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