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Polytope of Type {6,8,2}

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

to this polytope