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Polytope of Type {2,9,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,30184)
Rank : 4
Schlafli Type : {2,9,12}
Number of vertices, edges, etc : 2, 36, 216, 48
Order of s0s1s2s3 : 72
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 : {2,9,6}*864
   3-fold quotients : {2,3,12}*576
   6-fold quotients : {2,3,6}*288
   8-fold quotients : {2,9,6}*216
   9-fold quotients : {2,3,12}*192
   18-fold quotients : {2,3,6}*96
   24-fold quotients : {2,9,2}*72, {2,3,6}*72
   36-fold quotients : {2,3,3}*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)
( 11,667)( 12,668)( 13,672)( 14,671)( 15,670)( 16,669)( 17,674)( 18,673)
( 19,659)( 20,660)( 21,664)( 22,663)( 23,662)( 24,661)( 25,666)( 26,665)
( 27,715)( 28,716)( 29,720)( 30,719)( 31,718)( 32,717)( 33,722)( 34,721)
( 35,707)( 36,708)( 37,712)( 38,711)( 39,710)( 40,709)( 41,714)( 42,713)
( 43,699)( 44,700)( 45,704)( 46,703)( 47,702)( 48,701)( 49,706)( 50,705)
( 51,691)( 52,692)( 53,696)( 54,695)( 55,694)( 56,693)( 57,698)( 58,697)
( 59,683)( 60,684)( 61,688)( 62,687)( 63,686)( 64,685)( 65,690)( 66,689)
( 67,675)( 68,676)( 69,680)( 70,679)( 71,678)( 72,677)( 73,682)( 74,681)
( 75,795)( 76,796)( 77,800)( 78,799)( 79,798)( 80,797)( 81,802)( 82,801)
( 83,811)( 84,812)( 85,816)( 86,815)( 87,814)( 88,813)( 89,818)( 90,817)
( 91,803)( 92,804)( 93,808)( 94,807)( 95,806)( 96,805)( 97,810)( 98,809)
( 99,859)(100,860)(101,864)(102,863)(103,862)(104,861)(105,866)(106,865)
(107,851)(108,852)(109,856)(110,855)(111,854)(112,853)(113,858)(114,857)
(115,843)(116,844)(117,848)(118,847)(119,846)(120,845)(121,850)(122,849)
(123,835)(124,836)(125,840)(126,839)(127,838)(128,837)(129,842)(130,841)
(131,827)(132,828)(133,832)(134,831)(135,830)(136,829)(137,834)(138,833)
(139,819)(140,820)(141,824)(142,823)(143,822)(144,821)(145,826)(146,825)
(147,723)(148,724)(149,728)(150,727)(151,726)(152,725)(153,730)(154,729)
(155,739)(156,740)(157,744)(158,743)(159,742)(160,741)(161,746)(162,745)
(163,731)(164,732)(165,736)(166,735)(167,734)(168,733)(169,738)(170,737)
(171,787)(172,788)(173,792)(174,791)(175,790)(176,789)(177,794)(178,793)
(179,779)(180,780)(181,784)(182,783)(183,782)(184,781)(185,786)(186,785)
(187,771)(188,772)(189,776)(190,775)(191,774)(192,773)(193,778)(194,777)
(195,763)(196,764)(197,768)(198,767)(199,766)(200,765)(201,770)(202,769)
(203,755)(204,756)(205,760)(206,759)(207,758)(208,757)(209,762)(210,761)
(211,747)(212,748)(213,752)(214,751)(215,750)(216,749)(217,754)(218,753)
(219,436)(220,435)(221,439)(222,440)(223,437)(224,438)(225,441)(226,442)
(227,452)(228,451)(229,455)(230,456)(231,453)(232,454)(233,457)(234,458)
(235,444)(236,443)(237,447)(238,448)(239,445)(240,446)(241,449)(242,450)
(243,500)(244,499)(245,503)(246,504)(247,501)(248,502)(249,505)(250,506)
(251,492)(252,491)(253,495)(254,496)(255,493)(256,494)(257,497)(258,498)
(259,484)(260,483)(261,487)(262,488)(263,485)(264,486)(265,489)(266,490)
(267,476)(268,475)(269,479)(270,480)(271,477)(272,478)(273,481)(274,482)
(275,468)(276,467)(277,471)(278,472)(279,469)(280,470)(281,473)(282,474)
(283,460)(284,459)(285,463)(286,464)(287,461)(288,462)(289,465)(290,466)
(291,580)(292,579)(293,583)(294,584)(295,581)(296,582)(297,585)(298,586)
(299,596)(300,595)(301,599)(302,600)(303,597)(304,598)(305,601)(306,602)
(307,588)(308,587)(309,591)(310,592)(311,589)(312,590)(313,593)(314,594)
(315,644)(316,643)(317,647)(318,648)(319,645)(320,646)(321,649)(322,650)
(323,636)(324,635)(325,639)(326,640)(327,637)(328,638)(329,641)(330,642)
(331,628)(332,627)(333,631)(334,632)(335,629)(336,630)(337,633)(338,634)
(339,620)(340,619)(341,623)(342,624)(343,621)(344,622)(345,625)(346,626)
(347,612)(348,611)(349,615)(350,616)(351,613)(352,614)(353,617)(354,618)
(355,604)(356,603)(357,607)(358,608)(359,605)(360,606)(361,609)(362,610)
(363,508)(364,507)(365,511)(366,512)(367,509)(368,510)(369,513)(370,514)
(371,524)(372,523)(373,527)(374,528)(375,525)(376,526)(377,529)(378,530)
(379,516)(380,515)(381,519)(382,520)(383,517)(384,518)(385,521)(386,522)
(387,572)(388,571)(389,575)(390,576)(391,573)(392,574)(393,577)(394,578)
(395,564)(396,563)(397,567)(398,568)(399,565)(400,566)(401,569)(402,570)
(403,556)(404,555)(405,559)(406,560)(407,557)(408,558)(409,561)(410,562)
(411,548)(412,547)(413,551)(414,552)(415,549)(416,550)(417,553)(418,554)
(419,540)(420,539)(421,543)(422,544)(423,541)(424,542)(425,545)(426,546)
(427,532)(428,531)(429,535)(430,536)(431,533)(432,534)(433,537)(434,538);;
s2 := (  3,747)(  4,748)(  5,750)(  6,749)(  7,753)(  8,754)(  9,751)( 10,752)
( 11,763)( 12,764)( 13,766)( 14,765)( 15,769)( 16,770)( 17,767)( 18,768)
( 19,755)( 20,756)( 21,758)( 22,757)( 23,761)( 24,762)( 25,759)( 26,760)
( 27,723)( 28,724)( 29,726)( 30,725)( 31,729)( 32,730)( 33,727)( 34,728)
( 35,739)( 36,740)( 37,742)( 38,741)( 39,745)( 40,746)( 41,743)( 42,744)
( 43,731)( 44,732)( 45,734)( 46,733)( 47,737)( 48,738)( 49,735)( 50,736)
( 51,787)( 52,788)( 53,790)( 54,789)( 55,793)( 56,794)( 57,791)( 58,792)
( 59,779)( 60,780)( 61,782)( 62,781)( 63,785)( 64,786)( 65,783)( 66,784)
( 67,771)( 68,772)( 69,774)( 70,773)( 71,777)( 72,778)( 73,775)( 74,776)
( 75,675)( 76,676)( 77,678)( 78,677)( 79,681)( 80,682)( 81,679)( 82,680)
( 83,691)( 84,692)( 85,694)( 86,693)( 87,697)( 88,698)( 89,695)( 90,696)
( 91,683)( 92,684)( 93,686)( 94,685)( 95,689)( 96,690)( 97,687)( 98,688)
( 99,651)(100,652)(101,654)(102,653)(103,657)(104,658)(105,655)(106,656)
(107,667)(108,668)(109,670)(110,669)(111,673)(112,674)(113,671)(114,672)
(115,659)(116,660)(117,662)(118,661)(119,665)(120,666)(121,663)(122,664)
(123,715)(124,716)(125,718)(126,717)(127,721)(128,722)(129,719)(130,720)
(131,707)(132,708)(133,710)(134,709)(135,713)(136,714)(137,711)(138,712)
(139,699)(140,700)(141,702)(142,701)(143,705)(144,706)(145,703)(146,704)
(147,819)(148,820)(149,822)(150,821)(151,825)(152,826)(153,823)(154,824)
(155,835)(156,836)(157,838)(158,837)(159,841)(160,842)(161,839)(162,840)
(163,827)(164,828)(165,830)(166,829)(167,833)(168,834)(169,831)(170,832)
(171,795)(172,796)(173,798)(174,797)(175,801)(176,802)(177,799)(178,800)
(179,811)(180,812)(181,814)(182,813)(183,817)(184,818)(185,815)(186,816)
(187,803)(188,804)(189,806)(190,805)(191,809)(192,810)(193,807)(194,808)
(195,859)(196,860)(197,862)(198,861)(199,865)(200,866)(201,863)(202,864)
(203,851)(204,852)(205,854)(206,853)(207,857)(208,858)(209,855)(210,856)
(211,843)(212,844)(213,846)(214,845)(215,849)(216,850)(217,847)(218,848)
(219,532)(220,531)(221,533)(222,534)(223,538)(224,537)(225,536)(226,535)
(227,548)(228,547)(229,549)(230,550)(231,554)(232,553)(233,552)(234,551)
(235,540)(236,539)(237,541)(238,542)(239,546)(240,545)(241,544)(242,543)
(243,508)(244,507)(245,509)(246,510)(247,514)(248,513)(249,512)(250,511)
(251,524)(252,523)(253,525)(254,526)(255,530)(256,529)(257,528)(258,527)
(259,516)(260,515)(261,517)(262,518)(263,522)(264,521)(265,520)(266,519)
(267,572)(268,571)(269,573)(270,574)(271,578)(272,577)(273,576)(274,575)
(275,564)(276,563)(277,565)(278,566)(279,570)(280,569)(281,568)(282,567)
(283,556)(284,555)(285,557)(286,558)(287,562)(288,561)(289,560)(290,559)
(291,460)(292,459)(293,461)(294,462)(295,466)(296,465)(297,464)(298,463)
(299,476)(300,475)(301,477)(302,478)(303,482)(304,481)(305,480)(306,479)
(307,468)(308,467)(309,469)(310,470)(311,474)(312,473)(313,472)(314,471)
(315,436)(316,435)(317,437)(318,438)(319,442)(320,441)(321,440)(322,439)
(323,452)(324,451)(325,453)(326,454)(327,458)(328,457)(329,456)(330,455)
(331,444)(332,443)(333,445)(334,446)(335,450)(336,449)(337,448)(338,447)
(339,500)(340,499)(341,501)(342,502)(343,506)(344,505)(345,504)(346,503)
(347,492)(348,491)(349,493)(350,494)(351,498)(352,497)(353,496)(354,495)
(355,484)(356,483)(357,485)(358,486)(359,490)(360,489)(361,488)(362,487)
(363,604)(364,603)(365,605)(366,606)(367,610)(368,609)(369,608)(370,607)
(371,620)(372,619)(373,621)(374,622)(375,626)(376,625)(377,624)(378,623)
(379,612)(380,611)(381,613)(382,614)(383,618)(384,617)(385,616)(386,615)
(387,580)(388,579)(389,581)(390,582)(391,586)(392,585)(393,584)(394,583)
(395,596)(396,595)(397,597)(398,598)(399,602)(400,601)(401,600)(402,599)
(403,588)(404,587)(405,589)(406,590)(407,594)(408,593)(409,592)(410,591)
(411,644)(412,643)(413,645)(414,646)(415,650)(416,649)(417,648)(418,647)
(419,636)(420,635)(421,637)(422,638)(423,642)(424,641)(425,640)(426,639)
(427,628)(428,627)(429,629)(430,630)(431,634)(432,633)(433,632)(434,631);;
s3 := (  3,225)(  4,226)(  5,221)(  6,222)(  7,224)(  8,223)(  9,219)( 10,220)
( 11,233)( 12,234)( 13,229)( 14,230)( 15,232)( 16,231)( 17,227)( 18,228)
( 19,241)( 20,242)( 21,237)( 22,238)( 23,240)( 24,239)( 25,235)( 26,236)
( 27,249)( 28,250)( 29,245)( 30,246)( 31,248)( 32,247)( 33,243)( 34,244)
( 35,257)( 36,258)( 37,253)( 38,254)( 39,256)( 40,255)( 41,251)( 42,252)
( 43,265)( 44,266)( 45,261)( 46,262)( 47,264)( 48,263)( 49,259)( 50,260)
( 51,273)( 52,274)( 53,269)( 54,270)( 55,272)( 56,271)( 57,267)( 58,268)
( 59,281)( 60,282)( 61,277)( 62,278)( 63,280)( 64,279)( 65,275)( 66,276)
( 67,289)( 68,290)( 69,285)( 70,286)( 71,288)( 72,287)( 73,283)( 74,284)
( 75,369)( 76,370)( 77,365)( 78,366)( 79,368)( 80,367)( 81,363)( 82,364)
( 83,377)( 84,378)( 85,373)( 86,374)( 87,376)( 88,375)( 89,371)( 90,372)
( 91,385)( 92,386)( 93,381)( 94,382)( 95,384)( 96,383)( 97,379)( 98,380)
( 99,393)(100,394)(101,389)(102,390)(103,392)(104,391)(105,387)(106,388)
(107,401)(108,402)(109,397)(110,398)(111,400)(112,399)(113,395)(114,396)
(115,409)(116,410)(117,405)(118,406)(119,408)(120,407)(121,403)(122,404)
(123,417)(124,418)(125,413)(126,414)(127,416)(128,415)(129,411)(130,412)
(131,425)(132,426)(133,421)(134,422)(135,424)(136,423)(137,419)(138,420)
(139,433)(140,434)(141,429)(142,430)(143,432)(144,431)(145,427)(146,428)
(147,297)(148,298)(149,293)(150,294)(151,296)(152,295)(153,291)(154,292)
(155,305)(156,306)(157,301)(158,302)(159,304)(160,303)(161,299)(162,300)
(163,313)(164,314)(165,309)(166,310)(167,312)(168,311)(169,307)(170,308)
(171,321)(172,322)(173,317)(174,318)(175,320)(176,319)(177,315)(178,316)
(179,329)(180,330)(181,325)(182,326)(183,328)(184,327)(185,323)(186,324)
(187,337)(188,338)(189,333)(190,334)(191,336)(192,335)(193,331)(194,332)
(195,345)(196,346)(197,341)(198,342)(199,344)(200,343)(201,339)(202,340)
(203,353)(204,354)(205,349)(206,350)(207,352)(208,351)(209,347)(210,348)
(211,361)(212,362)(213,357)(214,358)(215,360)(216,359)(217,355)(218,356)
(435,657)(436,658)(437,653)(438,654)(439,656)(440,655)(441,651)(442,652)
(443,665)(444,666)(445,661)(446,662)(447,664)(448,663)(449,659)(450,660)
(451,673)(452,674)(453,669)(454,670)(455,672)(456,671)(457,667)(458,668)
(459,681)(460,682)(461,677)(462,678)(463,680)(464,679)(465,675)(466,676)
(467,689)(468,690)(469,685)(470,686)(471,688)(472,687)(473,683)(474,684)
(475,697)(476,698)(477,693)(478,694)(479,696)(480,695)(481,691)(482,692)
(483,705)(484,706)(485,701)(486,702)(487,704)(488,703)(489,699)(490,700)
(491,713)(492,714)(493,709)(494,710)(495,712)(496,711)(497,707)(498,708)
(499,721)(500,722)(501,717)(502,718)(503,720)(504,719)(505,715)(506,716)
(507,801)(508,802)(509,797)(510,798)(511,800)(512,799)(513,795)(514,796)
(515,809)(516,810)(517,805)(518,806)(519,808)(520,807)(521,803)(522,804)
(523,817)(524,818)(525,813)(526,814)(527,816)(528,815)(529,811)(530,812)
(531,825)(532,826)(533,821)(534,822)(535,824)(536,823)(537,819)(538,820)
(539,833)(540,834)(541,829)(542,830)(543,832)(544,831)(545,827)(546,828)
(547,841)(548,842)(549,837)(550,838)(551,840)(552,839)(553,835)(554,836)
(555,849)(556,850)(557,845)(558,846)(559,848)(560,847)(561,843)(562,844)
(563,857)(564,858)(565,853)(566,854)(567,856)(568,855)(569,851)(570,852)
(571,865)(572,866)(573,861)(574,862)(575,864)(576,863)(577,859)(578,860)
(579,729)(580,730)(581,725)(582,726)(583,728)(584,727)(585,723)(586,724)
(587,737)(588,738)(589,733)(590,734)(591,736)(592,735)(593,731)(594,732)
(595,745)(596,746)(597,741)(598,742)(599,744)(600,743)(601,739)(602,740)
(603,753)(604,754)(605,749)(606,750)(607,752)(608,751)(609,747)(610,748)
(611,761)(612,762)(613,757)(614,758)(615,760)(616,759)(617,755)(618,756)
(619,769)(620,770)(621,765)(622,766)(623,768)(624,767)(625,763)(626,764)
(627,777)(628,778)(629,773)(630,774)(631,776)(632,775)(633,771)(634,772)
(635,785)(636,786)(637,781)(638,782)(639,784)(640,783)(641,779)(642,780)
(643,793)(644,794)(645,789)(646,790)(647,792)(648,791)(649,787)(650,788);;
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*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)( 11,667)( 12,668)( 13,672)( 14,671)( 15,670)( 16,669)( 17,674)
( 18,673)( 19,659)( 20,660)( 21,664)( 22,663)( 23,662)( 24,661)( 25,666)
( 26,665)( 27,715)( 28,716)( 29,720)( 30,719)( 31,718)( 32,717)( 33,722)
( 34,721)( 35,707)( 36,708)( 37,712)( 38,711)( 39,710)( 40,709)( 41,714)
( 42,713)( 43,699)( 44,700)( 45,704)( 46,703)( 47,702)( 48,701)( 49,706)
( 50,705)( 51,691)( 52,692)( 53,696)( 54,695)( 55,694)( 56,693)( 57,698)
( 58,697)( 59,683)( 60,684)( 61,688)( 62,687)( 63,686)( 64,685)( 65,690)
( 66,689)( 67,675)( 68,676)( 69,680)( 70,679)( 71,678)( 72,677)( 73,682)
( 74,681)( 75,795)( 76,796)( 77,800)( 78,799)( 79,798)( 80,797)( 81,802)
( 82,801)( 83,811)( 84,812)( 85,816)( 86,815)( 87,814)( 88,813)( 89,818)
( 90,817)( 91,803)( 92,804)( 93,808)( 94,807)( 95,806)( 96,805)( 97,810)
( 98,809)( 99,859)(100,860)(101,864)(102,863)(103,862)(104,861)(105,866)
(106,865)(107,851)(108,852)(109,856)(110,855)(111,854)(112,853)(113,858)
(114,857)(115,843)(116,844)(117,848)(118,847)(119,846)(120,845)(121,850)
(122,849)(123,835)(124,836)(125,840)(126,839)(127,838)(128,837)(129,842)
(130,841)(131,827)(132,828)(133,832)(134,831)(135,830)(136,829)(137,834)
(138,833)(139,819)(140,820)(141,824)(142,823)(143,822)(144,821)(145,826)
(146,825)(147,723)(148,724)(149,728)(150,727)(151,726)(152,725)(153,730)
(154,729)(155,739)(156,740)(157,744)(158,743)(159,742)(160,741)(161,746)
(162,745)(163,731)(164,732)(165,736)(166,735)(167,734)(168,733)(169,738)
(170,737)(171,787)(172,788)(173,792)(174,791)(175,790)(176,789)(177,794)
(178,793)(179,779)(180,780)(181,784)(182,783)(183,782)(184,781)(185,786)
(186,785)(187,771)(188,772)(189,776)(190,775)(191,774)(192,773)(193,778)
(194,777)(195,763)(196,764)(197,768)(198,767)(199,766)(200,765)(201,770)
(202,769)(203,755)(204,756)(205,760)(206,759)(207,758)(208,757)(209,762)
(210,761)(211,747)(212,748)(213,752)(214,751)(215,750)(216,749)(217,754)
(218,753)(219,436)(220,435)(221,439)(222,440)(223,437)(224,438)(225,441)
(226,442)(227,452)(228,451)(229,455)(230,456)(231,453)(232,454)(233,457)
(234,458)(235,444)(236,443)(237,447)(238,448)(239,445)(240,446)(241,449)
(242,450)(243,500)(244,499)(245,503)(246,504)(247,501)(248,502)(249,505)
(250,506)(251,492)(252,491)(253,495)(254,496)(255,493)(256,494)(257,497)
(258,498)(259,484)(260,483)(261,487)(262,488)(263,485)(264,486)(265,489)
(266,490)(267,476)(268,475)(269,479)(270,480)(271,477)(272,478)(273,481)
(274,482)(275,468)(276,467)(277,471)(278,472)(279,469)(280,470)(281,473)
(282,474)(283,460)(284,459)(285,463)(286,464)(287,461)(288,462)(289,465)
(290,466)(291,580)(292,579)(293,583)(294,584)(295,581)(296,582)(297,585)
(298,586)(299,596)(300,595)(301,599)(302,600)(303,597)(304,598)(305,601)
(306,602)(307,588)(308,587)(309,591)(310,592)(311,589)(312,590)(313,593)
(314,594)(315,644)(316,643)(317,647)(318,648)(319,645)(320,646)(321,649)
(322,650)(323,636)(324,635)(325,639)(326,640)(327,637)(328,638)(329,641)
(330,642)(331,628)(332,627)(333,631)(334,632)(335,629)(336,630)(337,633)
(338,634)(339,620)(340,619)(341,623)(342,624)(343,621)(344,622)(345,625)
(346,626)(347,612)(348,611)(349,615)(350,616)(351,613)(352,614)(353,617)
(354,618)(355,604)(356,603)(357,607)(358,608)(359,605)(360,606)(361,609)
(362,610)(363,508)(364,507)(365,511)(366,512)(367,509)(368,510)(369,513)
(370,514)(371,524)(372,523)(373,527)(374,528)(375,525)(376,526)(377,529)
(378,530)(379,516)(380,515)(381,519)(382,520)(383,517)(384,518)(385,521)
(386,522)(387,572)(388,571)(389,575)(390,576)(391,573)(392,574)(393,577)
(394,578)(395,564)(396,563)(397,567)(398,568)(399,565)(400,566)(401,569)
(402,570)(403,556)(404,555)(405,559)(406,560)(407,557)(408,558)(409,561)
(410,562)(411,548)(412,547)(413,551)(414,552)(415,549)(416,550)(417,553)
(418,554)(419,540)(420,539)(421,543)(422,544)(423,541)(424,542)(425,545)
(426,546)(427,532)(428,531)(429,535)(430,536)(431,533)(432,534)(433,537)
(434,538);
s2 := Sym(866)!(  3,747)(  4,748)(  5,750)(  6,749)(  7,753)(  8,754)(  9,751)
( 10,752)( 11,763)( 12,764)( 13,766)( 14,765)( 15,769)( 16,770)( 17,767)
( 18,768)( 19,755)( 20,756)( 21,758)( 22,757)( 23,761)( 24,762)( 25,759)
( 26,760)( 27,723)( 28,724)( 29,726)( 30,725)( 31,729)( 32,730)( 33,727)
( 34,728)( 35,739)( 36,740)( 37,742)( 38,741)( 39,745)( 40,746)( 41,743)
( 42,744)( 43,731)( 44,732)( 45,734)( 46,733)( 47,737)( 48,738)( 49,735)
( 50,736)( 51,787)( 52,788)( 53,790)( 54,789)( 55,793)( 56,794)( 57,791)
( 58,792)( 59,779)( 60,780)( 61,782)( 62,781)( 63,785)( 64,786)( 65,783)
( 66,784)( 67,771)( 68,772)( 69,774)( 70,773)( 71,777)( 72,778)( 73,775)
( 74,776)( 75,675)( 76,676)( 77,678)( 78,677)( 79,681)( 80,682)( 81,679)
( 82,680)( 83,691)( 84,692)( 85,694)( 86,693)( 87,697)( 88,698)( 89,695)
( 90,696)( 91,683)( 92,684)( 93,686)( 94,685)( 95,689)( 96,690)( 97,687)
( 98,688)( 99,651)(100,652)(101,654)(102,653)(103,657)(104,658)(105,655)
(106,656)(107,667)(108,668)(109,670)(110,669)(111,673)(112,674)(113,671)
(114,672)(115,659)(116,660)(117,662)(118,661)(119,665)(120,666)(121,663)
(122,664)(123,715)(124,716)(125,718)(126,717)(127,721)(128,722)(129,719)
(130,720)(131,707)(132,708)(133,710)(134,709)(135,713)(136,714)(137,711)
(138,712)(139,699)(140,700)(141,702)(142,701)(143,705)(144,706)(145,703)
(146,704)(147,819)(148,820)(149,822)(150,821)(151,825)(152,826)(153,823)
(154,824)(155,835)(156,836)(157,838)(158,837)(159,841)(160,842)(161,839)
(162,840)(163,827)(164,828)(165,830)(166,829)(167,833)(168,834)(169,831)
(170,832)(171,795)(172,796)(173,798)(174,797)(175,801)(176,802)(177,799)
(178,800)(179,811)(180,812)(181,814)(182,813)(183,817)(184,818)(185,815)
(186,816)(187,803)(188,804)(189,806)(190,805)(191,809)(192,810)(193,807)
(194,808)(195,859)(196,860)(197,862)(198,861)(199,865)(200,866)(201,863)
(202,864)(203,851)(204,852)(205,854)(206,853)(207,857)(208,858)(209,855)
(210,856)(211,843)(212,844)(213,846)(214,845)(215,849)(216,850)(217,847)
(218,848)(219,532)(220,531)(221,533)(222,534)(223,538)(224,537)(225,536)
(226,535)(227,548)(228,547)(229,549)(230,550)(231,554)(232,553)(233,552)
(234,551)(235,540)(236,539)(237,541)(238,542)(239,546)(240,545)(241,544)
(242,543)(243,508)(244,507)(245,509)(246,510)(247,514)(248,513)(249,512)
(250,511)(251,524)(252,523)(253,525)(254,526)(255,530)(256,529)(257,528)
(258,527)(259,516)(260,515)(261,517)(262,518)(263,522)(264,521)(265,520)
(266,519)(267,572)(268,571)(269,573)(270,574)(271,578)(272,577)(273,576)
(274,575)(275,564)(276,563)(277,565)(278,566)(279,570)(280,569)(281,568)
(282,567)(283,556)(284,555)(285,557)(286,558)(287,562)(288,561)(289,560)
(290,559)(291,460)(292,459)(293,461)(294,462)(295,466)(296,465)(297,464)
(298,463)(299,476)(300,475)(301,477)(302,478)(303,482)(304,481)(305,480)
(306,479)(307,468)(308,467)(309,469)(310,470)(311,474)(312,473)(313,472)
(314,471)(315,436)(316,435)(317,437)(318,438)(319,442)(320,441)(321,440)
(322,439)(323,452)(324,451)(325,453)(326,454)(327,458)(328,457)(329,456)
(330,455)(331,444)(332,443)(333,445)(334,446)(335,450)(336,449)(337,448)
(338,447)(339,500)(340,499)(341,501)(342,502)(343,506)(344,505)(345,504)
(346,503)(347,492)(348,491)(349,493)(350,494)(351,498)(352,497)(353,496)
(354,495)(355,484)(356,483)(357,485)(358,486)(359,490)(360,489)(361,488)
(362,487)(363,604)(364,603)(365,605)(366,606)(367,610)(368,609)(369,608)
(370,607)(371,620)(372,619)(373,621)(374,622)(375,626)(376,625)(377,624)
(378,623)(379,612)(380,611)(381,613)(382,614)(383,618)(384,617)(385,616)
(386,615)(387,580)(388,579)(389,581)(390,582)(391,586)(392,585)(393,584)
(394,583)(395,596)(396,595)(397,597)(398,598)(399,602)(400,601)(401,600)
(402,599)(403,588)(404,587)(405,589)(406,590)(407,594)(408,593)(409,592)
(410,591)(411,644)(412,643)(413,645)(414,646)(415,650)(416,649)(417,648)
(418,647)(419,636)(420,635)(421,637)(422,638)(423,642)(424,641)(425,640)
(426,639)(427,628)(428,627)(429,629)(430,630)(431,634)(432,633)(433,632)
(434,631);
s3 := Sym(866)!(  3,225)(  4,226)(  5,221)(  6,222)(  7,224)(  8,223)(  9,219)
( 10,220)( 11,233)( 12,234)( 13,229)( 14,230)( 15,232)( 16,231)( 17,227)
( 18,228)( 19,241)( 20,242)( 21,237)( 22,238)( 23,240)( 24,239)( 25,235)
( 26,236)( 27,249)( 28,250)( 29,245)( 30,246)( 31,248)( 32,247)( 33,243)
( 34,244)( 35,257)( 36,258)( 37,253)( 38,254)( 39,256)( 40,255)( 41,251)
( 42,252)( 43,265)( 44,266)( 45,261)( 46,262)( 47,264)( 48,263)( 49,259)
( 50,260)( 51,273)( 52,274)( 53,269)( 54,270)( 55,272)( 56,271)( 57,267)
( 58,268)( 59,281)( 60,282)( 61,277)( 62,278)( 63,280)( 64,279)( 65,275)
( 66,276)( 67,289)( 68,290)( 69,285)( 70,286)( 71,288)( 72,287)( 73,283)
( 74,284)( 75,369)( 76,370)( 77,365)( 78,366)( 79,368)( 80,367)( 81,363)
( 82,364)( 83,377)( 84,378)( 85,373)( 86,374)( 87,376)( 88,375)( 89,371)
( 90,372)( 91,385)( 92,386)( 93,381)( 94,382)( 95,384)( 96,383)( 97,379)
( 98,380)( 99,393)(100,394)(101,389)(102,390)(103,392)(104,391)(105,387)
(106,388)(107,401)(108,402)(109,397)(110,398)(111,400)(112,399)(113,395)
(114,396)(115,409)(116,410)(117,405)(118,406)(119,408)(120,407)(121,403)
(122,404)(123,417)(124,418)(125,413)(126,414)(127,416)(128,415)(129,411)
(130,412)(131,425)(132,426)(133,421)(134,422)(135,424)(136,423)(137,419)
(138,420)(139,433)(140,434)(141,429)(142,430)(143,432)(144,431)(145,427)
(146,428)(147,297)(148,298)(149,293)(150,294)(151,296)(152,295)(153,291)
(154,292)(155,305)(156,306)(157,301)(158,302)(159,304)(160,303)(161,299)
(162,300)(163,313)(164,314)(165,309)(166,310)(167,312)(168,311)(169,307)
(170,308)(171,321)(172,322)(173,317)(174,318)(175,320)(176,319)(177,315)
(178,316)(179,329)(180,330)(181,325)(182,326)(183,328)(184,327)(185,323)
(186,324)(187,337)(188,338)(189,333)(190,334)(191,336)(192,335)(193,331)
(194,332)(195,345)(196,346)(197,341)(198,342)(199,344)(200,343)(201,339)
(202,340)(203,353)(204,354)(205,349)(206,350)(207,352)(208,351)(209,347)
(210,348)(211,361)(212,362)(213,357)(214,358)(215,360)(216,359)(217,355)
(218,356)(435,657)(436,658)(437,653)(438,654)(439,656)(440,655)(441,651)
(442,652)(443,665)(444,666)(445,661)(446,662)(447,664)(448,663)(449,659)
(450,660)(451,673)(452,674)(453,669)(454,670)(455,672)(456,671)(457,667)
(458,668)(459,681)(460,682)(461,677)(462,678)(463,680)(464,679)(465,675)
(466,676)(467,689)(468,690)(469,685)(470,686)(471,688)(472,687)(473,683)
(474,684)(475,697)(476,698)(477,693)(478,694)(479,696)(480,695)(481,691)
(482,692)(483,705)(484,706)(485,701)(486,702)(487,704)(488,703)(489,699)
(490,700)(491,713)(492,714)(493,709)(494,710)(495,712)(496,711)(497,707)
(498,708)(499,721)(500,722)(501,717)(502,718)(503,720)(504,719)(505,715)
(506,716)(507,801)(508,802)(509,797)(510,798)(511,800)(512,799)(513,795)
(514,796)(515,809)(516,810)(517,805)(518,806)(519,808)(520,807)(521,803)
(522,804)(523,817)(524,818)(525,813)(526,814)(527,816)(528,815)(529,811)
(530,812)(531,825)(532,826)(533,821)(534,822)(535,824)(536,823)(537,819)
(538,820)(539,833)(540,834)(541,829)(542,830)(543,832)(544,831)(545,827)
(546,828)(547,841)(548,842)(549,837)(550,838)(551,840)(552,839)(553,835)
(554,836)(555,849)(556,850)(557,845)(558,846)(559,848)(560,847)(561,843)
(562,844)(563,857)(564,858)(565,853)(566,854)(567,856)(568,855)(569,851)
(570,852)(571,865)(572,866)(573,861)(574,862)(575,864)(576,863)(577,859)
(578,860)(579,729)(580,730)(581,725)(582,726)(583,728)(584,727)(585,723)
(586,724)(587,737)(588,738)(589,733)(590,734)(591,736)(592,735)(593,731)
(594,732)(595,745)(596,746)(597,741)(598,742)(599,744)(600,743)(601,739)
(602,740)(603,753)(604,754)(605,749)(606,750)(607,752)(608,751)(609,747)
(610,748)(611,761)(612,762)(613,757)(614,758)(615,760)(616,759)(617,755)
(618,756)(619,769)(620,770)(621,765)(622,766)(623,768)(624,767)(625,763)
(626,764)(627,777)(628,778)(629,773)(630,774)(631,776)(632,775)(633,771)
(634,772)(635,785)(636,786)(637,781)(638,782)(639,784)(640,783)(641,779)
(642,780)(643,793)(644,794)(645,789)(646,790)(647,792)(648,791)(649,787)
(650,788);
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2 >; 
 

to this polytope