)(353,361)(354,362)(355,363)(356,364)(357,365)(358,366)(359,367)
(360,368)(369,377)(370,378)(371,379)(372,380)(373,381)(374,382)(375,383)
(376,384)(385,393)(386,394)(387,395)(388,396)(389,397)(390,398)(391,399)
(392,400)(401,409)(402,410)(403,411)(404,412)(405,413)(406,414)(407,415)
(408,416)(417,425)(418,426)(419,427)(420,428)(421,429)(422,430)(423,431)
(424,432)(433,441)(434,442)(435,443)(436,444)(437,445)(438,446)(439,447)
(440,448)(449,457)(450,458)(451,459)(452,460)(453,461)(454,462)(455,463)
(456,464)(465,473)(466,474)(467,475)(468,476)(469,477)(470,478)(471,479)
(472,480)(481,489)(482,490)(483,491)(484,492)(485,493)(486,494)(487,495)
(488,496)(497,505)(498,506)(499,507)(500,508)(501,509)(502,510)(503,511)
(504,512)(513,521)(514,522)(515,523)(516,524)(517,525)(518,526)(519,527)
(520,528)(529,537)(530,538)(531,539)(532,540)(533,541)(534,542)(535,543)
(536,544)(545,553)(546,554)(547,555)(548,556)(549,557)(550,558)(551,559)
(552,560)(561,569)(562,570)(563,571)(564,572)(565,573)(566,574)(567,575)
(568,576)(577,585)(578,586)(579,587)(580,588)(581,589)(582,590)(583,591)
(584,592)(593,601)(594,602)(595,603)(596,604)(597,605)(598,606)(599,607)
(600,608)(609,617)(610,618)(611,619)(612,620)(613,621)(614,622)(615,623)
(616,624)(625,633)(626,634)(627,635)(628,636)(629,637)(630,638)(631,639)
(632,640)(641,649)(642,650)(643,651)(644,652)(645,653)(646,654)(647,655)
(648,656)(657,665)(658,666)(659,667)(660,668)(661,669)(662,670)(663,671)
(664,672);
s1 := Sym(672)!(  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 97)
( 18, 98)( 19,100)( 20, 99)( 21,101)( 22,102)( 23,104)( 24,103)( 25,109)
( 26,110)( 27,112)( 28,111)( 29,105)( 30,106)( 31,108)( 32,107)( 33, 81)
( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)
( 42, 94)( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 65)
( 50, 66)( 51, 68)( 52, 67)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 77)
( 58, 78)( 59, 80)( 60, 79)( 61, 73)( 62, 74)( 63, 76)( 64, 75)(113,225)
(114,226)(115,228)(116,227)(117,229)(118,230)(119,232)(120,231)(121,237)
(122,238)(123,240)(124,239)(125,233)(126,234)(127,236)(128,235)(129,321)
(130,322)(131,324)(132,323)(133,325)(134,326)(135,328)(136,327)(137,333)
(138,334)(139,336)(140,335)(141,329)(142,330)(143,332)(144,331)(145,305)
(146,306)(147,308)(148,307)(149,309)(150,310)(151,312)(152,311)(153,317)
(154,318)(155,320)(156,319)(157,313)(158,314)(159,316)(160,315)(161,289)
(162,290)(163,292)(164,291)(165,293)(166,294)(167,296)(168,295)(169,301)
(170,302)(171,304)(172,303)(173,297)(174,298)(175,300)(176,299)(177,273)
(178,274)(179,276)(180,275)(181,277)(182,278)(183,280)(184,279)(185,285)
(186,286)(187,288)(188,287)(189,281)(190,282)(191,284)(192,283)(193,257)
(194,258)(195,260)(196,259)(197,261)(198,262)(199,264)(200,263)(201,269)
(202,270)(203,272)(204,271)(205,265)(206,266)(207,268)(208,267)(209,241)
(210,242)(211,244)(212,243)(213,245)(214,246)(215,248)(216,247)(217,253)
(218,254)(219,256)(220,255)(221,249)(222,250)(223,252)(224,251)(339,340)
(343,344)(345,349)(346,350)(347,352)(348,351)(353,433)(354,434)(355,436)
(356,435)(357,437)(358,438)(359,440)(360,439)(361,445)(362,446)(363,448)
(364,447)(365,441)(366,442)(367,444)(368,443)(369,417)(370,418)(371,420)
(372,419)(373,421)(374,422)(375,424)(376,423)(377,429)(378,430)(379,432)
(380,431)(381,425)(382,426)(383,428)(384,427)(385,401)(386,402)(387,404)
(388,403)(389,405)(390,406)(391,408)(392,407)(393,413)(394,414)(395,416)
(396,415)(397,409)(398,410)(399,412)(400,411)(449,561)(450,562)(451,564)
(452,563)(453,565)(454,566)(455,568)(456,567)(457,573)(458,574)(459,576)
(460,575)(461,569)(462,570)(463,572)(464,571)(465,657)(466,658)(467,660)
(468,659)(469,661)(470,662)(471,664)(472,663)(473,669)(474,670)(475,672)
(476,671)(477,665)(478,666)(479,668)(480,667)(481,641)(482,642)(483,644)
(484,643)(485,645)(486,646)(487,648)(488,647)(489,653)(490,654)(491,656)
(492,655)(493,649)(494,650)(495,652)(496,651)(497,625)(498,626)(499,628)
(500,627)(501,629)(502,630)(503,632)(504,631)(505,637)(506,638)(507,640)
(508,639)(509,633)(510,634)(511,636)(512,635)(513,609)(514,610)(515,612)
(516,611)(517,613)(518,614)(519,616)(520,615)(521,621)(522,622)(523,624)
(524,623)(525,617)(526,618)(527,620)(528,619)(529,593)(530,594)(531,596)
(532,595)(533,597)(534,598)(535,600)(536,599)(537,605)(538,606)(539,608)
(540,607)(541,601)(542,602)(543,604)(544,603)(545,577)(546,578)(547,580)
(548,579)(549,581)(550,582)(551,584)(552,583)(553,589)(554,590)(555,592)
(556,591)(557,585)(558,586)(559,588)(560,587);
s2 := Sym(672)!(  1,129)(  2,132)(  3,131)(  4,130)(  5,141)(  6,144)(  7,143)
(  8,142)(  9,137)( 10,140)( 11,139)( 12,138)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,113)( 18,116)( 19,115)( 20,114)( 21,125)( 22,128)( 23,127)
( 24,126)( 25,121)( 26,124)( 27,123)( 28,122)( 29,117)( 30,120)( 31,119)
( 32,118)( 33,209)( 34,212)( 35,211)( 36,210)( 37,221)( 38,224)( 39,223)
( 40,222)( 41,217)( 42,220)( 43,219)( 44,218)( 45,213)( 46,216)( 47,215)
( 48,214)( 49,193)( 50,196)( 51,195)( 52,194)( 53,205)( 54,208)( 55,207)
( 56,206)( 57,201)( 58,204)( 59,203)( 60,202)( 61,197)( 62,200)( 63,199)
( 64,198)( 65,177)( 66,180)( 67,179)( 68,178)( 69,189)( 70,192)( 71,191)
( 72,190)( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)
( 80,182)( 81,161)( 82,164)( 83,163)( 84,162)( 85,173)( 86,176)( 87,175)
( 88,174)( 89,169)( 90,172)( 91,171)( 92,170)( 93,165)( 94,168)( 95,167)
( 96,166)( 97,145)( 98,148)( 99,147)(100,146)(101,157)(102,160)(103,159)
(104,158)(105,153)(106,156)(107,155)(108,154)(109,149)(110,152)(111,151)
(112,150)(225,241)(226,244)(227,243)(228,242)(229,253)(230,256)(231,255)
(232,254)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)(239,247)
(240,246)(257,321)(258,324)(259,323)(260,322)(261,333)(262,336)(263,335)
(264,334)(265,329)(266,332)(267,331)(268,330)(269,325)(270,328)(271,327)
(272,326)(273,305)(274,308)(275,307)(276,306)(277,317)(278,320)(279,319)
(280,318)(281,313)(282,316)(283,315)(284,314)(285,309)(286,312)(287,311)
(288,310)(290,292)(293,301)(294,304)(295,303)(296,302)(298,300)(337,465)
(338,468)(339,467)(340,466)(341,477)(342,480)(343,479)(344,478)(345,473)
(346,476)(347,475)(348,474)(349,469)(350,472)(351,471)(352,470)(353,449)
(354,452)(355,451)(356,450)(357,461)(358,464)(359,463)(360,462)(361,457)
(362,460)(363,459)(364,458)(365,453)(366,456)(367,455)(368,454)(369,545)
(370,548)(371,547)(372,546)(373,557)(374,560)(375,559)(376,558)(377,553)
(378,556)(379,555)(380,554)(381,549)(382,552)(383,551)(384,550)(385,529)
(386,532)(387,531)(388,530)(389,541)(390,544)(391,543)(392,542)(393,537)
(394,540)(395,539)(396,538)(397,533)(398,536)(399,535)(400,534)(401,513)
(402,516)(403,515)(404,514)(405,525)(406,528)(407,527)(408,526)(409,521)
(410,524)(411,523)(412,522)(413,517)(414,520)(415,519)(416,518)(417,497)
(418,500)(419,499)(420,498)(421,509)(422,512)(423,511)(424,510)(425,505)
(426,508)(427,507)(428,506)(429,501)(430,504)(431,503)(432,502)(433,481)
(434,484)(435,483)(436,482)(437,493)(438,496)(439,495)(440,494)(441,489)
(442,492)(443,491)(444,490)(445,485)(446,488)(447,487)(448,486)(561,577)
(562,580)(563,579)(564,578)(565,589)(566,592)(567,591)(568,590)(569,585)
(570,588)(571,587)(572,586)(573,581)(574,584)(575,583)(576,582)(593,657)
(594,660)(595,659)(596,658)(597,669)(598,672)(599,671)(600,670)(601,665)
(602,668)(603,667)(604,666)(605,661)(606,664)(607,663)(608,662)(609,641)
(610,644)(611,643)(612,642)(613,653)(614,656)(615,655)(616,654)(617,649)
(618,652)(619,651)(620,650)(621,645)(622,648)(623,647)(624,646)(626,628)
(629,637)(630,640)(631,639)(632,638)(634,636);
s3 := Sym(672)!(  1,338)(  2,337)(  3,340)(  4,339)(  5,342)(  6,341)(  7,344)
(  8,343)(  9,346)( 10,345)( 11,348)( 12,347)( 13,350)( 14,349)( 15,352)
( 16,351)( 17,354)( 18,353)( 19,356)( 20,355)( 21,358)( 22,357)( 23,360)
( 24,359)( 25,362)( 26,361)( 27,364)( 28,363)( 29,366)( 30,365)( 31,368)
( 32,367)( 33,370)( 34,369)( 35,372)( 36,371)( 37,374)( 38,373)( 39,376)
( 40,375)( 41,378)( 42,377)( 43,380)( 44,379)( 45,382)( 46,381)( 47,384)
( 48,383)( 49,386)( 50,385)( 51,388)( 52,387)( 53,390)( 54,389)( 55,392)
( 56,391)( 57,394)( 58,393)( 59,396)( 60,395)( 61,398)( 62,397)( 63,400)
( 64,399)( 65,402)( 66,401)( 67,404)( 68,403)( 69,406)( 70,405)( 71,408)
( 72,407)( 73,410)( 74,409)( 75,412)( 76,411)( 77,414)( 78,413)( 79,416)
( 80,415)( 81,418)( 82,417)( 83,420)( 84,419)( 85,422)( 86,421)( 87,424)
( 88,423)( 89,426)( 90,425)( 91,428)( 92,427)( 93,430)( 94,429)( 95,432)
( 96,431)( 97,434)( 98,433)( 99,436)(100,435)(101,438)(102,437)(103,440)
(104,439)(105,442)(106,441)(107,444)(108,443)(109,446)(110,445)(111,448)
(112,447)(113,450)(114,449)(115,452)(116,451)(117,454)(118,453)(119,456)
(120,455)(121,458)(122,457)(123,460)(124,459)(125,462)(126,461)(127,464)
(128,463)(129,466)(130,465)(131,468)(132,467)(133,470)(134,469)(135,472)
(136,471)(137,474)(138,473)(139,476)(140,475)(141,478)(142,477)(143,480)
(144,479)(145,482)(146,481)(147,484)(148,483)(149,486)(150,485)(151,488)
(152,487)(153,490)(154,489)(155,492)(156,491)(157,494)(158,493)(159,496)
(160,495)(161,498)(162,497)(163,500)(164,499)(165,502)(166,501)(167,504)
(168,503)(169,506)(170,505)(171,508)(172,507)(173,510)(174,509)(175,512)
(176,511)(177,514)(178,513)(179,516)(180,515)(181,518)(182,517)(183,520)
(184,519)(185,522)(186,521)(187,524)(188,523)(189,526)(190,525)(191,528)
(192,527)(193,530)(194,529)(195,532)(196,531)(197,534)(198,533)(199,536)
(200,535)(201,538)(202,537)(203,540)(204,539)(205,542)(206,541)(207,544)
(208,543)(209,546)(210,545)(211,548)(212,547)(213,550)(214,549)(215,552)
(216,551)(217,554)(218,553)(219,556)(220,555)(221,558)(222,557)(223,560)
(224,559)(225,562)(226,561)(227,564)(228,563)(229,566)(230,565)(231,568)
(232,567)(233,570)(234,569)(235,572)(236,571)(237,574)(238,573)(239,576)
(240,575)(241,578)(242,577)(243,580)(244,579)(245,582)(246,581)(247,584)
(248,583)(249,586)(250,585)(251,588)(252,587)(253,590)(254,589)(255,592)
(256,591)(257,594)(258,593)(259,596)(260,595)(261,598)(262,597)(263,600)
(264,599)(265,602)(266,601)(267,604)(268,603)(269,606)(270,605)(271,608)
(272,607)(273,610)(274,609)(275,612)(276,611)(277,614)(278,613)(279,616)
(280,615)(281,618)(282,617)(283,620)(284,619)(285,622)(286,621)(287,624)
(288,623)(289,626)(290,625)(291,628)(292,627)(293,630)(294,629)(295,632)
(296,631)(297,634)(298,633)(299,636)(300,635)(301,638)(302,637)(303,640)
(304,639)(305,642)(306,641)(307,644)(308,643)(309,646)(310,645)(311,648)
(312,647)(313,650)(314,649)(315,652)(316,651)(317,654)(318,653)(319,656)
(320,655)(321,658)(322,657)(323,660)(324,659)(325,662)(326,661)(327,664)
(328,663)(329,666)(330,665)(331,668)(332,667)(333,670)(334,669)(335,672)
(336,671);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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