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Polytope of Type {6,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,12}*1728a
Also Known As : {{6,4}4,{4,12|2}}. if this polytope has another name.
Group : SmallGroup(1728,46611)
Rank : 4
Schlafli Type : {6,4,12}
Number of vertices, edges, etc : 18, 36, 72, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   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 : {6,4,12}*864, {6,4,6}*864b
   3-fold quotients : {6,4,4}*576
   4-fold quotients : {6,4,6}*432a
   6-fold quotients : {6,4,4}*288, {6,4,2}*288
   9-fold quotients : {2,4,12}*192a
   12-fold quotients : {6,4,2}*144
   18-fold quotients : {2,2,12}*96, {2,4,6}*96a
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,2,6}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,163)(  2,165)(  3,164)(  4,169)(  5,171)(  6,170)(  7,166)(  8,168)
(  9,167)( 10,172)( 11,174)( 12,173)( 13,178)( 14,180)( 15,179)( 16,175)
( 17,177)( 18,176)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)( 24,188)
( 25,184)( 26,186)( 27,185)( 28,190)( 29,192)( 30,191)( 31,196)( 32,198)
( 33,197)( 34,193)( 35,195)( 36,194)( 37,199)( 38,201)( 39,200)( 40,205)
( 41,207)( 42,206)( 43,202)( 44,204)( 45,203)( 46,208)( 47,210)( 48,209)
( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 55,109)( 56,111)
( 57,110)( 58,115)( 59,117)( 60,116)( 61,112)( 62,114)( 63,113)( 64,118)
( 65,120)( 66,119)( 67,124)( 68,126)( 69,125)( 70,121)( 71,123)( 72,122)
( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)( 79,130)( 80,132)
( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)( 86,144)( 87,143)( 88,139)
( 89,141)( 90,140)( 91,145)( 92,147)( 93,146)( 94,151)( 95,153)( 96,152)
( 97,148)( 98,150)( 99,149)(100,154)(101,156)(102,155)(103,160)(104,162)
(105,161)(106,157)(107,159)(108,158)(217,379)(218,381)(219,380)(220,385)
(221,387)(222,386)(223,382)(224,384)(225,383)(226,388)(227,390)(228,389)
(229,394)(230,396)(231,395)(232,391)(233,393)(234,392)(235,397)(236,399)
(237,398)(238,403)(239,405)(240,404)(241,400)(242,402)(243,401)(244,406)
(245,408)(246,407)(247,412)(248,414)(249,413)(250,409)(251,411)(252,410)
(253,415)(254,417)(255,416)(256,421)(257,423)(258,422)(259,418)(260,420)
(261,419)(262,424)(263,426)(264,425)(265,430)(266,432)(267,431)(268,427)
(269,429)(270,428)(271,325)(272,327)(273,326)(274,331)(275,333)(276,332)
(277,328)(278,330)(279,329)(280,334)(281,336)(282,335)(283,340)(284,342)
(285,341)(286,337)(287,339)(288,338)(289,343)(290,345)(291,344)(292,349)
(293,351)(294,350)(295,346)(296,348)(297,347)(298,352)(299,354)(300,353)
(301,358)(302,360)(303,359)(304,355)(305,357)(306,356)(307,361)(308,363)
(309,362)(310,367)(311,369)(312,368)(313,364)(314,366)(315,365)(316,370)
(317,372)(318,371)(319,376)(320,378)(321,377)(322,373)(323,375)(324,374)
(433,595)(434,597)(435,596)(436,601)(437,603)(438,602)(439,598)(440,600)
(441,599)(442,604)(443,606)(444,605)(445,610)(446,612)(447,611)(448,607)
(449,609)(450,608)(451,613)(452,615)(453,614)(454,619)(455,621)(456,620)
(457,616)(458,618)(459,617)(460,622)(461,624)(462,623)(463,628)(464,630)
(465,629)(466,625)(467,627)(468,626)(469,631)(470,633)(471,632)(472,637)
(473,639)(474,638)(475,634)(476,636)(477,635)(478,640)(479,642)(480,641)
(481,646)(482,648)(483,647)(484,643)(485,645)(486,644)(487,541)(488,543)
(489,542)(490,547)(491,549)(492,548)(493,544)(494,546)(495,545)(496,550)
(497,552)(498,551)(499,556)(500,558)(501,557)(502,553)(503,555)(504,554)
(505,559)(506,561)(507,560)(508,565)(509,567)(510,566)(511,562)(512,564)
(513,563)(514,568)(515,570)(516,569)(517,574)(518,576)(519,575)(520,571)
(521,573)(522,572)(523,577)(524,579)(525,578)(526,583)(527,585)(528,584)
(529,580)(530,582)(531,581)(532,586)(533,588)(534,587)(535,592)(536,594)
(537,593)(538,589)(539,591)(540,590)(649,811)(650,813)(651,812)(652,817)
(653,819)(654,818)(655,814)(656,816)(657,815)(658,820)(659,822)(660,821)
(661,826)(662,828)(663,827)(664,823)(665,825)(666,824)(667,829)(668,831)
(669,830)(670,835)(671,837)(672,836)(673,832)(674,834)(675,833)(676,838)
(677,840)(678,839)(679,844)(680,846)(681,845)(682,841)(683,843)(684,842)
(685,847)(686,849)(687,848)(688,853)(689,855)(690,854)(691,850)(692,852)
(693,851)(694,856)(695,858)(696,857)(697,862)(698,864)(699,863)(700,859)
(701,861)(702,860)(703,757)(704,759)(705,758)(706,763)(707,765)(708,764)
(709,760)(710,762)(711,761)(712,766)(713,768)(714,767)(715,772)(716,774)
(717,773)(718,769)(719,771)(720,770)(721,775)(722,777)(723,776)(724,781)
(725,783)(726,782)(727,778)(728,780)(729,779)(730,784)(731,786)(732,785)
(733,790)(734,792)(735,791)(736,787)(737,789)(738,788)(739,793)(740,795)
(741,794)(742,799)(743,801)(744,800)(745,796)(746,798)(747,797)(748,802)
(749,804)(750,803)(751,808)(752,810)(753,809)(754,805)(755,807)(756,806);;
s1 := (  1,  4)(  3,  9)(  5,  8)( 10, 13)( 12, 18)( 14, 17)( 19, 22)( 21, 27)
( 23, 26)( 28, 31)( 30, 36)( 32, 35)( 37, 40)( 39, 45)( 41, 44)( 46, 49)
( 48, 54)( 50, 53)( 55, 58)( 57, 63)( 59, 62)( 64, 67)( 66, 72)( 68, 71)
( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91, 94)( 93, 99)
( 95, 98)(100,103)(102,108)(104,107)(109,112)(111,117)(113,116)(118,121)
(120,126)(122,125)(127,130)(129,135)(131,134)(136,139)(138,144)(140,143)
(145,148)(147,153)(149,152)(154,157)(156,162)(158,161)(163,166)(165,171)
(167,170)(172,175)(174,180)(176,179)(181,184)(183,189)(185,188)(190,193)
(192,198)(194,197)(199,202)(201,207)(203,206)(208,211)(210,216)(212,215)
(217,220)(219,225)(221,224)(226,229)(228,234)(230,233)(235,238)(237,243)
(239,242)(244,247)(246,252)(248,251)(253,256)(255,261)(257,260)(262,265)
(264,270)(266,269)(271,274)(273,279)(275,278)(280,283)(282,288)(284,287)
(289,292)(291,297)(293,296)(298,301)(300,306)(302,305)(307,310)(309,315)
(311,314)(316,319)(318,324)(320,323)(325,328)(327,333)(329,332)(334,337)
(336,342)(338,341)(343,346)(345,351)(347,350)(352,355)(354,360)(356,359)
(361,364)(363,369)(365,368)(370,373)(372,378)(374,377)(379,382)(381,387)
(383,386)(388,391)(390,396)(392,395)(397,400)(399,405)(401,404)(406,409)
(408,414)(410,413)(415,418)(417,423)(419,422)(424,427)(426,432)(428,431)
(433,490)(434,488)(435,495)(436,487)(437,494)(438,492)(439,493)(440,491)
(441,489)(442,499)(443,497)(444,504)(445,496)(446,503)(447,501)(448,502)
(449,500)(450,498)(451,508)(452,506)(453,513)(454,505)(455,512)(456,510)
(457,511)(458,509)(459,507)(460,517)(461,515)(462,522)(463,514)(464,521)
(465,519)(466,520)(467,518)(468,516)(469,526)(470,524)(471,531)(472,523)
(473,530)(474,528)(475,529)(476,527)(477,525)(478,535)(479,533)(480,540)
(481,532)(482,539)(483,537)(484,538)(485,536)(486,534)(541,598)(542,596)
(543,603)(544,595)(545,602)(546,600)(547,601)(548,599)(549,597)(550,607)
(551,605)(552,612)(553,604)(554,611)(555,609)(556,610)(557,608)(558,606)
(559,616)(560,614)(561,621)(562,613)(563,620)(564,618)(565,619)(566,617)
(567,615)(568,625)(569,623)(570,630)(571,622)(572,629)(573,627)(574,628)
(575,626)(576,624)(577,634)(578,632)(579,639)(580,631)(581,638)(582,636)
(583,637)(584,635)(585,633)(586,643)(587,641)(588,648)(589,640)(590,647)
(591,645)(592,646)(593,644)(594,642)(649,706)(650,704)(651,711)(652,703)
(653,710)(654,708)(655,709)(656,707)(657,705)(658,715)(659,713)(660,720)
(661,712)(662,719)(663,717)(664,718)(665,716)(666,714)(667,724)(668,722)
(669,729)(670,721)(671,728)(672,726)(673,727)(674,725)(675,723)(676,733)
(677,731)(678,738)(679,730)(680,737)(681,735)(682,736)(683,734)(684,732)
(685,742)(686,740)(687,747)(688,739)(689,746)(690,744)(691,745)(692,743)
(693,741)(694,751)(695,749)(696,756)(697,748)(698,755)(699,753)(700,754)
(701,752)(702,750)(757,814)(758,812)(759,819)(760,811)(761,818)(762,816)
(763,817)(764,815)(765,813)(766,823)(767,821)(768,828)(769,820)(770,827)
(771,825)(772,826)(773,824)(774,822)(775,832)(776,830)(777,837)(778,829)
(779,836)(780,834)(781,835)(782,833)(783,831)(784,841)(785,839)(786,846)
(787,838)(788,845)(789,843)(790,844)(791,842)(792,840)(793,850)(794,848)
(795,855)(796,847)(797,854)(798,852)(799,853)(800,851)(801,849)(802,859)
(803,857)(804,864)(805,856)(806,863)(807,861)(808,862)(809,860)(810,858);;
s2 := (  1,649)(  2,650)(  3,651)(  4,657)(  5,655)(  6,656)(  7,653)(  8,654)
(  9,652)( 10,667)( 11,668)( 12,669)( 13,675)( 14,673)( 15,674)( 16,671)
( 17,672)( 18,670)( 19,658)( 20,659)( 21,660)( 22,666)( 23,664)( 24,665)
( 25,662)( 26,663)( 27,661)( 28,676)( 29,677)( 30,678)( 31,684)( 32,682)
( 33,683)( 34,680)( 35,681)( 36,679)( 37,694)( 38,695)( 39,696)( 40,702)
( 41,700)( 42,701)( 43,698)( 44,699)( 45,697)( 46,685)( 47,686)( 48,687)
( 49,693)( 50,691)( 51,692)( 52,689)( 53,690)( 54,688)( 55,703)( 56,704)
( 57,705)( 58,711)( 59,709)( 60,710)( 61,707)( 62,708)( 63,706)( 64,721)
( 65,722)( 66,723)( 67,729)( 68,727)( 69,728)( 70,725)( 71,726)( 72,724)
( 73,712)( 74,713)( 75,714)( 76,720)( 77,718)( 78,719)( 79,716)( 80,717)
( 81,715)( 82,730)( 83,731)( 84,732)( 85,738)( 86,736)( 87,737)( 88,734)
( 89,735)( 90,733)( 91,748)( 92,749)( 93,750)( 94,756)( 95,754)( 96,755)
( 97,752)( 98,753)( 99,751)(100,739)(101,740)(102,741)(103,747)(104,745)
(105,746)(106,743)(107,744)(108,742)(109,757)(110,758)(111,759)(112,765)
(113,763)(114,764)(115,761)(116,762)(117,760)(118,775)(119,776)(120,777)
(121,783)(122,781)(123,782)(124,779)(125,780)(126,778)(127,766)(128,767)
(129,768)(130,774)(131,772)(132,773)(133,770)(134,771)(135,769)(136,784)
(137,785)(138,786)(139,792)(140,790)(141,791)(142,788)(143,789)(144,787)
(145,802)(146,803)(147,804)(148,810)(149,808)(150,809)(151,806)(152,807)
(153,805)(154,793)(155,794)(156,795)(157,801)(158,799)(159,800)(160,797)
(161,798)(162,796)(163,811)(164,812)(165,813)(166,819)(167,817)(168,818)
(169,815)(170,816)(171,814)(172,829)(173,830)(174,831)(175,837)(176,835)
(177,836)(178,833)(179,834)(180,832)(181,820)(182,821)(183,822)(184,828)
(185,826)(186,827)(187,824)(188,825)(189,823)(190,838)(191,839)(192,840)
(193,846)(194,844)(195,845)(196,842)(197,843)(198,841)(199,856)(200,857)
(201,858)(202,864)(203,862)(204,863)(205,860)(206,861)(207,859)(208,847)
(209,848)(210,849)(211,855)(212,853)(213,854)(214,851)(215,852)(216,850)
(217,460)(218,461)(219,462)(220,468)(221,466)(222,467)(223,464)(224,465)
(225,463)(226,478)(227,479)(228,480)(229,486)(230,484)(231,485)(232,482)
(233,483)(234,481)(235,469)(236,470)(237,471)(238,477)(239,475)(240,476)
(241,473)(242,474)(243,472)(244,433)(245,434)(246,435)(247,441)(248,439)
(249,440)(250,437)(251,438)(252,436)(253,451)(254,452)(255,453)(256,459)
(257,457)(258,458)(259,455)(260,456)(261,454)(262,442)(263,443)(264,444)
(265,450)(266,448)(267,449)(268,446)(269,447)(270,445)(271,514)(272,515)
(273,516)(274,522)(275,520)(276,521)(277,518)(278,519)(279,517)(280,532)
(281,533)(282,534)(283,540)(284,538)(285,539)(286,536)(287,537)(288,535)
(289,523)(290,524)(291,525)(292,531)(293,529)(294,530)(295,527)(296,528)
(297,526)(298,487)(299,488)(300,489)(301,495)(302,493)(303,494)(304,491)
(305,492)(306,490)(307,505)(308,506)(309,507)(310,513)(311,511)(312,512)
(313,509)(314,510)(315,508)(316,496)(317,497)(318,498)(319,504)(320,502)
(321,503)(322,500)(323,501)(324,499)(325,568)(326,569)(327,570)(328,576)
(329,574)(330,575)(331,572)(332,573)(333,571)(334,586)(335,587)(336,588)
(337,594)(338,592)(339,593)(340,590)(341,591)(342,589)(343,577)(344,578)
(345,579)(346,585)(347,583)(348,584)(349,581)(350,582)(351,580)(352,541)
(353,542)(354,543)(355,549)(356,547)(357,548)(358,545)(359,546)(360,544)
(361,559)(362,560)(363,561)(364,567)(365,565)(366,566)(367,563)(368,564)
(369,562)(370,550)(371,551)(372,552)(373,558)(374,556)(375,557)(376,554)
(377,555)(378,553)(379,622)(380,623)(381,624)(382,630)(383,628)(384,629)
(385,626)(386,627)(387,625)(388,640)(389,641)(390,642)(391,648)(392,646)
(393,647)(394,644)(395,645)(396,643)(397,631)(398,632)(399,633)(400,639)
(401,637)(402,638)(403,635)(404,636)(405,634)(406,595)(407,596)(408,597)
(409,603)(410,601)(411,602)(412,599)(413,600)(414,598)(415,613)(416,614)
(417,615)(418,621)(419,619)(420,620)(421,617)(422,618)(423,616)(424,604)
(425,605)(426,606)(427,612)(428,610)(429,611)(430,608)(431,609)(432,607);;
s3 := (  1,226)(  2,227)(  3,228)(  4,229)(  5,230)(  6,231)(  7,232)(  8,233)
(  9,234)( 10,217)( 11,218)( 12,219)( 13,220)( 14,221)( 15,222)( 16,223)
( 17,224)( 18,225)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,253)( 29,254)( 30,255)( 31,256)( 32,257)
( 33,258)( 34,259)( 35,260)( 36,261)( 37,244)( 38,245)( 39,246)( 40,247)
( 41,248)( 42,249)( 43,250)( 44,251)( 45,252)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,280)( 56,281)
( 57,282)( 58,283)( 59,284)( 60,285)( 61,286)( 62,287)( 63,288)( 64,271)
( 65,272)( 66,273)( 67,274)( 68,275)( 69,276)( 70,277)( 71,278)( 72,279)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)( 88,313)
( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)( 96,303)
( 97,304)( 98,305)( 99,306)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,334)(110,335)(111,336)(112,337)
(113,338)(114,339)(115,340)(116,341)(117,342)(118,325)(119,326)(120,327)
(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,343)(128,344)
(129,345)(130,346)(131,347)(132,348)(133,349)(134,350)(135,351)(136,361)
(137,362)(138,363)(139,364)(140,365)(141,366)(142,367)(143,368)(144,369)
(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)(152,359)
(153,360)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)(160,376)
(161,377)(162,378)(163,388)(164,389)(165,390)(166,391)(167,392)(168,393)
(169,394)(170,395)(171,396)(172,379)(173,380)(174,381)(175,382)(176,383)
(177,384)(178,385)(179,386)(180,387)(181,397)(182,398)(183,399)(184,400)
(185,401)(186,402)(187,403)(188,404)(189,405)(190,415)(191,416)(192,417)
(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)(200,407)
(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)(208,424)
(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)(216,432)
(433,658)(434,659)(435,660)(436,661)(437,662)(438,663)(439,664)(440,665)
(441,666)(442,649)(443,650)(444,651)(445,652)(446,653)(447,654)(448,655)
(449,656)(450,657)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,685)(461,686)(462,687)(463,688)(464,689)
(465,690)(466,691)(467,692)(468,693)(469,676)(470,677)(471,678)(472,679)
(473,680)(474,681)(475,682)(476,683)(477,684)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,712)(488,713)
(489,714)(490,715)(491,716)(492,717)(493,718)(494,719)(495,720)(496,703)
(497,704)(498,705)(499,706)(500,707)(501,708)(502,709)(503,710)(504,711)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,739)(515,740)(516,741)(517,742)(518,743)(519,744)(520,745)
(521,746)(522,747)(523,730)(524,731)(525,732)(526,733)(527,734)(528,735)
(529,736)(530,737)(531,738)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,766)(542,767)(543,768)(544,769)
(545,770)(546,771)(547,772)(548,773)(549,774)(550,757)(551,758)(552,759)
(553,760)(554,761)(555,762)(556,763)(557,764)(558,765)(559,775)(560,776)
(561,777)(562,778)(563,779)(564,780)(565,781)(566,782)(567,783)(568,793)
(569,794)(570,795)(571,796)(572,797)(573,798)(574,799)(575,800)(576,801)
(577,784)(578,785)(579,786)(580,787)(581,788)(582,789)(583,790)(584,791)
(585,792)(586,802)(587,803)(588,804)(589,805)(590,806)(591,807)(592,808)
(593,809)(594,810)(595,820)(596,821)(597,822)(598,823)(599,824)(600,825)
(601,826)(602,827)(603,828)(604,811)(605,812)(606,813)(607,814)(608,815)
(609,816)(610,817)(611,818)(612,819)(613,829)(614,830)(615,831)(616,832)
(617,833)(618,834)(619,835)(620,836)(621,837)(622,847)(623,848)(624,849)
(625,850)(626,851)(627,852)(628,853)(629,854)(630,855)(631,838)(632,839)
(633,840)(634,841)(635,842)(636,843)(637,844)(638,845)(639,846)(640,856)
(641,857)(642,858)(643,859)(644,860)(645,861)(646,862)(647,863)(648,864);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,163)(  2,165)(  3,164)(  4,169)(  5,171)(  6,170)(  7,166)
(  8,168)(  9,167)( 10,172)( 11,174)( 12,173)( 13,178)( 14,180)( 15,179)
( 16,175)( 17,177)( 18,176)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)
( 24,188)( 25,184)( 26,186)( 27,185)( 28,190)( 29,192)( 30,191)( 31,196)
( 32,198)( 33,197)( 34,193)( 35,195)( 36,194)( 37,199)( 38,201)( 39,200)
( 40,205)( 41,207)( 42,206)( 43,202)( 44,204)( 45,203)( 46,208)( 47,210)
( 48,209)( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 55,109)
( 56,111)( 57,110)( 58,115)( 59,117)( 60,116)( 61,112)( 62,114)( 63,113)
( 64,118)( 65,120)( 66,119)( 67,124)( 68,126)( 69,125)( 70,121)( 71,123)
( 72,122)( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)( 79,130)
( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)( 86,144)( 87,143)
( 88,139)( 89,141)( 90,140)( 91,145)( 92,147)( 93,146)( 94,151)( 95,153)
( 96,152)( 97,148)( 98,150)( 99,149)(100,154)(101,156)(102,155)(103,160)
(104,162)(105,161)(106,157)(107,159)(108,158)(217,379)(218,381)(219,380)
(220,385)(221,387)(222,386)(223,382)(224,384)(225,383)(226,388)(227,390)
(228,389)(229,394)(230,396)(231,395)(232,391)(233,393)(234,392)(235,397)
(236,399)(237,398)(238,403)(239,405)(240,404)(241,400)(242,402)(243,401)
(244,406)(245,408)(246,407)(247,412)(248,414)(249,413)(250,409)(251,411)
(252,410)(253,415)(254,417)(255,416)(256,421)(257,423)(258,422)(259,418)
(260,420)(261,419)(262,424)(263,426)(264,425)(265,430)(266,432)(267,431)
(268,427)(269,429)(270,428)(271,325)(272,327)(273,326)(274,331)(275,333)
(276,332)(277,328)(278,330)(279,329)(280,334)(281,336)(282,335)(283,340)
(284,342)(285,341)(286,337)(287,339)(288,338)(289,343)(290,345)(291,344)
(292,349)(293,351)(294,350)(295,346)(296,348)(297,347)(298,352)(299,354)
(300,353)(301,358)(302,360)(303,359)(304,355)(305,357)(306,356)(307,361)
(308,363)(309,362)(310,367)(311,369)(312,368)(313,364)(314,366)(315,365)
(316,370)(317,372)(318,371)(319,376)(320,378)(321,377)(322,373)(323,375)
(324,374)(433,595)(434,597)(435,596)(436,601)(437,603)(438,602)(439,598)
(440,600)(441,599)(442,604)(443,606)(444,605)(445,610)(446,612)(447,611)
(448,607)(449,609)(450,608)(451,613)(452,615)(453,614)(454,619)(455,621)
(456,620)(457,616)(458,618)(459,617)(460,622)(461,624)(462,623)(463,628)
(464,630)(465,629)(466,625)(467,627)(468,626)(469,631)(470,633)(471,632)
(472,637)(473,639)(474,638)(475,634)(476,636)(477,635)(478,640)(479,642)
(480,641)(481,646)(482,648)(483,647)(484,643)(485,645)(486,644)(487,541)
(488,543)(489,542)(490,547)(491,549)(492,548)(493,544)(494,546)(495,545)
(496,550)(497,552)(498,551)(499,556)(500,558)(501,557)(502,553)(503,555)
(504,554)(505,559)(506,561)(507,560)(508,565)(509,567)(510,566)(511,562)
(512,564)(513,563)(514,568)(515,570)(516,569)(517,574)(518,576)(519,575)
(520,571)(521,573)(522,572)(523,577)(524,579)(525,578)(526,583)(527,585)
(528,584)(529,580)(530,582)(531,581)(532,586)(533,588)(534,587)(535,592)
(536,594)(537,593)(538,589)(539,591)(540,590)(649,811)(650,813)(651,812)
(652,817)(653,819)(654,818)(655,814)(656,816)(657,815)(658,820)(659,822)
(660,821)(661,826)(662,828)(663,827)(664,823)(665,825)(666,824)(667,829)
(668,831)(669,830)(670,835)(671,837)(672,836)(673,832)(674,834)(675,833)
(676,838)(677,840)(678,839)(679,844)(680,846)(681,845)(682,841)(683,843)
(684,842)(685,847)(686,849)(687,848)(688,853)(689,855)(690,854)(691,850)
(692,852)(693,851)(694,856)(695,858)(696,857)(697,862)(698,864)(699,863)
(700,859)(701,861)(702,860)(703,757)(704,759)(705,758)(706,763)(707,765)
(708,764)(709,760)(710,762)(711,761)(712,766)(713,768)(714,767)(715,772)
(716,774)(717,773)(718,769)(719,771)(720,770)(721,775)(722,777)(723,776)
(724,781)(725,783)(726,782)(727,778)(728,780)(729,779)(730,784)(731,786)
(732,785)(733,790)(734,792)(735,791)(736,787)(737,789)(738,788)(739,793)
(740,795)(741,794)(742,799)(743,801)(744,800)(745,796)(746,798)(747,797)
(748,802)(749,804)(750,803)(751,808)(752,810)(753,809)(754,805)(755,807)
(756,806);
s1 := Sym(864)!(  1,  4)(  3,  9)(  5,  8)( 10, 13)( 12, 18)( 14, 17)( 19, 22)
( 21, 27)( 23, 26)( 28, 31)( 30, 36)( 32, 35)( 37, 40)( 39, 45)( 41, 44)
( 46, 49)( 48, 54)( 50, 53)( 55, 58)( 57, 63)( 59, 62)( 64, 67)( 66, 72)
( 68, 71)( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91, 94)
( 93, 99)( 95, 98)(100,103)(102,108)(104,107)(109,112)(111,117)(113,116)
(118,121)(120,126)(122,125)(127,130)(129,135)(131,134)(136,139)(138,144)
(140,143)(145,148)(147,153)(149,152)(154,157)(156,162)(158,161)(163,166)
(165,171)(167,170)(172,175)(174,180)(176,179)(181,184)(183,189)(185,188)
(190,193)(192,198)(194,197)(199,202)(201,207)(203,206)(208,211)(210,216)
(212,215)(217,220)(219,225)(221,224)(226,229)(228,234)(230,233)(235,238)
(237,243)(239,242)(244,247)(246,252)(248,251)(253,256)(255,261)(257,260)
(262,265)(264,270)(266,269)(271,274)(273,279)(275,278)(280,283)(282,288)
(284,287)(289,292)(291,297)(293,296)(298,301)(300,306)(302,305)(307,310)
(309,315)(311,314)(316,319)(318,324)(320,323)(325,328)(327,333)(329,332)
(334,337)(336,342)(338,341)(343,346)(345,351)(347,350)(352,355)(354,360)
(356,359)(361,364)(363,369)(365,368)(370,373)(372,378)(374,377)(379,382)
(381,387)(383,386)(388,391)(390,396)(392,395)(397,400)(399,405)(401,404)
(406,409)(408,414)(410,413)(415,418)(417,423)(419,422)(424,427)(426,432)
(428,431)(433,490)(434,488)(435,495)(436,487)(437,494)(438,492)(439,493)
(440,491)(441,489)(442,499)(443,497)(444,504)(445,496)(446,503)(447,501)
(448,502)(449,500)(450,498)(451,508)(452,506)(453,513)(454,505)(455,512)
(456,510)(457,511)(458,509)(459,507)(460,517)(461,515)(462,522)(463,514)
(464,521)(465,519)(466,520)(467,518)(468,516)(469,526)(470,524)(471,531)
(472,523)(473,530)(474,528)(475,529)(476,527)(477,525)(478,535)(479,533)
(480,540)(481,532)(482,539)(483,537)(484,538)(485,536)(486,534)(541,598)
(542,596)(543,603)(544,595)(545,602)(546,600)(547,601)(548,599)(549,597)
(550,607)(551,605)(552,612)(553,604)(554,611)(555,609)(556,610)(557,608)
(558,606)(559,616)(560,614)(561,621)(562,613)(563,620)(564,618)(565,619)
(566,617)(567,615)(568,625)(569,623)(570,630)(571,622)(572,629)(573,627)
(574,628)(575,626)(576,624)(577,634)(578,632)(579,639)(580,631)(581,638)
(582,636)(583,637)(584,635)(585,633)(586,643)(587,641)(588,648)(589,640)
(590,647)(591,645)(592,646)(593,644)(594,642)(649,706)(650,704)(651,711)
(652,703)(653,710)(654,708)(655,709)(656,707)(657,705)(658,715)(659,713)
(660,720)(661,712)(662,719)(663,717)(664,718)(665,716)(666,714)(667,724)
(668,722)(669,729)(670,721)(671,728)(672,726)(673,727)(674,725)(675,723)
(676,733)(677,731)(678,738)(679,730)(680,737)(681,735)(682,736)(683,734)
(684,732)(685,742)(686,740)(687,747)(688,739)(689,746)(690,744)(691,745)
(692,743)(693,741)(694,751)(695,749)(696,756)(697,748)(698,755)(699,753)
(700,754)(701,752)(702,750)(757,814)(758,812)(759,819)(760,811)(761,818)
(762,816)(763,817)(764,815)(765,813)(766,823)(767,821)(768,828)(769,820)
(770,827)(771,825)(772,826)(773,824)(774,822)(775,832)(776,830)(777,837)
(778,829)(779,836)(780,834)(781,835)(782,833)(783,831)(784,841)(785,839)
(786,846)(787,838)(788,845)(789,843)(790,844)(791,842)(792,840)(793,850)
(794,848)(795,855)(796,847)(797,854)(798,852)(799,853)(800,851)(801,849)
(802,859)(803,857)(804,864)(805,856)(806,863)(807,861)(808,862)(809,860)
(810,858);
s2 := Sym(864)!(  1,649)(  2,650)(  3,651)(  4,657)(  5,655)(  6,656)(  7,653)
(  8,654)(  9,652)( 10,667)( 11,668)( 12,669)( 13,675)( 14,673)( 15,674)
( 16,671)( 17,672)( 18,670)( 19,658)( 20,659)( 21,660)( 22,666)( 23,664)
( 24,665)( 25,662)( 26,663)( 27,661)( 28,676)( 29,677)( 30,678)( 31,684)
( 32,682)( 33,683)( 34,680)( 35,681)( 36,679)( 37,694)( 38,695)( 39,696)
( 40,702)( 41,700)( 42,701)( 43,698)( 44,699)( 45,697)( 46,685)( 47,686)
( 48,687)( 49,693)( 50,691)( 51,692)( 52,689)( 53,690)( 54,688)( 55,703)
( 56,704)( 57,705)( 58,711)( 59,709)( 60,710)( 61,707)( 62,708)( 63,706)
( 64,721)( 65,722)( 66,723)( 67,729)( 68,727)( 69,728)( 70,725)( 71,726)
( 72,724)( 73,712)( 74,713)( 75,714)( 76,720)( 77,718)( 78,719)( 79,716)
( 80,717)( 81,715)( 82,730)( 83,731)( 84,732)( 85,738)( 86,736)( 87,737)
( 88,734)( 89,735)( 90,733)( 91,748)( 92,749)( 93,750)( 94,756)( 95,754)
( 96,755)( 97,752)( 98,753)( 99,751)(100,739)(101,740)(102,741)(103,747)
(104,745)(105,746)(106,743)(107,744)(108,742)(109,757)(110,758)(111,759)
(112,765)(113,763)(114,764)(115,761)(116,762)(117,760)(118,775)(119,776)
(120,777)(121,783)(122,781)(123,782)(124,779)(125,780)(126,778)(127,766)
(128,767)(129,768)(130,774)(131,772)(132,773)(133,770)(134,771)(135,769)
(136,784)(137,785)(138,786)(139,792)(140,790)(141,791)(142,788)(143,789)
(144,787)(145,802)(146,803)(147,804)(148,810)(149,808)(150,809)(151,806)
(152,807)(153,805)(154,793)(155,794)(156,795)(157,801)(158,799)(159,800)
(160,797)(161,798)(162,796)(163,811)(164,812)(165,813)(166,819)(167,817)
(168,818)(169,815)(170,816)(171,814)(172,829)(173,830)(174,831)(175,837)
(176,835)(177,836)(178,833)(179,834)(180,832)(181,820)(182,821)(183,822)
(184,828)(185,826)(186,827)(187,824)(188,825)(189,823)(190,838)(191,839)
(192,840)(193,846)(194,844)(195,845)(196,842)(197,843)(198,841)(199,856)
(200,857)(201,858)(202,864)(203,862)(204,863)(205,860)(206,861)(207,859)
(208,847)(209,848)(210,849)(211,855)(212,853)(213,854)(214,851)(215,852)
(216,850)(217,460)(218,461)(219,462)(220,468)(221,466)(222,467)(223,464)
(224,465)(225,463)(226,478)(227,479)(228,480)(229,486)(230,484)(231,485)
(232,482)(233,483)(234,481)(235,469)(236,470)(237,471)(238,477)(239,475)
(240,476)(241,473)(242,474)(243,472)(244,433)(245,434)(246,435)(247,441)
(248,439)(249,440)(250,437)(251,438)(252,436)(253,451)(254,452)(255,453)
(256,459)(257,457)(258,458)(259,455)(260,456)(261,454)(262,442)(263,443)
(264,444)(265,450)(266,448)(267,449)(268,446)(269,447)(270,445)(271,514)
(272,515)(273,516)(274,522)(275,520)(276,521)(277,518)(278,519)(279,517)
(280,532)(281,533)(282,534)(283,540)(284,538)(285,539)(286,536)(287,537)
(288,535)(289,523)(290,524)(291,525)(292,531)(293,529)(294,530)(295,527)
(296,528)(297,526)(298,487)(299,488)(300,489)(301,495)(302,493)(303,494)
(304,491)(305,492)(306,490)(307,505)(308,506)(309,507)(310,513)(311,511)
(312,512)(313,509)(314,510)(315,508)(316,496)(317,497)(318,498)(319,504)
(320,502)(321,503)(322,500)(323,501)(324,499)(325,568)(326,569)(327,570)
(328,576)(329,574)(330,575)(331,572)(332,573)(333,571)(334,586)(335,587)
(336,588)(337,594)(338,592)(339,593)(340,590)(341,591)(342,589)(343,577)
(344,578)(345,579)(346,585)(347,583)(348,584)(349,581)(350,582)(351,580)
(352,541)(353,542)(354,543)(355,549)(356,547)(357,548)(358,545)(359,546)
(360,544)(361,559)(362,560)(363,561)(364,567)(365,565)(366,566)(367,563)
(368,564)(369,562)(370,550)(371,551)(372,552)(373,558)(374,556)(375,557)
(376,554)(377,555)(378,553)(379,622)(380,623)(381,624)(382,630)(383,628)
(384,629)(385,626)(386,627)(387,625)(388,640)(389,641)(390,642)(391,648)
(392,646)(393,647)(394,644)(395,645)(396,643)(397,631)(398,632)(399,633)
(400,639)(401,637)(402,638)(403,635)(404,636)(405,634)(406,595)(407,596)
(408,597)(409,603)(410,601)(411,602)(412,599)(413,600)(414,598)(415,613)
(416,614)(417,615)(418,621)(419,619)(420,620)(421,617)(422,618)(423,616)
(424,604)(425,605)(426,606)(427,612)(428,610)(429,611)(430,608)(431,609)
(432,607);
s3 := Sym(864)!(  1,226)(  2,227)(  3,228)(  4,229)(  5,230)(  6,231)(  7,232)
(  8,233)(  9,234)( 10,217)( 11,218)( 12,219)( 13,220)( 14,221)( 15,222)
( 16,223)( 17,224)( 18,225)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,253)( 29,254)( 30,255)( 31,256)
( 32,257)( 33,258)( 34,259)( 35,260)( 36,261)( 37,244)( 38,245)( 39,246)
( 40,247)( 41,248)( 42,249)( 43,250)( 44,251)( 45,252)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,280)
( 56,281)( 57,282)( 58,283)( 59,284)( 60,285)( 61,286)( 62,287)( 63,288)
( 64,271)( 65,272)( 66,273)( 67,274)( 68,275)( 69,276)( 70,277)( 71,278)
( 72,279)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)
( 88,313)( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)
( 96,303)( 97,304)( 98,305)( 99,306)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,334)(110,335)(111,336)
(112,337)(113,338)(114,339)(115,340)(116,341)(117,342)(118,325)(119,326)
(120,327)(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,343)
(128,344)(129,345)(130,346)(131,347)(132,348)(133,349)(134,350)(135,351)
(136,361)(137,362)(138,363)(139,364)(140,365)(141,366)(142,367)(143,368)
(144,369)(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)
(152,359)(153,360)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)
(160,376)(161,377)(162,378)(163,388)(164,389)(165,390)(166,391)(167,392)
(168,393)(169,394)(170,395)(171,396)(172,379)(173,380)(174,381)(175,382)
(176,383)(177,384)(178,385)(179,386)(180,387)(181,397)(182,398)(183,399)
(184,400)(185,401)(186,402)(187,403)(188,404)(189,405)(190,415)(191,416)
(192,417)(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)
(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)
(208,424)(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)
(216,432)(433,658)(434,659)(435,660)(436,661)(437,662)(438,663)(439,664)
(440,665)(441,666)(442,649)(443,650)(444,651)(445,652)(446,653)(447,654)
(448,655)(449,656)(450,657)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,685)(461,686)(462,687)(463,688)
(464,689)(465,690)(466,691)(467,692)(468,693)(469,676)(470,677)(471,678)
(472,679)(473,680)(474,681)(475,682)(476,683)(477,684)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,712)
(488,713)(489,714)(490,715)(491,716)(492,717)(493,718)(494,719)(495,720)
(496,703)(497,704)(498,705)(499,706)(500,707)(501,708)(502,709)(503,710)
(504,711)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,739)(515,740)(516,741)(517,742)(518,743)(519,744)
(520,745)(521,746)(522,747)(523,730)(524,731)(525,732)(526,733)(527,734)
(528,735)(529,736)(530,737)(531,738)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,766)(542,767)(543,768)
(544,769)(545,770)(546,771)(547,772)(548,773)(549,774)(550,757)(551,758)
(552,759)(553,760)(554,761)(555,762)(556,763)(557,764)(558,765)(559,775)
(560,776)(561,777)(562,778)(563,779)(564,780)(565,781)(566,782)(567,783)
(568,793)(569,794)(570,795)(571,796)(572,797)(573,798)(574,799)(575,800)
(576,801)(577,784)(578,785)(579,786)(580,787)(581,788)(582,789)(583,790)
(584,791)(585,792)(586,802)(587,803)(588,804)(589,805)(590,806)(591,807)
(592,808)(593,809)(594,810)(595,820)(596,821)(597,822)(598,823)(599,824)
(600,825)(601,826)(602,827)(603,828)(604,811)(605,812)(606,813)(607,814)
(608,815)(609,816)(610,817)(611,818)(612,819)(613,829)(614,830)(615,831)
(616,832)(617,833)(618,834)(619,835)(620,836)(621,837)(622,847)(623,848)
(624,849)(625,850)(626,851)(627,852)(628,853)(629,854)(630,855)(631,838)
(632,839)(633,840)(634,841)(635,842)(636,843)(637,844)(638,845)(639,846)
(640,856)(641,857)(642,858)(643,859)(644,860)(645,861)(646,862)(647,863)
(648,864);
poly := sub<Sym(864)|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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope