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Polytope of Type {2,6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,8}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46634)
Rank : 4
Schlafli Type : {2,6,8}
Number of vertices, edges, etc : 2, 54, 216, 72
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,6,4}*864b
   3-fold quotients : {2,6,8}*576
   6-fold quotients : {2,6,4}*288
   9-fold quotients : {2,6,8}*192
   12-fold quotients : {2,6,4}*144
   18-fold quotients : {2,6,4}*96a
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3,489)(  4,491)(  5,490)(  6,495)(  7,497)(  8,496)(  9,492)( 10,494)
( 11,493)( 12,507)( 13,509)( 14,508)( 15,513)( 16,515)( 17,514)( 18,510)
( 19,512)( 20,511)( 21,498)( 22,500)( 23,499)( 24,504)( 25,506)( 26,505)
( 27,501)( 28,503)( 29,502)( 30,516)( 31,518)( 32,517)( 33,522)( 34,524)
( 35,523)( 36,519)( 37,521)( 38,520)( 39,534)( 40,536)( 41,535)( 42,540)
( 43,542)( 44,541)( 45,537)( 46,539)( 47,538)( 48,525)( 49,527)( 50,526)
( 51,531)( 52,533)( 53,532)( 54,528)( 55,530)( 56,529)( 57,462)( 58,464)
( 59,463)( 60,468)( 61,470)( 62,469)( 63,465)( 64,467)( 65,466)( 66,480)
( 67,482)( 68,481)( 69,486)( 70,488)( 71,487)( 72,483)( 73,485)( 74,484)
( 75,471)( 76,473)( 77,472)( 78,477)( 79,479)( 80,478)( 81,474)( 82,476)
( 83,475)( 84,435)( 85,437)( 86,436)( 87,441)( 88,443)( 89,442)( 90,438)
( 91,440)( 92,439)( 93,453)( 94,455)( 95,454)( 96,459)( 97,461)( 98,460)
( 99,456)(100,458)(101,457)(102,444)(103,446)(104,445)(105,450)(106,452)
(107,451)(108,447)(109,449)(110,448)(111,624)(112,626)(113,625)(114,630)
(115,632)(116,631)(117,627)(118,629)(119,628)(120,642)(121,644)(122,643)
(123,648)(124,650)(125,649)(126,645)(127,647)(128,646)(129,633)(130,635)
(131,634)(132,639)(133,641)(134,640)(135,636)(136,638)(137,637)(138,597)
(139,599)(140,598)(141,603)(142,605)(143,604)(144,600)(145,602)(146,601)
(147,615)(148,617)(149,616)(150,621)(151,623)(152,622)(153,618)(154,620)
(155,619)(156,606)(157,608)(158,607)(159,612)(160,614)(161,613)(162,609)
(163,611)(164,610)(165,543)(166,545)(167,544)(168,549)(169,551)(170,550)
(171,546)(172,548)(173,547)(174,561)(175,563)(176,562)(177,567)(178,569)
(179,568)(180,564)(181,566)(182,565)(183,552)(184,554)(185,553)(186,558)
(187,560)(188,559)(189,555)(190,557)(191,556)(192,570)(193,572)(194,571)
(195,576)(196,578)(197,577)(198,573)(199,575)(200,574)(201,588)(202,590)
(203,589)(204,594)(205,596)(206,595)(207,591)(208,593)(209,592)(210,579)
(211,581)(212,580)(213,585)(214,587)(215,586)(216,582)(217,584)(218,583)
(219,732)(220,734)(221,733)(222,738)(223,740)(224,739)(225,735)(226,737)
(227,736)(228,750)(229,752)(230,751)(231,756)(232,758)(233,757)(234,753)
(235,755)(236,754)(237,741)(238,743)(239,742)(240,747)(241,749)(242,748)
(243,744)(244,746)(245,745)(246,705)(247,707)(248,706)(249,711)(250,713)
(251,712)(252,708)(253,710)(254,709)(255,723)(256,725)(257,724)(258,729)
(259,731)(260,730)(261,726)(262,728)(263,727)(264,714)(265,716)(266,715)
(267,720)(268,722)(269,721)(270,717)(271,719)(272,718)(273,651)(274,653)
(275,652)(276,657)(277,659)(278,658)(279,654)(280,656)(281,655)(282,669)
(283,671)(284,670)(285,675)(286,677)(287,676)(288,672)(289,674)(290,673)
(291,660)(292,662)(293,661)(294,666)(295,668)(296,667)(297,663)(298,665)
(299,664)(300,678)(301,680)(302,679)(303,684)(304,686)(305,685)(306,681)
(307,683)(308,682)(309,696)(310,698)(311,697)(312,702)(313,704)(314,703)
(315,699)(316,701)(317,700)(318,687)(319,689)(320,688)(321,693)(322,695)
(323,694)(324,690)(325,692)(326,691)(327,813)(328,815)(329,814)(330,819)
(331,821)(332,820)(333,816)(334,818)(335,817)(336,831)(337,833)(338,832)
(339,837)(340,839)(341,838)(342,834)(343,836)(344,835)(345,822)(346,824)
(347,823)(348,828)(349,830)(350,829)(351,825)(352,827)(353,826)(354,840)
(355,842)(356,841)(357,846)(358,848)(359,847)(360,843)(361,845)(362,844)
(363,858)(364,860)(365,859)(366,864)(367,866)(368,865)(369,861)(370,863)
(371,862)(372,849)(373,851)(374,850)(375,855)(376,857)(377,856)(378,852)
(379,854)(380,853)(381,786)(382,788)(383,787)(384,792)(385,794)(386,793)
(387,789)(388,791)(389,790)(390,804)(391,806)(392,805)(393,810)(394,812)
(395,811)(396,807)(397,809)(398,808)(399,795)(400,797)(401,796)(402,801)
(403,803)(404,802)(405,798)(406,800)(407,799)(408,759)(409,761)(410,760)
(411,765)(412,767)(413,766)(414,762)(415,764)(416,763)(417,777)(418,779)
(419,778)(420,783)(421,785)(422,784)(423,780)(424,782)(425,781)(426,768)
(427,770)(428,769)(429,774)(430,776)(431,775)(432,771)(433,773)(434,772);;
s2 := (  3,661)(  4,660)(  5,662)(  6,655)(  7,654)(  8,656)(  9,676)( 10,675)
( 11,677)( 12,652)( 13,651)( 14,653)( 15,673)( 16,672)( 17,674)( 18,667)
( 19,666)( 20,668)( 21,670)( 22,669)( 23,671)( 24,664)( 25,663)( 26,665)
( 27,658)( 28,657)( 29,659)( 30,688)( 31,687)( 32,689)( 33,682)( 34,681)
( 35,683)( 36,703)( 37,702)( 38,704)( 39,679)( 40,678)( 41,680)( 42,700)
( 43,699)( 44,701)( 45,694)( 46,693)( 47,695)( 48,697)( 49,696)( 50,698)
( 51,691)( 52,690)( 53,692)( 54,685)( 55,684)( 56,686)( 57,742)( 58,741)
( 59,743)( 60,736)( 61,735)( 62,737)( 63,757)( 64,756)( 65,758)( 66,733)
( 67,732)( 68,734)( 69,754)( 70,753)( 71,755)( 72,748)( 73,747)( 74,749)
( 75,751)( 76,750)( 77,752)( 78,745)( 79,744)( 80,746)( 81,739)( 82,738)
( 83,740)( 84,715)( 85,714)( 86,716)( 87,709)( 88,708)( 89,710)( 90,730)
( 91,729)( 92,731)( 93,706)( 94,705)( 95,707)( 96,727)( 97,726)( 98,728)
( 99,721)(100,720)(101,722)(102,724)(103,723)(104,725)(105,718)(106,717)
(107,719)(108,712)(109,711)(110,713)(111,850)(112,849)(113,851)(114,844)
(115,843)(116,845)(117,865)(118,864)(119,866)(120,841)(121,840)(122,842)
(123,862)(124,861)(125,863)(126,856)(127,855)(128,857)(129,859)(130,858)
(131,860)(132,853)(133,852)(134,854)(135,847)(136,846)(137,848)(138,823)
(139,822)(140,824)(141,817)(142,816)(143,818)(144,838)(145,837)(146,839)
(147,814)(148,813)(149,815)(150,835)(151,834)(152,836)(153,829)(154,828)
(155,830)(156,832)(157,831)(158,833)(159,826)(160,825)(161,827)(162,820)
(163,819)(164,821)(165,796)(166,795)(167,797)(168,790)(169,789)(170,791)
(171,811)(172,810)(173,812)(174,787)(175,786)(176,788)(177,808)(178,807)
(179,809)(180,802)(181,801)(182,803)(183,805)(184,804)(185,806)(186,799)
(187,798)(188,800)(189,793)(190,792)(191,794)(192,769)(193,768)(194,770)
(195,763)(196,762)(197,764)(198,784)(199,783)(200,785)(201,760)(202,759)
(203,761)(204,781)(205,780)(206,782)(207,775)(208,774)(209,776)(210,778)
(211,777)(212,779)(213,772)(214,771)(215,773)(216,766)(217,765)(218,767)
(219,472)(220,471)(221,473)(222,466)(223,465)(224,467)(225,487)(226,486)
(227,488)(228,463)(229,462)(230,464)(231,484)(232,483)(233,485)(234,478)
(235,477)(236,479)(237,481)(238,480)(239,482)(240,475)(241,474)(242,476)
(243,469)(244,468)(245,470)(246,445)(247,444)(248,446)(249,439)(250,438)
(251,440)(252,460)(253,459)(254,461)(255,436)(256,435)(257,437)(258,457)
(259,456)(260,458)(261,451)(262,450)(263,452)(264,454)(265,453)(266,455)
(267,448)(268,447)(269,449)(270,442)(271,441)(272,443)(273,499)(274,498)
(275,500)(276,493)(277,492)(278,494)(279,514)(280,513)(281,515)(282,490)
(283,489)(284,491)(285,511)(286,510)(287,512)(288,505)(289,504)(290,506)
(291,508)(292,507)(293,509)(294,502)(295,501)(296,503)(297,496)(298,495)
(299,497)(300,526)(301,525)(302,527)(303,520)(304,519)(305,521)(306,541)
(307,540)(308,542)(309,517)(310,516)(311,518)(312,538)(313,537)(314,539)
(315,532)(316,531)(317,533)(318,535)(319,534)(320,536)(321,529)(322,528)
(323,530)(324,523)(325,522)(326,524)(327,607)(328,606)(329,608)(330,601)
(331,600)(332,602)(333,622)(334,621)(335,623)(336,598)(337,597)(338,599)
(339,619)(340,618)(341,620)(342,613)(343,612)(344,614)(345,616)(346,615)
(347,617)(348,610)(349,609)(350,611)(351,604)(352,603)(353,605)(354,634)
(355,633)(356,635)(357,628)(358,627)(359,629)(360,649)(361,648)(362,650)
(363,625)(364,624)(365,626)(366,646)(367,645)(368,647)(369,640)(370,639)
(371,641)(372,643)(373,642)(374,644)(375,637)(376,636)(377,638)(378,631)
(379,630)(380,632)(381,553)(382,552)(383,554)(384,547)(385,546)(386,548)
(387,568)(388,567)(389,569)(390,544)(391,543)(392,545)(393,565)(394,564)
(395,566)(396,559)(397,558)(398,560)(399,562)(400,561)(401,563)(402,556)
(403,555)(404,557)(405,550)(406,549)(407,551)(408,580)(409,579)(410,581)
(411,574)(412,573)(413,575)(414,595)(415,594)(416,596)(417,571)(418,570)
(419,572)(420,592)(421,591)(422,593)(423,586)(424,585)(425,587)(426,589)
(427,588)(428,590)(429,583)(430,582)(431,584)(432,577)(433,576)(434,578);;
s3 := (  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)( 10,118)
( 11,119)( 12,135)( 13,136)( 14,137)( 15,129)( 16,130)( 17,131)( 18,132)
( 19,133)( 20,134)( 21,123)( 22,124)( 23,125)( 24,126)( 25,127)( 26,128)
( 27,120)( 28,121)( 29,122)( 30,138)( 31,139)( 32,140)( 33,141)( 34,142)
( 35,143)( 36,144)( 37,145)( 38,146)( 39,162)( 40,163)( 41,164)( 42,156)
( 43,157)( 44,158)( 45,159)( 46,160)( 47,161)( 48,150)( 49,151)( 50,152)
( 51,153)( 52,154)( 53,155)( 54,147)( 55,148)( 56,149)( 57,192)( 58,193)
( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)( 66,216)
( 67,217)( 68,218)( 69,210)( 70,211)( 71,212)( 72,213)( 73,214)( 74,215)
( 75,204)( 76,205)( 77,206)( 78,207)( 79,208)( 80,209)( 81,201)( 82,202)
( 83,203)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)
( 91,172)( 92,173)( 93,189)( 94,190)( 95,191)( 96,183)( 97,184)( 98,185)
( 99,186)(100,187)(101,188)(102,177)(103,178)(104,179)(105,180)(106,181)
(107,182)(108,174)(109,175)(110,176)(219,327)(220,328)(221,329)(222,330)
(223,331)(224,332)(225,333)(226,334)(227,335)(228,351)(229,352)(230,353)
(231,345)(232,346)(233,347)(234,348)(235,349)(236,350)(237,339)(238,340)
(239,341)(240,342)(241,343)(242,344)(243,336)(244,337)(245,338)(246,354)
(247,355)(248,356)(249,357)(250,358)(251,359)(252,360)(253,361)(254,362)
(255,378)(256,379)(257,380)(258,372)(259,373)(260,374)(261,375)(262,376)
(263,377)(264,366)(265,367)(266,368)(267,369)(268,370)(269,371)(270,363)
(271,364)(272,365)(273,408)(274,409)(275,410)(276,411)(277,412)(278,413)
(279,414)(280,415)(281,416)(282,432)(283,433)(284,434)(285,426)(286,427)
(287,428)(288,429)(289,430)(290,431)(291,420)(292,421)(293,422)(294,423)
(295,424)(296,425)(297,417)(298,418)(299,419)(300,381)(301,382)(302,383)
(303,384)(304,385)(305,386)(306,387)(307,388)(308,389)(309,405)(310,406)
(311,407)(312,399)(313,400)(314,401)(315,402)(316,403)(317,404)(318,393)
(319,394)(320,395)(321,396)(322,397)(323,398)(324,390)(325,391)(326,392)
(435,543)(436,544)(437,545)(438,546)(439,547)(440,548)(441,549)(442,550)
(443,551)(444,567)(445,568)(446,569)(447,561)(448,562)(449,563)(450,564)
(451,565)(452,566)(453,555)(454,556)(455,557)(456,558)(457,559)(458,560)
(459,552)(460,553)(461,554)(462,570)(463,571)(464,572)(465,573)(466,574)
(467,575)(468,576)(469,577)(470,578)(471,594)(472,595)(473,596)(474,588)
(475,589)(476,590)(477,591)(478,592)(479,593)(480,582)(481,583)(482,584)
(483,585)(484,586)(485,587)(486,579)(487,580)(488,581)(489,624)(490,625)
(491,626)(492,627)(493,628)(494,629)(495,630)(496,631)(497,632)(498,648)
(499,649)(500,650)(501,642)(502,643)(503,644)(504,645)(505,646)(506,647)
(507,636)(508,637)(509,638)(510,639)(511,640)(512,641)(513,633)(514,634)
(515,635)(516,597)(517,598)(518,599)(519,600)(520,601)(521,602)(522,603)
(523,604)(524,605)(525,621)(526,622)(527,623)(528,615)(529,616)(530,617)
(531,618)(532,619)(533,620)(534,609)(535,610)(536,611)(537,612)(538,613)
(539,614)(540,606)(541,607)(542,608)(651,759)(652,760)(653,761)(654,762)
(655,763)(656,764)(657,765)(658,766)(659,767)(660,783)(661,784)(662,785)
(663,777)(664,778)(665,779)(666,780)(667,781)(668,782)(669,771)(670,772)
(671,773)(672,774)(673,775)(674,776)(675,768)(676,769)(677,770)(678,786)
(679,787)(680,788)(681,789)(682,790)(683,791)(684,792)(685,793)(686,794)
(687,810)(688,811)(689,812)(690,804)(691,805)(692,806)(693,807)(694,808)
(695,809)(696,798)(697,799)(698,800)(699,801)(700,802)(701,803)(702,795)
(703,796)(704,797)(705,840)(706,841)(707,842)(708,843)(709,844)(710,845)
(711,846)(712,847)(713,848)(714,864)(715,865)(716,866)(717,858)(718,859)
(719,860)(720,861)(721,862)(722,863)(723,852)(724,853)(725,854)(726,855)
(727,856)(728,857)(729,849)(730,850)(731,851)(732,813)(733,814)(734,815)
(735,816)(736,817)(737,818)(738,819)(739,820)(740,821)(741,837)(742,838)
(743,839)(744,831)(745,832)(746,833)(747,834)(748,835)(749,836)(750,825)
(751,826)(752,827)(753,828)(754,829)(755,830)(756,822)(757,823)(758,824);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(866)!(1,2);
s1 := Sym(866)!(  3,489)(  4,491)(  5,490)(  6,495)(  7,497)(  8,496)(  9,492)
( 10,494)( 11,493)( 12,507)( 13,509)( 14,508)( 15,513)( 16,515)( 17,514)
( 18,510)( 19,512)( 20,511)( 21,498)( 22,500)( 23,499)( 24,504)( 25,506)
( 26,505)( 27,501)( 28,503)( 29,502)( 30,516)( 31,518)( 32,517)( 33,522)
( 34,524)( 35,523)( 36,519)( 37,521)( 38,520)( 39,534)( 40,536)( 41,535)
( 42,540)( 43,542)( 44,541)( 45,537)( 46,539)( 47,538)( 48,525)( 49,527)
( 50,526)( 51,531)( 52,533)( 53,532)( 54,528)( 55,530)( 56,529)( 57,462)
( 58,464)( 59,463)( 60,468)( 61,470)( 62,469)( 63,465)( 64,467)( 65,466)
( 66,480)( 67,482)( 68,481)( 69,486)( 70,488)( 71,487)( 72,483)( 73,485)
( 74,484)( 75,471)( 76,473)( 77,472)( 78,477)( 79,479)( 80,478)( 81,474)
( 82,476)( 83,475)( 84,435)( 85,437)( 86,436)( 87,441)( 88,443)( 89,442)
( 90,438)( 91,440)( 92,439)( 93,453)( 94,455)( 95,454)( 96,459)( 97,461)
( 98,460)( 99,456)(100,458)(101,457)(102,444)(103,446)(104,445)(105,450)
(106,452)(107,451)(108,447)(109,449)(110,448)(111,624)(112,626)(113,625)
(114,630)(115,632)(116,631)(117,627)(118,629)(119,628)(120,642)(121,644)
(122,643)(123,648)(124,650)(125,649)(126,645)(127,647)(128,646)(129,633)
(130,635)(131,634)(132,639)(133,641)(134,640)(135,636)(136,638)(137,637)
(138,597)(139,599)(140,598)(141,603)(142,605)(143,604)(144,600)(145,602)
(146,601)(147,615)(148,617)(149,616)(150,621)(151,623)(152,622)(153,618)
(154,620)(155,619)(156,606)(157,608)(158,607)(159,612)(160,614)(161,613)
(162,609)(163,611)(164,610)(165,543)(166,545)(167,544)(168,549)(169,551)
(170,550)(171,546)(172,548)(173,547)(174,561)(175,563)(176,562)(177,567)
(178,569)(179,568)(180,564)(181,566)(182,565)(183,552)(184,554)(185,553)
(186,558)(187,560)(188,559)(189,555)(190,557)(191,556)(192,570)(193,572)
(194,571)(195,576)(196,578)(197,577)(198,573)(199,575)(200,574)(201,588)
(202,590)(203,589)(204,594)(205,596)(206,595)(207,591)(208,593)(209,592)
(210,579)(211,581)(212,580)(213,585)(214,587)(215,586)(216,582)(217,584)
(218,583)(219,732)(220,734)(221,733)(222,738)(223,740)(224,739)(225,735)
(226,737)(227,736)(228,750)(229,752)(230,751)(231,756)(232,758)(233,757)
(234,753)(235,755)(236,754)(237,741)(238,743)(239,742)(240,747)(241,749)
(242,748)(243,744)(244,746)(245,745)(246,705)(247,707)(248,706)(249,711)
(250,713)(251,712)(252,708)(253,710)(254,709)(255,723)(256,725)(257,724)
(258,729)(259,731)(260,730)(261,726)(262,728)(263,727)(264,714)(265,716)
(266,715)(267,720)(268,722)(269,721)(270,717)(271,719)(272,718)(273,651)
(274,653)(275,652)(276,657)(277,659)(278,658)(279,654)(280,656)(281,655)
(282,669)(283,671)(284,670)(285,675)(286,677)(287,676)(288,672)(289,674)
(290,673)(291,660)(292,662)(293,661)(294,666)(295,668)(296,667)(297,663)
(298,665)(299,664)(300,678)(301,680)(302,679)(303,684)(304,686)(305,685)
(306,681)(307,683)(308,682)(309,696)(310,698)(311,697)(312,702)(313,704)
(314,703)(315,699)(316,701)(317,700)(318,687)(319,689)(320,688)(321,693)
(322,695)(323,694)(324,690)(325,692)(326,691)(327,813)(328,815)(329,814)
(330,819)(331,821)(332,820)(333,816)(334,818)(335,817)(336,831)(337,833)
(338,832)(339,837)(340,839)(341,838)(342,834)(343,836)(344,835)(345,822)
(346,824)(347,823)(348,828)(349,830)(350,829)(351,825)(352,827)(353,826)
(354,840)(355,842)(356,841)(357,846)(358,848)(359,847)(360,843)(361,845)
(362,844)(363,858)(364,860)(365,859)(366,864)(367,866)(368,865)(369,861)
(370,863)(371,862)(372,849)(373,851)(374,850)(375,855)(376,857)(377,856)
(378,852)(379,854)(380,853)(381,786)(382,788)(383,787)(384,792)(385,794)
(386,793)(387,789)(388,791)(389,790)(390,804)(391,806)(392,805)(393,810)
(394,812)(395,811)(396,807)(397,809)(398,808)(399,795)(400,797)(401,796)
(402,801)(403,803)(404,802)(405,798)(406,800)(407,799)(408,759)(409,761)
(410,760)(411,765)(412,767)(413,766)(414,762)(415,764)(416,763)(417,777)
(418,779)(419,778)(420,783)(421,785)(422,784)(423,780)(424,782)(425,781)
(426,768)(427,770)(428,769)(429,774)(430,776)(431,775)(432,771)(433,773)
(434,772);
s2 := Sym(866)!(  3,661)(  4,660)(  5,662)(  6,655)(  7,654)(  8,656)(  9,676)
( 10,675)( 11,677)( 12,652)( 13,651)( 14,653)( 15,673)( 16,672)( 17,674)
( 18,667)( 19,666)( 20,668)( 21,670)( 22,669)( 23,671)( 24,664)( 25,663)
( 26,665)( 27,658)( 28,657)( 29,659)( 30,688)( 31,687)( 32,689)( 33,682)
( 34,681)( 35,683)( 36,703)( 37,702)( 38,704)( 39,679)( 40,678)( 41,680)
( 42,700)( 43,699)( 44,701)( 45,694)( 46,693)( 47,695)( 48,697)( 49,696)
( 50,698)( 51,691)( 52,690)( 53,692)( 54,685)( 55,684)( 56,686)( 57,742)
( 58,741)( 59,743)( 60,736)( 61,735)( 62,737)( 63,757)( 64,756)( 65,758)
( 66,733)( 67,732)( 68,734)( 69,754)( 70,753)( 71,755)( 72,748)( 73,747)
( 74,749)( 75,751)( 76,750)( 77,752)( 78,745)( 79,744)( 80,746)( 81,739)
( 82,738)( 83,740)( 84,715)( 85,714)( 86,716)( 87,709)( 88,708)( 89,710)
( 90,730)( 91,729)( 92,731)( 93,706)( 94,705)( 95,707)( 96,727)( 97,726)
( 98,728)( 99,721)(100,720)(101,722)(102,724)(103,723)(104,725)(105,718)
(106,717)(107,719)(108,712)(109,711)(110,713)(111,850)(112,849)(113,851)
(114,844)(115,843)(116,845)(117,865)(118,864)(119,866)(120,841)(121,840)
(122,842)(123,862)(124,861)(125,863)(126,856)(127,855)(128,857)(129,859)
(130,858)(131,860)(132,853)(133,852)(134,854)(135,847)(136,846)(137,848)
(138,823)(139,822)(140,824)(141,817)(142,816)(143,818)(144,838)(145,837)
(146,839)(147,814)(148,813)(149,815)(150,835)(151,834)(152,836)(153,829)
(154,828)(155,830)(156,832)(157,831)(158,833)(159,826)(160,825)(161,827)
(162,820)(163,819)(164,821)(165,796)(166,795)(167,797)(168,790)(169,789)
(170,791)(171,811)(172,810)(173,812)(174,787)(175,786)(176,788)(177,808)
(178,807)(179,809)(180,802)(181,801)(182,803)(183,805)(184,804)(185,806)
(186,799)(187,798)(188,800)(189,793)(190,792)(191,794)(192,769)(193,768)
(194,770)(195,763)(196,762)(197,764)(198,784)(199,783)(200,785)(201,760)
(202,759)(203,761)(204,781)(205,780)(206,782)(207,775)(208,774)(209,776)
(210,778)(211,777)(212,779)(213,772)(214,771)(215,773)(216,766)(217,765)
(218,767)(219,472)(220,471)(221,473)(222,466)(223,465)(224,467)(225,487)
(226,486)(227,488)(228,463)(229,462)(230,464)(231,484)(232,483)(233,485)
(234,478)(235,477)(236,479)(237,481)(238,480)(239,482)(240,475)(241,474)
(242,476)(243,469)(244,468)(245,470)(246,445)(247,444)(248,446)(249,439)
(250,438)(251,440)(252,460)(253,459)(254,461)(255,436)(256,435)(257,437)
(258,457)(259,456)(260,458)(261,451)(262,450)(263,452)(264,454)(265,453)
(266,455)(267,448)(268,447)(269,449)(270,442)(271,441)(272,443)(273,499)
(274,498)(275,500)(276,493)(277,492)(278,494)(279,514)(280,513)(281,515)
(282,490)(283,489)(284,491)(285,511)(286,510)(287,512)(288,505)(289,504)
(290,506)(291,508)(292,507)(293,509)(294,502)(295,501)(296,503)(297,496)
(298,495)(299,497)(300,526)(301,525)(302,527)(303,520)(304,519)(305,521)
(306,541)(307,540)(308,542)(309,517)(310,516)(311,518)(312,538)(313,537)
(314,539)(315,532)(316,531)(317,533)(318,535)(319,534)(320,536)(321,529)
(322,528)(323,530)(324,523)(325,522)(326,524)(327,607)(328,606)(329,608)
(330,601)(331,600)(332,602)(333,622)(334,621)(335,623)(336,598)(337,597)
(338,599)(339,619)(340,618)(341,620)(342,613)(343,612)(344,614)(345,616)
(346,615)(347,617)(348,610)(349,609)(350,611)(351,604)(352,603)(353,605)
(354,634)(355,633)(356,635)(357,628)(358,627)(359,629)(360,649)(361,648)
(362,650)(363,625)(364,624)(365,626)(366,646)(367,645)(368,647)(369,640)
(370,639)(371,641)(372,643)(373,642)(374,644)(375,637)(376,636)(377,638)
(378,631)(379,630)(380,632)(381,553)(382,552)(383,554)(384,547)(385,546)
(386,548)(387,568)(388,567)(389,569)(390,544)(391,543)(392,545)(393,565)
(394,564)(395,566)(396,559)(397,558)(398,560)(399,562)(400,561)(401,563)
(402,556)(403,555)(404,557)(405,550)(406,549)(407,551)(408,580)(409,579)
(410,581)(411,574)(412,573)(413,575)(414,595)(415,594)(416,596)(417,571)
(418,570)(419,572)(420,592)(421,591)(422,593)(423,586)(424,585)(425,587)
(426,589)(427,588)(428,590)(429,583)(430,582)(431,584)(432,577)(433,576)
(434,578);
s3 := Sym(866)!(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)(  9,117)
( 10,118)( 11,119)( 12,135)( 13,136)( 14,137)( 15,129)( 16,130)( 17,131)
( 18,132)( 19,133)( 20,134)( 21,123)( 22,124)( 23,125)( 24,126)( 25,127)
( 26,128)( 27,120)( 28,121)( 29,122)( 30,138)( 31,139)( 32,140)( 33,141)
( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,162)( 40,163)( 41,164)
( 42,156)( 43,157)( 44,158)( 45,159)( 46,160)( 47,161)( 48,150)( 49,151)
( 50,152)( 51,153)( 52,154)( 53,155)( 54,147)( 55,148)( 56,149)( 57,192)
( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)( 65,200)
( 66,216)( 67,217)( 68,218)( 69,210)( 70,211)( 71,212)( 72,213)( 73,214)
( 74,215)( 75,204)( 76,205)( 77,206)( 78,207)( 79,208)( 80,209)( 81,201)
( 82,202)( 83,203)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)( 89,170)
( 90,171)( 91,172)( 92,173)( 93,189)( 94,190)( 95,191)( 96,183)( 97,184)
( 98,185)( 99,186)(100,187)(101,188)(102,177)(103,178)(104,179)(105,180)
(106,181)(107,182)(108,174)(109,175)(110,176)(219,327)(220,328)(221,329)
(222,330)(223,331)(224,332)(225,333)(226,334)(227,335)(228,351)(229,352)
(230,353)(231,345)(232,346)(233,347)(234,348)(235,349)(236,350)(237,339)
(238,340)(239,341)(240,342)(241,343)(242,344)(243,336)(244,337)(245,338)
(246,354)(247,355)(248,356)(249,357)(250,358)(251,359)(252,360)(253,361)
(254,362)(255,378)(256,379)(257,380)(258,372)(259,373)(260,374)(261,375)
(262,376)(263,377)(264,366)(265,367)(266,368)(267,369)(268,370)(269,371)
(270,363)(271,364)(272,365)(273,408)(274,409)(275,410)(276,411)(277,412)
(278,413)(279,414)(280,415)(281,416)(282,432)(283,433)(284,434)(285,426)
(286,427)(287,428)(288,429)(289,430)(290,431)(291,420)(292,421)(293,422)
(294,423)(295,424)(296,425)(297,417)(298,418)(299,419)(300,381)(301,382)
(302,383)(303,384)(304,385)(305,386)(306,387)(307,388)(308,389)(309,405)
(310,406)(311,407)(312,399)(313,400)(314,401)(315,402)(316,403)(317,404)
(318,393)(319,394)(320,395)(321,396)(322,397)(323,398)(324,390)(325,391)
(326,392)(435,543)(436,544)(437,545)(438,546)(439,547)(440,548)(441,549)
(442,550)(443,551)(444,567)(445,568)(446,569)(447,561)(448,562)(449,563)
(450,564)(451,565)(452,566)(453,555)(454,556)(455,557)(456,558)(457,559)
(458,560)(459,552)(460,553)(461,554)(462,570)(463,571)(464,572)(465,573)
(466,574)(467,575)(468,576)(469,577)(470,578)(471,594)(472,595)(473,596)
(474,588)(475,589)(476,590)(477,591)(478,592)(479,593)(480,582)(481,583)
(482,584)(483,585)(484,586)(485,587)(486,579)(487,580)(488,581)(489,624)
(490,625)(491,626)(492,627)(493,628)(494,629)(495,630)(496,631)(497,632)
(498,648)(499,649)(500,650)(501,642)(502,643)(503,644)(504,645)(505,646)
(506,647)(507,636)(508,637)(509,638)(510,639)(511,640)(512,641)(513,633)
(514,634)(515,635)(516,597)(517,598)(518,599)(519,600)(520,601)(521,602)
(522,603)(523,604)(524,605)(525,621)(526,622)(527,623)(528,615)(529,616)
(530,617)(531,618)(532,619)(533,620)(534,609)(535,610)(536,611)(537,612)
(538,613)(539,614)(540,606)(541,607)(542,608)(651,759)(652,760)(653,761)
(654,762)(655,763)(656,764)(657,765)(658,766)(659,767)(660,783)(661,784)
(662,785)(663,777)(664,778)(665,779)(666,780)(667,781)(668,782)(669,771)
(670,772)(671,773)(672,774)(673,775)(674,776)(675,768)(676,769)(677,770)
(678,786)(679,787)(680,788)(681,789)(682,790)(683,791)(684,792)(685,793)
(686,794)(687,810)(688,811)(689,812)(690,804)(691,805)(692,806)(693,807)
(694,808)(695,809)(696,798)(697,799)(698,800)(699,801)(700,802)(701,803)
(702,795)(703,796)(704,797)(705,840)(706,841)(707,842)(708,843)(709,844)
(710,845)(711,846)(712,847)(713,848)(714,864)(715,865)(716,866)(717,858)
(718,859)(719,860)(720,861)(721,862)(722,863)(723,852)(724,853)(725,854)
(726,855)(727,856)(728,857)(729,849)(730,850)(731,851)(732,813)(733,814)
(734,815)(735,816)(736,817)(737,818)(738,819)(739,820)(740,821)(741,837)
(742,838)(743,839)(744,831)(745,832)(746,833)(747,834)(748,835)(749,836)
(750,825)(751,826)(752,827)(753,828)(754,829)(755,830)(756,822)(757,823)
(758,824);
poly := sub<Sym(866)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >; 
 

to this polytope