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Polytope of Type {2,24,3}

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

to this polytope