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Polytope of Type {2,3,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,24}*1728
if this polytope has a name.
Group : SmallGroup(1728,30267)
Rank : 4
Schlafli Type : {2,3,24}
Number of vertices, edges, etc : 2, 18, 216, 144
Order of s0s1s2s3 : 12
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 : {2,3,12}*864
   3-fold quotients : {2,3,24}*576
   6-fold quotients : {2,3,12}*288
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {2,3,8}*192
   18-fold quotients : {2,3,4}*96
   24-fold quotients : {2,3,6}*72
   36-fold quotients : {2,3,4}*48
   72-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3,651)(  4,652)(  5,656)(  6,655)(  7,654)(  8,653)(  9,658)( 10,657)
( 11,659)( 12,660)( 13,664)( 14,663)( 15,662)( 16,661)( 17,666)( 18,665)
( 19,667)( 20,668)( 21,672)( 22,671)( 23,670)( 24,669)( 25,674)( 26,673)
( 27,699)( 28,700)( 29,704)( 30,703)( 31,702)( 32,701)( 33,706)( 34,705)
( 35,707)( 36,708)( 37,712)( 38,711)( 39,710)( 40,709)( 41,714)( 42,713)
( 43,715)( 44,716)( 45,720)( 46,719)( 47,718)( 48,717)( 49,722)( 50,721)
( 51,675)( 52,676)( 53,680)( 54,679)( 55,678)( 56,677)( 57,682)( 58,681)
( 59,683)( 60,684)( 61,688)( 62,687)( 63,686)( 64,685)( 65,690)( 66,689)
( 67,691)( 68,692)( 69,696)( 70,695)( 71,694)( 72,693)( 73,698)( 74,697)
( 75,795)( 76,796)( 77,800)( 78,799)( 79,798)( 80,797)( 81,802)( 82,801)
( 83,803)( 84,804)( 85,808)( 86,807)( 87,806)( 88,805)( 89,810)( 90,809)
( 91,811)( 92,812)( 93,816)( 94,815)( 95,814)( 96,813)( 97,818)( 98,817)
( 99,843)(100,844)(101,848)(102,847)(103,846)(104,845)(105,850)(106,849)
(107,851)(108,852)(109,856)(110,855)(111,854)(112,853)(113,858)(114,857)
(115,859)(116,860)(117,864)(118,863)(119,862)(120,861)(121,866)(122,865)
(123,819)(124,820)(125,824)(126,823)(127,822)(128,821)(129,826)(130,825)
(131,827)(132,828)(133,832)(134,831)(135,830)(136,829)(137,834)(138,833)
(139,835)(140,836)(141,840)(142,839)(143,838)(144,837)(145,842)(146,841)
(147,723)(148,724)(149,728)(150,727)(151,726)(152,725)(153,730)(154,729)
(155,731)(156,732)(157,736)(158,735)(159,734)(160,733)(161,738)(162,737)
(163,739)(164,740)(165,744)(166,743)(167,742)(168,741)(169,746)(170,745)
(171,771)(172,772)(173,776)(174,775)(175,774)(176,773)(177,778)(178,777)
(179,779)(180,780)(181,784)(182,783)(183,782)(184,781)(185,786)(186,785)
(187,787)(188,788)(189,792)(190,791)(191,790)(192,789)(193,794)(194,793)
(195,747)(196,748)(197,752)(198,751)(199,750)(200,749)(201,754)(202,753)
(203,755)(204,756)(205,760)(206,759)(207,758)(208,757)(209,762)(210,761)
(211,763)(212,764)(213,768)(214,767)(215,766)(216,765)(217,770)(218,769)
(219,436)(220,435)(221,439)(222,440)(223,437)(224,438)(225,441)(226,442)
(227,444)(228,443)(229,447)(230,448)(231,445)(232,446)(233,449)(234,450)
(235,452)(236,451)(237,455)(238,456)(239,453)(240,454)(241,457)(242,458)
(243,484)(244,483)(245,487)(246,488)(247,485)(248,486)(249,489)(250,490)
(251,492)(252,491)(253,495)(254,496)(255,493)(256,494)(257,497)(258,498)
(259,500)(260,499)(261,503)(262,504)(263,501)(264,502)(265,505)(266,506)
(267,460)(268,459)(269,463)(270,464)(271,461)(272,462)(273,465)(274,466)
(275,468)(276,467)(277,471)(278,472)(279,469)(280,470)(281,473)(282,474)
(283,476)(284,475)(285,479)(286,480)(287,477)(288,478)(289,481)(290,482)
(291,580)(292,579)(293,583)(294,584)(295,581)(296,582)(297,585)(298,586)
(299,588)(300,587)(301,591)(302,592)(303,589)(304,590)(305,593)(306,594)
(307,596)(308,595)(309,599)(310,600)(311,597)(312,598)(313,601)(314,602)
(315,628)(316,627)(317,631)(318,632)(319,629)(320,630)(321,633)(322,634)
(323,636)(324,635)(325,639)(326,640)(327,637)(328,638)(329,641)(330,642)
(331,644)(332,643)(333,647)(334,648)(335,645)(336,646)(337,649)(338,650)
(339,604)(340,603)(341,607)(342,608)(343,605)(344,606)(345,609)(346,610)
(347,612)(348,611)(349,615)(350,616)(351,613)(352,614)(353,617)(354,618)
(355,620)(356,619)(357,623)(358,624)(359,621)(360,622)(361,625)(362,626)
(363,508)(364,507)(365,511)(366,512)(367,509)(368,510)(369,513)(370,514)
(371,516)(372,515)(373,519)(374,520)(375,517)(376,518)(377,521)(378,522)
(379,524)(380,523)(381,527)(382,528)(383,525)(384,526)(385,529)(386,530)
(387,556)(388,555)(389,559)(390,560)(391,557)(392,558)(393,561)(394,562)
(395,564)(396,563)(397,567)(398,568)(399,565)(400,566)(401,569)(402,570)
(403,572)(404,571)(405,575)(406,576)(407,573)(408,574)(409,577)(410,578)
(411,532)(412,531)(413,535)(414,536)(415,533)(416,534)(417,537)(418,538)
(419,540)(420,539)(421,543)(422,544)(423,541)(424,542)(425,545)(426,546)
(427,548)(428,547)(429,551)(430,552)(431,549)(432,550)(433,553)(434,554);;
s2 := (  3,827)(  4,828)(  5,833)(  6,834)(  7,832)(  8,831)(  9,829)( 10,830)
( 11,835)( 12,836)( 13,841)( 14,842)( 15,840)( 16,839)( 17,837)( 18,838)
( 19,819)( 20,820)( 21,825)( 22,826)( 23,824)( 24,823)( 25,821)( 26,822)
( 27,811)( 28,812)( 29,817)( 30,818)( 31,816)( 32,815)( 33,813)( 34,814)
( 35,795)( 36,796)( 37,801)( 38,802)( 39,800)( 40,799)( 41,797)( 42,798)
( 43,803)( 44,804)( 45,809)( 46,810)( 47,808)( 48,807)( 49,805)( 50,806)
( 51,843)( 52,844)( 53,849)( 54,850)( 55,848)( 56,847)( 57,845)( 58,846)
( 59,851)( 60,852)( 61,857)( 62,858)( 63,856)( 64,855)( 65,853)( 66,854)
( 67,859)( 68,860)( 69,865)( 70,866)( 71,864)( 72,863)( 73,861)( 74,862)
( 75,755)( 76,756)( 77,761)( 78,762)( 79,760)( 80,759)( 81,757)( 82,758)
( 83,763)( 84,764)( 85,769)( 86,770)( 87,768)( 88,767)( 89,765)( 90,766)
( 91,747)( 92,748)( 93,753)( 94,754)( 95,752)( 96,751)( 97,749)( 98,750)
( 99,739)(100,740)(101,745)(102,746)(103,744)(104,743)(105,741)(106,742)
(107,723)(108,724)(109,729)(110,730)(111,728)(112,727)(113,725)(114,726)
(115,731)(116,732)(117,737)(118,738)(119,736)(120,735)(121,733)(122,734)
(123,771)(124,772)(125,777)(126,778)(127,776)(128,775)(129,773)(130,774)
(131,779)(132,780)(133,785)(134,786)(135,784)(136,783)(137,781)(138,782)
(139,787)(140,788)(141,793)(142,794)(143,792)(144,791)(145,789)(146,790)
(147,683)(148,684)(149,689)(150,690)(151,688)(152,687)(153,685)(154,686)
(155,691)(156,692)(157,697)(158,698)(159,696)(160,695)(161,693)(162,694)
(163,675)(164,676)(165,681)(166,682)(167,680)(168,679)(169,677)(170,678)
(171,667)(172,668)(173,673)(174,674)(175,672)(176,671)(177,669)(178,670)
(179,651)(180,652)(181,657)(182,658)(183,656)(184,655)(185,653)(186,654)
(187,659)(188,660)(189,665)(190,666)(191,664)(192,663)(193,661)(194,662)
(195,699)(196,700)(197,705)(198,706)(199,704)(200,703)(201,701)(202,702)
(203,707)(204,708)(205,713)(206,714)(207,712)(208,711)(209,709)(210,710)
(211,715)(212,716)(213,721)(214,722)(215,720)(216,719)(217,717)(218,718)
(219,612)(220,611)(221,618)(222,617)(223,615)(224,616)(225,614)(226,613)
(227,620)(228,619)(229,626)(230,625)(231,623)(232,624)(233,622)(234,621)
(235,604)(236,603)(237,610)(238,609)(239,607)(240,608)(241,606)(242,605)
(243,596)(244,595)(245,602)(246,601)(247,599)(248,600)(249,598)(250,597)
(251,580)(252,579)(253,586)(254,585)(255,583)(256,584)(257,582)(258,581)
(259,588)(260,587)(261,594)(262,593)(263,591)(264,592)(265,590)(266,589)
(267,628)(268,627)(269,634)(270,633)(271,631)(272,632)(273,630)(274,629)
(275,636)(276,635)(277,642)(278,641)(279,639)(280,640)(281,638)(282,637)
(283,644)(284,643)(285,650)(286,649)(287,647)(288,648)(289,646)(290,645)
(291,540)(292,539)(293,546)(294,545)(295,543)(296,544)(297,542)(298,541)
(299,548)(300,547)(301,554)(302,553)(303,551)(304,552)(305,550)(306,549)
(307,532)(308,531)(309,538)(310,537)(311,535)(312,536)(313,534)(314,533)
(315,524)(316,523)(317,530)(318,529)(319,527)(320,528)(321,526)(322,525)
(323,508)(324,507)(325,514)(326,513)(327,511)(328,512)(329,510)(330,509)
(331,516)(332,515)(333,522)(334,521)(335,519)(336,520)(337,518)(338,517)
(339,556)(340,555)(341,562)(342,561)(343,559)(344,560)(345,558)(346,557)
(347,564)(348,563)(349,570)(350,569)(351,567)(352,568)(353,566)(354,565)
(355,572)(356,571)(357,578)(358,577)(359,575)(360,576)(361,574)(362,573)
(363,468)(364,467)(365,474)(366,473)(367,471)(368,472)(369,470)(370,469)
(371,476)(372,475)(373,482)(374,481)(375,479)(376,480)(377,478)(378,477)
(379,460)(380,459)(381,466)(382,465)(383,463)(384,464)(385,462)(386,461)
(387,452)(388,451)(389,458)(390,457)(391,455)(392,456)(393,454)(394,453)
(395,436)(396,435)(397,442)(398,441)(399,439)(400,440)(401,438)(402,437)
(403,444)(404,443)(405,450)(406,449)(407,447)(408,448)(409,446)(410,445)
(411,484)(412,483)(413,490)(414,489)(415,487)(416,488)(417,486)(418,485)
(419,492)(420,491)(421,498)(422,497)(423,495)(424,496)(425,494)(426,493)
(427,500)(428,499)(429,506)(430,505)(431,503)(432,504)(433,502)(434,501);;
s3 := (  3,441)(  4,442)(  5,439)(  6,440)(  7,438)(  8,437)(  9,436)( 10,435)
( 11,457)( 12,458)( 13,455)( 14,456)( 15,454)( 16,453)( 17,452)( 18,451)
( 19,449)( 20,450)( 21,447)( 22,448)( 23,446)( 24,445)( 25,444)( 26,443)
( 27,489)( 28,490)( 29,487)( 30,488)( 31,486)( 32,485)( 33,484)( 34,483)
( 35,505)( 36,506)( 37,503)( 38,504)( 39,502)( 40,501)( 41,500)( 42,499)
( 43,497)( 44,498)( 45,495)( 46,496)( 47,494)( 48,493)( 49,492)( 50,491)
( 51,465)( 52,466)( 53,463)( 54,464)( 55,462)( 56,461)( 57,460)( 58,459)
( 59,481)( 60,482)( 61,479)( 62,480)( 63,478)( 64,477)( 65,476)( 66,475)
( 67,473)( 68,474)( 69,471)( 70,472)( 71,470)( 72,469)( 73,468)( 74,467)
( 75,513)( 76,514)( 77,511)( 78,512)( 79,510)( 80,509)( 81,508)( 82,507)
( 83,529)( 84,530)( 85,527)( 86,528)( 87,526)( 88,525)( 89,524)( 90,523)
( 91,521)( 92,522)( 93,519)( 94,520)( 95,518)( 96,517)( 97,516)( 98,515)
( 99,561)(100,562)(101,559)(102,560)(103,558)(104,557)(105,556)(106,555)
(107,577)(108,578)(109,575)(110,576)(111,574)(112,573)(113,572)(114,571)
(115,569)(116,570)(117,567)(118,568)(119,566)(120,565)(121,564)(122,563)
(123,537)(124,538)(125,535)(126,536)(127,534)(128,533)(129,532)(130,531)
(131,553)(132,554)(133,551)(134,552)(135,550)(136,549)(137,548)(138,547)
(139,545)(140,546)(141,543)(142,544)(143,542)(144,541)(145,540)(146,539)
(147,585)(148,586)(149,583)(150,584)(151,582)(152,581)(153,580)(154,579)
(155,601)(156,602)(157,599)(158,600)(159,598)(160,597)(161,596)(162,595)
(163,593)(164,594)(165,591)(166,592)(167,590)(168,589)(169,588)(170,587)
(171,633)(172,634)(173,631)(174,632)(175,630)(176,629)(177,628)(178,627)
(179,649)(180,650)(181,647)(182,648)(183,646)(184,645)(185,644)(186,643)
(187,641)(188,642)(189,639)(190,640)(191,638)(192,637)(193,636)(194,635)
(195,609)(196,610)(197,607)(198,608)(199,606)(200,605)(201,604)(202,603)
(203,625)(204,626)(205,623)(206,624)(207,622)(208,621)(209,620)(210,619)
(211,617)(212,618)(213,615)(214,616)(215,614)(216,613)(217,612)(218,611)
(219,658)(220,657)(221,656)(222,655)(223,653)(224,654)(225,651)(226,652)
(227,674)(228,673)(229,672)(230,671)(231,669)(232,670)(233,667)(234,668)
(235,666)(236,665)(237,664)(238,663)(239,661)(240,662)(241,659)(242,660)
(243,706)(244,705)(245,704)(246,703)(247,701)(248,702)(249,699)(250,700)
(251,722)(252,721)(253,720)(254,719)(255,717)(256,718)(257,715)(258,716)
(259,714)(260,713)(261,712)(262,711)(263,709)(264,710)(265,707)(266,708)
(267,682)(268,681)(269,680)(270,679)(271,677)(272,678)(273,675)(274,676)
(275,698)(276,697)(277,696)(278,695)(279,693)(280,694)(281,691)(282,692)
(283,690)(284,689)(285,688)(286,687)(287,685)(288,686)(289,683)(290,684)
(291,730)(292,729)(293,728)(294,727)(295,725)(296,726)(297,723)(298,724)
(299,746)(300,745)(301,744)(302,743)(303,741)(304,742)(305,739)(306,740)
(307,738)(308,737)(309,736)(310,735)(311,733)(312,734)(313,731)(314,732)
(315,778)(316,777)(317,776)(318,775)(319,773)(320,774)(321,771)(322,772)
(323,794)(324,793)(325,792)(326,791)(327,789)(328,790)(329,787)(330,788)
(331,786)(332,785)(333,784)(334,783)(335,781)(336,782)(337,779)(338,780)
(339,754)(340,753)(341,752)(342,751)(343,749)(344,750)(345,747)(346,748)
(347,770)(348,769)(349,768)(350,767)(351,765)(352,766)(353,763)(354,764)
(355,762)(356,761)(357,760)(358,759)(359,757)(360,758)(361,755)(362,756)
(363,802)(364,801)(365,800)(366,799)(367,797)(368,798)(369,795)(370,796)
(371,818)(372,817)(373,816)(374,815)(375,813)(376,814)(377,811)(378,812)
(379,810)(380,809)(381,808)(382,807)(383,805)(384,806)(385,803)(386,804)
(387,850)(388,849)(389,848)(390,847)(391,845)(392,846)(393,843)(394,844)
(395,866)(396,865)(397,864)(398,863)(399,861)(400,862)(401,859)(402,860)
(403,858)(404,857)(405,856)(406,855)(407,853)(408,854)(409,851)(410,852)
(411,826)(412,825)(413,824)(414,823)(415,821)(416,822)(417,819)(418,820)
(419,842)(420,841)(421,840)(422,839)(423,837)(424,838)(425,835)(426,836)
(427,834)(428,833)(429,832)(430,831)(431,829)(432,830)(433,827)(434,828);;
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, s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(866)!(1,2);
s1 := Sym(866)!(  3,651)(  4,652)(  5,656)(  6,655)(  7,654)(  8,653)(  9,658)
( 10,657)( 11,659)( 12,660)( 13,664)( 14,663)( 15,662)( 16,661)( 17,666)
( 18,665)( 19,667)( 20,668)( 21,672)( 22,671)( 23,670)( 24,669)( 25,674)
( 26,673)( 27,699)( 28,700)( 29,704)( 30,703)( 31,702)( 32,701)( 33,706)
( 34,705)( 35,707)( 36,708)( 37,712)( 38,711)( 39,710)( 40,709)( 41,714)
( 42,713)( 43,715)( 44,716)( 45,720)( 46,719)( 47,718)( 48,717)( 49,722)
( 50,721)( 51,675)( 52,676)( 53,680)( 54,679)( 55,678)( 56,677)( 57,682)
( 58,681)( 59,683)( 60,684)( 61,688)( 62,687)( 63,686)( 64,685)( 65,690)
( 66,689)( 67,691)( 68,692)( 69,696)( 70,695)( 71,694)( 72,693)( 73,698)
( 74,697)( 75,795)( 76,796)( 77,800)( 78,799)( 79,798)( 80,797)( 81,802)
( 82,801)( 83,803)( 84,804)( 85,808)( 86,807)( 87,806)( 88,805)( 89,810)
( 90,809)( 91,811)( 92,812)( 93,816)( 94,815)( 95,814)( 96,813)( 97,818)
( 98,817)( 99,843)(100,844)(101,848)(102,847)(103,846)(104,845)(105,850)
(106,849)(107,851)(108,852)(109,856)(110,855)(111,854)(112,853)(113,858)
(114,857)(115,859)(116,860)(117,864)(118,863)(119,862)(120,861)(121,866)
(122,865)(123,819)(124,820)(125,824)(126,823)(127,822)(128,821)(129,826)
(130,825)(131,827)(132,828)(133,832)(134,831)(135,830)(136,829)(137,834)
(138,833)(139,835)(140,836)(141,840)(142,839)(143,838)(144,837)(145,842)
(146,841)(147,723)(148,724)(149,728)(150,727)(151,726)(152,725)(153,730)
(154,729)(155,731)(156,732)(157,736)(158,735)(159,734)(160,733)(161,738)
(162,737)(163,739)(164,740)(165,744)(166,743)(167,742)(168,741)(169,746)
(170,745)(171,771)(172,772)(173,776)(174,775)(175,774)(176,773)(177,778)
(178,777)(179,779)(180,780)(181,784)(182,783)(183,782)(184,781)(185,786)
(186,785)(187,787)(188,788)(189,792)(190,791)(191,790)(192,789)(193,794)
(194,793)(195,747)(196,748)(197,752)(198,751)(199,750)(200,749)(201,754)
(202,753)(203,755)(204,756)(205,760)(206,759)(207,758)(208,757)(209,762)
(210,761)(211,763)(212,764)(213,768)(214,767)(215,766)(216,765)(217,770)
(218,769)(219,436)(220,435)(221,439)(222,440)(223,437)(224,438)(225,441)
(226,442)(227,444)(228,443)(229,447)(230,448)(231,445)(232,446)(233,449)
(234,450)(235,452)(236,451)(237,455)(238,456)(239,453)(240,454)(241,457)
(242,458)(243,484)(244,483)(245,487)(246,488)(247,485)(248,486)(249,489)
(250,490)(251,492)(252,491)(253,495)(254,496)(255,493)(256,494)(257,497)
(258,498)(259,500)(260,499)(261,503)(262,504)(263,501)(264,502)(265,505)
(266,506)(267,460)(268,459)(269,463)(270,464)(271,461)(272,462)(273,465)
(274,466)(275,468)(276,467)(277,471)(278,472)(279,469)(280,470)(281,473)
(282,474)(283,476)(284,475)(285,479)(286,480)(287,477)(288,478)(289,481)
(290,482)(291,580)(292,579)(293,583)(294,584)(295,581)(296,582)(297,585)
(298,586)(299,588)(300,587)(301,591)(302,592)(303,589)(304,590)(305,593)
(306,594)(307,596)(308,595)(309,599)(310,600)(311,597)(312,598)(313,601)
(314,602)(315,628)(316,627)(317,631)(318,632)(319,629)(320,630)(321,633)
(322,634)(323,636)(324,635)(325,639)(326,640)(327,637)(328,638)(329,641)
(330,642)(331,644)(332,643)(333,647)(334,648)(335,645)(336,646)(337,649)
(338,650)(339,604)(340,603)(341,607)(342,608)(343,605)(344,606)(345,609)
(346,610)(347,612)(348,611)(349,615)(350,616)(351,613)(352,614)(353,617)
(354,618)(355,620)(356,619)(357,623)(358,624)(359,621)(360,622)(361,625)
(362,626)(363,508)(364,507)(365,511)(366,512)(367,509)(368,510)(369,513)
(370,514)(371,516)(372,515)(373,519)(374,520)(375,517)(376,518)(377,521)
(378,522)(379,524)(380,523)(381,527)(382,528)(383,525)(384,526)(385,529)
(386,530)(387,556)(388,555)(389,559)(390,560)(391,557)(392,558)(393,561)
(394,562)(395,564)(396,563)(397,567)(398,568)(399,565)(400,566)(401,569)
(402,570)(403,572)(404,571)(405,575)(406,576)(407,573)(408,574)(409,577)
(410,578)(411,532)(412,531)(413,535)(414,536)(415,533)(416,534)(417,537)
(418,538)(419,540)(420,539)(421,543)(422,544)(423,541)(424,542)(425,545)
(426,546)(427,548)(428,547)(429,551)(430,552)(431,549)(432,550)(433,553)
(434,554);
s2 := Sym(866)!(  3,827)(  4,828)(  5,833)(  6,834)(  7,832)(  8,831)(  9,829)
( 10,830)( 11,835)( 12,836)( 13,841)( 14,842)( 15,840)( 16,839)( 17,837)
( 18,838)( 19,819)( 20,820)( 21,825)( 22,826)( 23,824)( 24,823)( 25,821)
( 26,822)( 27,811)( 28,812)( 29,817)( 30,818)( 31,816)( 32,815)( 33,813)
( 34,814)( 35,795)( 36,796)( 37,801)( 38,802)( 39,800)( 40,799)( 41,797)
( 42,798)( 43,803)( 44,804)( 45,809)( 46,810)( 47,808)( 48,807)( 49,805)
( 50,806)( 51,843)( 52,844)( 53,849)( 54,850)( 55,848)( 56,847)( 57,845)
( 58,846)( 59,851)( 60,852)( 61,857)( 62,858)( 63,856)( 64,855)( 65,853)
( 66,854)( 67,859)( 68,860)( 69,865)( 70,866)( 71,864)( 72,863)( 73,861)
( 74,862)( 75,755)( 76,756)( 77,761)( 78,762)( 79,760)( 80,759)( 81,757)
( 82,758)( 83,763)( 84,764)( 85,769)( 86,770)( 87,768)( 88,767)( 89,765)
( 90,766)( 91,747)( 92,748)( 93,753)( 94,754)( 95,752)( 96,751)( 97,749)
( 98,750)( 99,739)(100,740)(101,745)(102,746)(103,744)(104,743)(105,741)
(106,742)(107,723)(108,724)(109,729)(110,730)(111,728)(112,727)(113,725)
(114,726)(115,731)(116,732)(117,737)(118,738)(119,736)(120,735)(121,733)
(122,734)(123,771)(124,772)(125,777)(126,778)(127,776)(128,775)(129,773)
(130,774)(131,779)(132,780)(133,785)(134,786)(135,784)(136,783)(137,781)
(138,782)(139,787)(140,788)(141,793)(142,794)(143,792)(144,791)(145,789)
(146,790)(147,683)(148,684)(149,689)(150,690)(151,688)(152,687)(153,685)
(154,686)(155,691)(156,692)(157,697)(158,698)(159,696)(160,695)(161,693)
(162,694)(163,675)(164,676)(165,681)(166,682)(167,680)(168,679)(169,677)
(170,678)(171,667)(172,668)(173,673)(174,674)(175,672)(176,671)(177,669)
(178,670)(179,651)(180,652)(181,657)(182,658)(183,656)(184,655)(185,653)
(186,654)(187,659)(188,660)(189,665)(190,666)(191,664)(192,663)(193,661)
(194,662)(195,699)(196,700)(197,705)(198,706)(199,704)(200,703)(201,701)
(202,702)(203,707)(204,708)(205,713)(206,714)(207,712)(208,711)(209,709)
(210,710)(211,715)(212,716)(213,721)(214,722)(215,720)(216,719)(217,717)
(218,718)(219,612)(220,611)(221,618)(222,617)(223,615)(224,616)(225,614)
(226,613)(227,620)(228,619)(229,626)(230,625)(231,623)(232,624)(233,622)
(234,621)(235,604)(236,603)(237,610)(238,609)(239,607)(240,608)(241,606)
(242,605)(243,596)(244,595)(245,602)(246,601)(247,599)(248,600)(249,598)
(250,597)(251,580)(252,579)(253,586)(254,585)(255,583)(256,584)(257,582)
(258,581)(259,588)(260,587)(261,594)(262,593)(263,591)(264,592)(265,590)
(266,589)(267,628)(268,627)(269,634)(270,633)(271,631)(272,632)(273,630)
(274,629)(275,636)(276,635)(277,642)(278,641)(279,639)(280,640)(281,638)
(282,637)(283,644)(284,643)(285,650)(286,649)(287,647)(288,648)(289,646)
(290,645)(291,540)(292,539)(293,546)(294,545)(295,543)(296,544)(297,542)
(298,541)(299,548)(300,547)(301,554)(302,553)(303,551)(304,552)(305,550)
(306,549)(307,532)(308,531)(309,538)(310,537)(311,535)(312,536)(313,534)
(314,533)(315,524)(316,523)(317,530)(318,529)(319,527)(320,528)(321,526)
(322,525)(323,508)(324,507)(325,514)(326,513)(327,511)(328,512)(329,510)
(330,509)(331,516)(332,515)(333,522)(334,521)(335,519)(336,520)(337,518)
(338,517)(339,556)(340,555)(341,562)(342,561)(343,559)(344,560)(345,558)
(346,557)(347,564)(348,563)(349,570)(350,569)(351,567)(352,568)(353,566)
(354,565)(355,572)(356,571)(357,578)(358,577)(359,575)(360,576)(361,574)
(362,573)(363,468)(364,467)(365,474)(366,473)(367,471)(368,472)(369,470)
(370,469)(371,476)(372,475)(373,482)(374,481)(375,479)(376,480)(377,478)
(378,477)(379,460)(380,459)(381,466)(382,465)(383,463)(384,464)(385,462)
(386,461)(387,452)(388,451)(389,458)(390,457)(391,455)(392,456)(393,454)
(394,453)(395,436)(396,435)(397,442)(398,441)(399,439)(400,440)(401,438)
(402,437)(403,444)(404,443)(405,450)(406,449)(407,447)(408,448)(409,446)
(410,445)(411,484)(412,483)(413,490)(414,489)(415,487)(416,488)(417,486)
(418,485)(419,492)(420,491)(421,498)(422,497)(423,495)(424,496)(425,494)
(426,493)(427,500)(428,499)(429,506)(430,505)(431,503)(432,504)(433,502)
(434,501);
s3 := Sym(866)!(  3,441)(  4,442)(  5,439)(  6,440)(  7,438)(  8,437)(  9,436)
( 10,435)( 11,457)( 12,458)( 13,455)( 14,456)( 15,454)( 16,453)( 17,452)
( 18,451)( 19,449)( 20,450)( 21,447)( 22,448)( 23,446)( 24,445)( 25,444)
( 26,443)( 27,489)( 28,490)( 29,487)( 30,488)( 31,486)( 32,485)( 33,484)
( 34,483)( 35,505)( 36,506)( 37,503)( 38,504)( 39,502)( 40,501)( 41,500)
( 42,499)( 43,497)( 44,498)( 45,495)( 46,496)( 47,494)( 48,493)( 49,492)
( 50,491)( 51,465)( 52,466)( 53,463)( 54,464)( 55,462)( 56,461)( 57,460)
( 58,459)( 59,481)( 60,482)( 61,479)( 62,480)( 63,478)( 64,477)( 65,476)
( 66,475)( 67,473)( 68,474)( 69,471)( 70,472)( 71,470)( 72,469)( 73,468)
( 74,467)( 75,513)( 76,514)( 77,511)( 78,512)( 79,510)( 80,509)( 81,508)
( 82,507)( 83,529)( 84,530)( 85,527)( 86,528)( 87,526)( 88,525)( 89,524)
( 90,523)( 91,521)( 92,522)( 93,519)( 94,520)( 95,518)( 96,517)( 97,516)
( 98,515)( 99,561)(100,562)(101,559)(102,560)(103,558)(104,557)(105,556)
(106,555)(107,577)(108,578)(109,575)(110,576)(111,574)(112,573)(113,572)
(114,571)(115,569)(116,570)(117,567)(118,568)(119,566)(120,565)(121,564)
(122,563)(123,537)(124,538)(125,535)(126,536)(127,534)(128,533)(129,532)
(130,531)(131,553)(132,554)(133,551)(134,552)(135,550)(136,549)(137,548)
(138,547)(139,545)(140,546)(141,543)(142,544)(143,542)(144,541)(145,540)
(146,539)(147,585)(148,586)(149,583)(150,584)(151,582)(152,581)(153,580)
(154,579)(155,601)(156,602)(157,599)(158,600)(159,598)(160,597)(161,596)
(162,595)(163,593)(164,594)(165,591)(166,592)(167,590)(168,589)(169,588)
(170,587)(171,633)(172,634)(173,631)(174,632)(175,630)(176,629)(177,628)
(178,627)(179,649)(180,650)(181,647)(182,648)(183,646)(184,645)(185,644)
(186,643)(187,641)(188,642)(189,639)(190,640)(191,638)(192,637)(193,636)
(194,635)(195,609)(196,610)(197,607)(198,608)(199,606)(200,605)(201,604)
(202,603)(203,625)(204,626)(205,623)(206,624)(207,622)(208,621)(209,620)
(210,619)(211,617)(212,618)(213,615)(214,616)(215,614)(216,613)(217,612)
(218,611)(219,658)(220,657)(221,656)(222,655)(223,653)(224,654)(225,651)
(226,652)(227,674)(228,673)(229,672)(230,671)(231,669)(232,670)(233,667)
(234,668)(235,666)(236,665)(237,664)(238,663)(239,661)(240,662)(241,659)
(242,660)(243,706)(244,705)(245,704)(246,703)(247,701)(248,702)(249,699)
(250,700)(251,722)(252,721)(253,720)(254,719)(255,717)(256,718)(257,715)
(258,716)(259,714)(260,713)(261,712)(262,711)(263,709)(264,710)(265,707)
(266,708)(267,682)(268,681)(269,680)(270,679)(271,677)(272,678)(273,675)
(274,676)(275,698)(276,697)(277,696)(278,695)(279,693)(280,694)(281,691)
(282,692)(283,690)(284,689)(285,688)(286,687)(287,685)(288,686)(289,683)
(290,684)(291,730)(292,729)(293,728)(294,727)(295,725)(296,726)(297,723)
(298,724)(299,746)(300,745)(301,744)(302,743)(303,741)(304,742)(305,739)
(306,740)(307,738)(308,737)(309,736)(310,735)(311,733)(312,734)(313,731)
(314,732)(315,778)(316,777)(317,776)(318,775)(319,773)(320,774)(321,771)
(322,772)(323,794)(324,793)(325,792)(326,791)(327,789)(328,790)(329,787)
(330,788)(331,786)(332,785)(333,784)(334,783)(335,781)(336,782)(337,779)
(338,780)(339,754)(340,753)(341,752)(342,751)(343,749)(344,750)(345,747)
(346,748)(347,770)(348,769)(349,768)(350,767)(351,765)(352,766)(353,763)
(354,764)(355,762)(356,761)(357,760)(358,759)(359,757)(360,758)(361,755)
(362,756)(363,802)(364,801)(365,800)(366,799)(367,797)(368,798)(369,795)
(370,796)(371,818)(372,817)(373,816)(374,815)(375,813)(376,814)(377,811)
(378,812)(379,810)(380,809)(381,808)(382,807)(383,805)(384,806)(385,803)
(386,804)(387,850)(388,849)(389,848)(390,847)(391,845)(392,846)(393,843)
(394,844)(395,866)(396,865)(397,864)(398,863)(399,861)(400,862)(401,859)
(402,860)(403,858)(404,857)(405,856)(406,855)(407,853)(408,854)(409,851)
(410,852)(411,826)(412,825)(413,824)(414,823)(415,821)(416,822)(417,819)
(418,820)(419,842)(420,841)(421,840)(422,839)(423,837)(424,838)(425,835)
(426,836)(427,834)(428,833)(429,832)(430,831)(431,829)(432,830)(433,827)
(434,828);
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, s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2 >; 
 

to this polytope