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Polytope of Type {24,6,2}

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

to this polytope