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Polytope of Type {3,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*1728
if this polytope has a name.
Group : SmallGroup(1728,46356)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 12, 36, 144, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {3,6,6}*864
   3-fold quotients : {3,6,12}*576, {3,6,4}*576a
   4-fold quotients : {3,6,12}*432b
   6-fold quotients : {3,6,6}*288, {3,6,2}*288
   8-fold quotients : {3,6,6}*216b
   9-fold quotients : {3,6,4}*192
   12-fold quotients : {3,2,12}*144, {3,6,4}*144
   18-fold quotients : {3,6,2}*96
   24-fold quotients : {3,2,6}*72, {3,6,2}*72
   36-fold quotients : {3,2,4}*48, {3,3,2}*48
   48-fold quotients : {3,2,3}*36
   72-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,649)(  2,650)(  3,652)(  4,651)(  5,657)(  6,658)(  7,660)(  8,659)
(  9,653)( 10,654)( 11,656)( 12,655)( 13,661)( 14,662)( 15,664)( 16,663)
( 17,669)( 18,670)( 19,672)( 20,671)( 21,665)( 22,666)( 23,668)( 24,667)
( 25,673)( 26,674)( 27,676)( 28,675)( 29,681)( 30,682)( 31,684)( 32,683)
( 33,677)( 34,678)( 35,680)( 36,679)( 37,721)( 38,722)( 39,724)( 40,723)
( 41,729)( 42,730)( 43,732)( 44,731)( 45,725)( 46,726)( 47,728)( 48,727)
( 49,733)( 50,734)( 51,736)( 52,735)( 53,741)( 54,742)( 55,744)( 56,743)
( 57,737)( 58,738)( 59,740)( 60,739)( 61,745)( 62,746)( 63,748)( 64,747)
( 65,753)( 66,754)( 67,756)( 68,755)( 69,749)( 70,750)( 71,752)( 72,751)
( 73,685)( 74,686)( 75,688)( 76,687)( 77,693)( 78,694)( 79,696)( 80,695)
( 81,689)( 82,690)( 83,692)( 84,691)( 85,697)( 86,698)( 87,700)( 88,699)
( 89,705)( 90,706)( 91,708)( 92,707)( 93,701)( 94,702)( 95,704)( 96,703)
( 97,709)( 98,710)( 99,712)(100,711)(101,717)(102,718)(103,720)(104,719)
(105,713)(106,714)(107,716)(108,715)(109,757)(110,758)(111,760)(112,759)
(113,765)(114,766)(115,768)(116,767)(117,761)(118,762)(119,764)(120,763)
(121,769)(122,770)(123,772)(124,771)(125,777)(126,778)(127,780)(128,779)
(129,773)(130,774)(131,776)(132,775)(133,781)(134,782)(135,784)(136,783)
(137,789)(138,790)(139,792)(140,791)(141,785)(142,786)(143,788)(144,787)
(145,829)(146,830)(147,832)(148,831)(149,837)(150,838)(151,840)(152,839)
(153,833)(154,834)(155,836)(156,835)(157,841)(158,842)(159,844)(160,843)
(161,849)(162,850)(163,852)(164,851)(165,845)(166,846)(167,848)(168,847)
(169,853)(170,854)(171,856)(172,855)(173,861)(174,862)(175,864)(176,863)
(177,857)(178,858)(179,860)(180,859)(181,793)(182,794)(183,796)(184,795)
(185,801)(186,802)(187,804)(188,803)(189,797)(190,798)(191,800)(192,799)
(193,805)(194,806)(195,808)(196,807)(197,813)(198,814)(199,816)(200,815)
(201,809)(202,810)(203,812)(204,811)(205,817)(206,818)(207,820)(208,819)
(209,825)(210,826)(211,828)(212,827)(213,821)(214,822)(215,824)(216,823)
(217,541)(218,542)(219,544)(220,543)(221,549)(222,550)(223,552)(224,551)
(225,545)(226,546)(227,548)(228,547)(229,553)(230,554)(231,556)(232,555)
(233,561)(234,562)(235,564)(236,563)(237,557)(238,558)(239,560)(240,559)
(241,565)(242,566)(243,568)(244,567)(245,573)(246,574)(247,576)(248,575)
(249,569)(250,570)(251,572)(252,571)(253,613)(254,614)(255,616)(256,615)
(257,621)(258,622)(259,624)(260,623)(261,617)(262,618)(263,620)(264,619)
(265,625)(266,626)(267,628)(268,627)(269,633)(270,634)(271,636)(272,635)
(273,629)(274,630)(275,632)(276,631)(277,637)(278,638)(279,640)(280,639)
(281,645)(282,646)(283,648)(284,647)(285,641)(286,642)(287,644)(288,643)
(289,577)(290,578)(291,580)(292,579)(293,585)(294,586)(295,588)(296,587)
(297,581)(298,582)(299,584)(300,583)(301,589)(302,590)(303,592)(304,591)
(305,597)(306,598)(307,600)(308,599)(309,593)(310,594)(311,596)(312,595)
(313,601)(314,602)(315,604)(316,603)(317,609)(318,610)(319,612)(320,611)
(321,605)(322,606)(323,608)(324,607)(325,433)(326,434)(327,436)(328,435)
(329,441)(330,442)(331,444)(332,443)(333,437)(334,438)(335,440)(336,439)
(337,445)(338,446)(339,448)(340,447)(341,453)(342,454)(343,456)(344,455)
(345,449)(346,450)(347,452)(348,451)(349,457)(350,458)(351,460)(352,459)
(353,465)(354,466)(355,468)(356,467)(357,461)(358,462)(359,464)(360,463)
(361,505)(362,506)(363,508)(364,507)(365,513)(366,514)(367,516)(368,515)
(369,509)(370,510)(371,512)(372,511)(373,517)(374,518)(375,520)(376,519)
(377,525)(378,526)(379,528)(380,527)(381,521)(382,522)(383,524)(384,523)
(385,529)(386,530)(387,532)(388,531)(389,537)(390,538)(391,540)(392,539)
(393,533)(394,534)(395,536)(396,535)(397,469)(398,470)(399,472)(400,471)
(401,477)(402,478)(403,480)(404,479)(405,473)(406,474)(407,476)(408,475)
(409,481)(410,482)(411,484)(412,483)(413,489)(414,490)(415,492)(416,491)
(417,485)(418,486)(419,488)(420,487)(421,493)(422,494)(423,496)(424,495)
(425,501)(426,502)(427,504)(428,503)(429,497)(430,498)(431,500)(432,499);;
s1 := (  1,689)(  2,692)(  3,691)(  4,690)(  5,685)(  6,688)(  7,687)(  8,686)
(  9,693)( 10,696)( 11,695)( 12,694)( 13,701)( 14,704)( 15,703)( 16,702)
( 17,697)( 18,700)( 19,699)( 20,698)( 21,705)( 22,708)( 23,707)( 24,706)
( 25,713)( 26,716)( 27,715)( 28,714)( 29,709)( 30,712)( 31,711)( 32,710)
( 33,717)( 34,720)( 35,719)( 36,718)( 37,653)( 38,656)( 39,655)( 40,654)
( 41,649)( 42,652)( 43,651)( 44,650)( 45,657)( 46,660)( 47,659)( 48,658)
( 49,665)( 50,668)( 51,667)( 52,666)( 53,661)( 54,664)( 55,663)( 56,662)
( 57,669)( 58,672)( 59,671)( 60,670)( 61,677)( 62,680)( 63,679)( 64,678)
( 65,673)( 66,676)( 67,675)( 68,674)( 69,681)( 70,684)( 71,683)( 72,682)
( 73,725)( 74,728)( 75,727)( 76,726)( 77,721)( 78,724)( 79,723)( 80,722)
( 81,729)( 82,732)( 83,731)( 84,730)( 85,737)( 86,740)( 87,739)( 88,738)
( 89,733)( 90,736)( 91,735)( 92,734)( 93,741)( 94,744)( 95,743)( 96,742)
( 97,749)( 98,752)( 99,751)(100,750)(101,745)(102,748)(103,747)(104,746)
(105,753)(106,756)(107,755)(108,754)(109,797)(110,800)(111,799)(112,798)
(113,793)(114,796)(115,795)(116,794)(117,801)(118,804)(119,803)(120,802)
(121,809)(122,812)(123,811)(124,810)(125,805)(126,808)(127,807)(128,806)
(129,813)(130,816)(131,815)(132,814)(133,821)(134,824)(135,823)(136,822)
(137,817)(138,820)(139,819)(140,818)(141,825)(142,828)(143,827)(144,826)
(145,761)(146,764)(147,763)(148,762)(149,757)(150,760)(151,759)(152,758)
(153,765)(154,768)(155,767)(156,766)(157,773)(158,776)(159,775)(160,774)
(161,769)(162,772)(163,771)(164,770)(165,777)(166,780)(167,779)(168,778)
(169,785)(170,788)(171,787)(172,786)(173,781)(174,784)(175,783)(176,782)
(177,789)(178,792)(179,791)(180,790)(181,833)(182,836)(183,835)(184,834)
(185,829)(186,832)(187,831)(188,830)(189,837)(190,840)(191,839)(192,838)
(193,845)(194,848)(195,847)(196,846)(197,841)(198,844)(199,843)(200,842)
(201,849)(202,852)(203,851)(204,850)(205,857)(206,860)(207,859)(208,858)
(209,853)(210,856)(211,855)(212,854)(213,861)(214,864)(215,863)(216,862)
(217,581)(218,584)(219,583)(220,582)(221,577)(222,580)(223,579)(224,578)
(225,585)(226,588)(227,587)(228,586)(229,593)(230,596)(231,595)(232,594)
(233,589)(234,592)(235,591)(236,590)(237,597)(238,600)(239,599)(240,598)
(241,605)(242,608)(243,607)(244,606)(245,601)(246,604)(247,603)(248,602)
(249,609)(250,612)(251,611)(252,610)(253,545)(254,548)(255,547)(256,546)
(257,541)(258,544)(259,543)(260,542)(261,549)(262,552)(263,551)(264,550)
(265,557)(266,560)(267,559)(268,558)(269,553)(270,556)(271,555)(272,554)
(273,561)(274,564)(275,563)(276,562)(277,569)(278,572)(279,571)(280,570)
(281,565)(282,568)(283,567)(284,566)(285,573)(286,576)(287,575)(288,574)
(289,617)(290,620)(291,619)(292,618)(293,613)(294,616)(295,615)(296,614)
(297,621)(298,624)(299,623)(300,622)(301,629)(302,632)(303,631)(304,630)
(305,625)(306,628)(307,627)(308,626)(309,633)(310,636)(311,635)(312,634)
(313,641)(314,644)(315,643)(316,642)(317,637)(318,640)(319,639)(320,638)
(321,645)(322,648)(323,647)(324,646)(325,473)(326,476)(327,475)(328,474)
(329,469)(330,472)(331,471)(332,470)(333,477)(334,480)(335,479)(336,478)
(337,485)(338,488)(339,487)(340,486)(341,481)(342,484)(343,483)(344,482)
(345,489)(346,492)(347,491)(348,490)(349,497)(350,500)(351,499)(352,498)
(353,493)(354,496)(355,495)(356,494)(357,501)(358,504)(359,503)(360,502)
(361,437)(362,440)(363,439)(364,438)(365,433)(366,436)(367,435)(368,434)
(369,441)(370,444)(371,443)(372,442)(373,449)(374,452)(375,451)(376,450)
(377,445)(378,448)(379,447)(380,446)(381,453)(382,456)(383,455)(384,454)
(385,461)(386,464)(387,463)(388,462)(389,457)(390,460)(391,459)(392,458)
(393,465)(394,468)(395,467)(396,466)(397,509)(398,512)(399,511)(400,510)
(401,505)(402,508)(403,507)(404,506)(405,513)(406,516)(407,515)(408,514)
(409,521)(410,524)(411,523)(412,522)(413,517)(414,520)(415,519)(416,518)
(417,525)(418,528)(419,527)(420,526)(421,533)(422,536)(423,535)(424,534)
(425,529)(426,532)(427,531)(428,530)(429,537)(430,540)(431,539)(432,538);;
s2 := (  1,218)(  2,217)(  3,219)(  4,220)(  5,222)(  6,221)(  7,223)(  8,224)
(  9,226)( 10,225)( 11,227)( 12,228)( 13,242)( 14,241)( 15,243)( 16,244)
( 17,246)( 18,245)( 19,247)( 20,248)( 21,250)( 22,249)( 23,251)( 24,252)
( 25,230)( 26,229)( 27,231)( 28,232)( 29,234)( 30,233)( 31,235)( 32,236)
( 33,238)( 34,237)( 35,239)( 36,240)( 37,290)( 38,289)( 39,291)( 40,292)
( 41,294)( 42,293)( 43,295)( 44,296)( 45,298)( 46,297)( 47,299)( 48,300)
( 49,314)( 50,313)( 51,315)( 52,316)( 53,318)( 54,317)( 55,319)( 56,320)
( 57,322)( 58,321)( 59,323)( 60,324)( 61,302)( 62,301)( 63,303)( 64,304)
( 65,306)( 66,305)( 67,307)( 68,308)( 69,310)( 70,309)( 71,311)( 72,312)
( 73,254)( 74,253)( 75,255)( 76,256)( 77,258)( 78,257)( 79,259)( 80,260)
( 81,262)( 82,261)( 83,263)( 84,264)( 85,278)( 86,277)( 87,279)( 88,280)
( 89,282)( 90,281)( 91,283)( 92,284)( 93,286)( 94,285)( 95,287)( 96,288)
( 97,266)( 98,265)( 99,267)(100,268)(101,270)(102,269)(103,271)(104,272)
(105,274)(106,273)(107,275)(108,276)(109,326)(110,325)(111,327)(112,328)
(113,330)(114,329)(115,331)(116,332)(117,334)(118,333)(119,335)(120,336)
(121,350)(122,349)(123,351)(124,352)(125,354)(126,353)(127,355)(128,356)
(129,358)(130,357)(131,359)(132,360)(133,338)(134,337)(135,339)(136,340)
(137,342)(138,341)(139,343)(140,344)(141,346)(142,345)(143,347)(144,348)
(145,398)(146,397)(147,399)(148,400)(149,402)(150,401)(151,403)(152,404)
(153,406)(154,405)(155,407)(156,408)(157,422)(158,421)(159,423)(160,424)
(161,426)(162,425)(163,427)(164,428)(165,430)(166,429)(167,431)(168,432)
(169,410)(170,409)(171,411)(172,412)(173,414)(174,413)(175,415)(176,416)
(177,418)(178,417)(179,419)(180,420)(181,362)(182,361)(183,363)(184,364)
(185,366)(186,365)(187,367)(188,368)(189,370)(190,369)(191,371)(192,372)
(193,386)(194,385)(195,387)(196,388)(197,390)(198,389)(199,391)(200,392)
(201,394)(202,393)(203,395)(204,396)(205,374)(206,373)(207,375)(208,376)
(209,378)(210,377)(211,379)(212,380)(213,382)(214,381)(215,383)(216,384)
(433,758)(434,757)(435,759)(436,760)(437,762)(438,761)(439,763)(440,764)
(441,766)(442,765)(443,767)(444,768)(445,782)(446,781)(447,783)(448,784)
(449,786)(450,785)(451,787)(452,788)(453,790)(454,789)(455,791)(456,792)
(457,770)(458,769)(459,771)(460,772)(461,774)(462,773)(463,775)(464,776)
(465,778)(466,777)(467,779)(468,780)(469,830)(470,829)(471,831)(472,832)
(473,834)(474,833)(475,835)(476,836)(477,838)(478,837)(479,839)(480,840)
(481,854)(482,853)(483,855)(484,856)(485,858)(486,857)(487,859)(488,860)
(489,862)(490,861)(491,863)(492,864)(493,842)(494,841)(495,843)(496,844)
(497,846)(498,845)(499,847)(500,848)(501,850)(502,849)(503,851)(504,852)
(505,794)(506,793)(507,795)(508,796)(509,798)(510,797)(511,799)(512,800)
(513,802)(514,801)(515,803)(516,804)(517,818)(518,817)(519,819)(520,820)
(521,822)(522,821)(523,823)(524,824)(525,826)(526,825)(527,827)(528,828)
(529,806)(530,805)(531,807)(532,808)(533,810)(534,809)(535,811)(536,812)
(537,814)(538,813)(539,815)(540,816)(541,650)(542,649)(543,651)(544,652)
(545,654)(546,653)(547,655)(548,656)(549,658)(550,657)(551,659)(552,660)
(553,674)(554,673)(555,675)(556,676)(557,678)(558,677)(559,679)(560,680)
(561,682)(562,681)(563,683)(564,684)(565,662)(566,661)(567,663)(568,664)
(569,666)(570,665)(571,667)(572,668)(573,670)(574,669)(575,671)(576,672)
(577,722)(578,721)(579,723)(580,724)(581,726)(582,725)(583,727)(584,728)
(585,730)(586,729)(587,731)(588,732)(589,746)(590,745)(591,747)(592,748)
(593,750)(594,749)(595,751)(596,752)(597,754)(598,753)(599,755)(600,756)
(601,734)(602,733)(603,735)(604,736)(605,738)(606,737)(607,739)(608,740)
(609,742)(610,741)(611,743)(612,744)(613,686)(614,685)(615,687)(616,688)
(617,690)(618,689)(619,691)(620,692)(621,694)(622,693)(623,695)(624,696)
(625,710)(626,709)(627,711)(628,712)(629,714)(630,713)(631,715)(632,716)
(633,718)(634,717)(635,719)(636,720)(637,698)(638,697)(639,699)(640,700)
(641,702)(642,701)(643,703)(644,704)(645,706)(646,705)(647,707)(648,708);;
s3 := (  1, 13)(  2, 14)(  3, 15)(  4, 16)(  5, 17)(  6, 18)(  7, 19)(  8, 20)
(  9, 21)( 10, 22)( 11, 23)( 12, 24)( 37, 49)( 38, 50)( 39, 51)( 40, 52)
( 41, 53)( 42, 54)( 43, 55)( 44, 56)( 45, 57)( 46, 58)( 47, 59)( 48, 60)
( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 91)( 80, 92)
( 81, 93)( 82, 94)( 83, 95)( 84, 96)(109,121)(110,122)(111,123)(112,124)
(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)(120,132)
(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)(152,164)
(153,165)(154,166)(155,167)(156,168)(181,193)(182,194)(183,195)(184,196)
(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)(192,204)
(217,337)(218,338)(219,339)(220,340)(221,341)(222,342)(223,343)(224,344)
(225,345)(226,346)(227,347)(228,348)(229,325)(230,326)(231,327)(232,328)
(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)(240,336)
(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)(247,355)(248,356)
(249,357)(250,358)(251,359)(252,360)(253,373)(254,374)(255,375)(256,376)
(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)(264,384)
(265,361)(266,362)(267,363)(268,364)(269,365)(270,366)(271,367)(272,368)
(273,369)(274,370)(275,371)(276,372)(277,385)(278,386)(279,387)(280,388)
(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)(288,396)
(289,409)(290,410)(291,411)(292,412)(293,413)(294,414)(295,415)(296,416)
(297,417)(298,418)(299,419)(300,420)(301,397)(302,398)(303,399)(304,400)
(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)(312,408)
(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)(320,428)
(321,429)(322,430)(323,431)(324,432)(433,553)(434,554)(435,555)(436,556)
(437,557)(438,558)(439,559)(440,560)(441,561)(442,562)(443,563)(444,564)
(445,541)(446,542)(447,543)(448,544)(449,545)(450,546)(451,547)(452,548)
(453,549)(454,550)(455,551)(456,552)(457,565)(458,566)(459,567)(460,568)
(461,569)(462,570)(463,571)(464,572)(465,573)(466,574)(467,575)(468,576)
(469,589)(470,590)(471,591)(472,592)(473,593)(474,594)(475,595)(476,596)
(477,597)(478,598)(479,599)(480,600)(481,577)(482,578)(483,579)(484,580)
(485,581)(486,582)(487,583)(488,584)(489,585)(490,586)(491,587)(492,588)
(493,601)(494,602)(495,603)(496,604)(497,605)(498,606)(499,607)(500,608)
(501,609)(502,610)(503,611)(504,612)(505,625)(506,626)(507,627)(508,628)
(509,629)(510,630)(511,631)(512,632)(513,633)(514,634)(515,635)(516,636)
(517,613)(518,614)(519,615)(520,616)(521,617)(522,618)(523,619)(524,620)
(525,621)(526,622)(527,623)(528,624)(529,637)(530,638)(531,639)(532,640)
(533,641)(534,642)(535,643)(536,644)(537,645)(538,646)(539,647)(540,648)
(649,661)(650,662)(651,663)(652,664)(653,665)(654,666)(655,667)(656,668)
(657,669)(658,670)(659,671)(660,672)(685,697)(686,698)(687,699)(688,700)
(689,701)(690,702)(691,703)(692,704)(693,705)(694,706)(695,707)(696,708)
(721,733)(722,734)(723,735)(724,736)(725,737)(726,738)(727,739)(728,740)
(729,741)(730,742)(731,743)(732,744)(757,769)(758,770)(759,771)(760,772)
(761,773)(762,774)(763,775)(764,776)(765,777)(766,778)(767,779)(768,780)
(793,805)(794,806)(795,807)(796,808)(797,809)(798,810)(799,811)(800,812)
(801,813)(802,814)(803,815)(804,816)(829,841)(830,842)(831,843)(832,844)
(833,845)(834,846)(835,847)(836,848)(837,849)(838,850)(839,851)(840,852);;
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, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,649)(  2,650)(  3,652)(  4,651)(  5,657)(  6,658)(  7,660)
(  8,659)(  9,653)( 10,654)( 11,656)( 12,655)( 13,661)( 14,662)( 15,664)
( 16,663)( 17,669)( 18,670)( 19,672)( 20,671)( 21,665)( 22,666)( 23,668)
( 24,667)( 25,673)( 26,674)( 27,676)( 28,675)( 29,681)( 30,682)( 31,684)
( 32,683)( 33,677)( 34,678)( 35,680)( 36,679)( 37,721)( 38,722)( 39,724)
( 40,723)( 41,729)( 42,730)( 43,732)( 44,731)( 45,725)( 46,726)( 47,728)
( 48,727)( 49,733)( 50,734)( 51,736)( 52,735)( 53,741)( 54,742)( 55,744)
( 56,743)( 57,737)( 58,738)( 59,740)( 60,739)( 61,745)( 62,746)( 63,748)
( 64,747)( 65,753)( 66,754)( 67,756)( 68,755)( 69,749)( 70,750)( 71,752)
( 72,751)( 73,685)( 74,686)( 75,688)( 76,687)( 77,693)( 78,694)( 79,696)
( 80,695)( 81,689)( 82,690)( 83,692)( 84,691)( 85,697)( 86,698)( 87,700)
( 88,699)( 89,705)( 90,706)( 91,708)( 92,707)( 93,701)( 94,702)( 95,704)
( 96,703)( 97,709)( 98,710)( 99,712)(100,711)(101,717)(102,718)(103,720)
(104,719)(105,713)(106,714)(107,716)(108,715)(109,757)(110,758)(111,760)
(112,759)(113,765)(114,766)(115,768)(116,767)(117,761)(118,762)(119,764)
(120,763)(121,769)(122,770)(123,772)(124,771)(125,777)(126,778)(127,780)
(128,779)(129,773)(130,774)(131,776)(132,775)(133,781)(134,782)(135,784)
(136,783)(137,789)(138,790)(139,792)(140,791)(141,785)(142,786)(143,788)
(144,787)(145,829)(146,830)(147,832)(148,831)(149,837)(150,838)(151,840)
(152,839)(153,833)(154,834)(155,836)(156,835)(157,841)(158,842)(159,844)
(160,843)(161,849)(162,850)(163,852)(164,851)(165,845)(166,846)(167,848)
(168,847)(169,853)(170,854)(171,856)(172,855)(173,861)(174,862)(175,864)
(176,863)(177,857)(178,858)(179,860)(180,859)(181,793)(182,794)(183,796)
(184,795)(185,801)(186,802)(187,804)(188,803)(189,797)(190,798)(191,800)
(192,799)(193,805)(194,806)(195,808)(196,807)(197,813)(198,814)(199,816)
(200,815)(201,809)(202,810)(203,812)(204,811)(205,817)(206,818)(207,820)
(208,819)(209,825)(210,826)(211,828)(212,827)(213,821)(214,822)(215,824)
(216,823)(217,541)(218,542)(219,544)(220,543)(221,549)(222,550)(223,552)
(224,551)(225,545)(226,546)(227,548)(228,547)(229,553)(230,554)(231,556)
(232,555)(233,561)(234,562)(235,564)(236,563)(237,557)(238,558)(239,560)
(240,559)(241,565)(242,566)(243,568)(244,567)(245,573)(246,574)(247,576)
(248,575)(249,569)(250,570)(251,572)(252,571)(253,613)(254,614)(255,616)
(256,615)(257,621)(258,622)(259,624)(260,623)(261,617)(262,618)(263,620)
(264,619)(265,625)(266,626)(267,628)(268,627)(269,633)(270,634)(271,636)
(272,635)(273,629)(274,630)(275,632)(276,631)(277,637)(278,638)(279,640)
(280,639)(281,645)(282,646)(283,648)(284,647)(285,641)(286,642)(287,644)
(288,643)(289,577)(290,578)(291,580)(292,579)(293,585)(294,586)(295,588)
(296,587)(297,581)(298,582)(299,584)(300,583)(301,589)(302,590)(303,592)
(304,591)(305,597)(306,598)(307,600)(308,599)(309,593)(310,594)(311,596)
(312,595)(313,601)(314,602)(315,604)(316,603)(317,609)(318,610)(319,612)
(320,611)(321,605)(322,606)(323,608)(324,607)(325,433)(326,434)(327,436)
(328,435)(329,441)(330,442)(331,444)(332,443)(333,437)(334,438)(335,440)
(336,439)(337,445)(338,446)(339,448)(340,447)(341,453)(342,454)(343,456)
(344,455)(345,449)(346,450)(347,452)(348,451)(349,457)(350,458)(351,460)
(352,459)(353,465)(354,466)(355,468)(356,467)(357,461)(358,462)(359,464)
(360,463)(361,505)(362,506)(363,508)(364,507)(365,513)(366,514)(367,516)
(368,515)(369,509)(370,510)(371,512)(372,511)(373,517)(374,518)(375,520)
(376,519)(377,525)(378,526)(379,528)(380,527)(381,521)(382,522)(383,524)
(384,523)(385,529)(386,530)(387,532)(388,531)(389,537)(390,538)(391,540)
(392,539)(393,533)(394,534)(395,536)(396,535)(397,469)(398,470)(399,472)
(400,471)(401,477)(402,478)(403,480)(404,479)(405,473)(406,474)(407,476)
(408,475)(409,481)(410,482)(411,484)(412,483)(413,489)(414,490)(415,492)
(416,491)(417,485)(418,486)(419,488)(420,487)(421,493)(422,494)(423,496)
(424,495)(425,501)(426,502)(427,504)(428,503)(429,497)(430,498)(431,500)
(432,499);
s1 := Sym(864)!(  1,689)(  2,692)(  3,691)(  4,690)(  5,685)(  6,688)(  7,687)
(  8,686)(  9,693)( 10,696)( 11,695)( 12,694)( 13,701)( 14,704)( 15,703)
( 16,702)( 17,697)( 18,700)( 19,699)( 20,698)( 21,705)( 22,708)( 23,707)
( 24,706)( 25,713)( 26,716)( 27,715)( 28,714)( 29,709)( 30,712)( 31,711)
( 32,710)( 33,717)( 34,720)( 35,719)( 36,718)( 37,653)( 38,656)( 39,655)
( 40,654)( 41,649)( 42,652)( 43,651)( 44,650)( 45,657)( 46,660)( 47,659)
( 48,658)( 49,665)( 50,668)( 51,667)( 52,666)( 53,661)( 54,664)( 55,663)
( 56,662)( 57,669)( 58,672)( 59,671)( 60,670)( 61,677)( 62,680)( 63,679)
( 64,678)( 65,673)( 66,676)( 67,675)( 68,674)( 69,681)( 70,684)( 71,683)
( 72,682)( 73,725)( 74,728)( 75,727)( 76,726)( 77,721)( 78,724)( 79,723)
( 80,722)( 81,729)( 82,732)( 83,731)( 84,730)( 85,737)( 86,740)( 87,739)
( 88,738)( 89,733)( 90,736)( 91,735)( 92,734)( 93,741)( 94,744)( 95,743)
( 96,742)( 97,749)( 98,752)( 99,751)(100,750)(101,745)(102,748)(103,747)
(104,746)(105,753)(106,756)(107,755)(108,754)(109,797)(110,800)(111,799)
(112,798)(113,793)(114,796)(115,795)(116,794)(117,801)(118,804)(119,803)
(120,802)(121,809)(122,812)(123,811)(124,810)(125,805)(126,808)(127,807)
(128,806)(129,813)(130,816)(131,815)(132,814)(133,821)(134,824)(135,823)
(136,822)(137,817)(138,820)(139,819)(140,818)(141,825)(142,828)(143,827)
(144,826)(145,761)(146,764)(147,763)(148,762)(149,757)(150,760)(151,759)
(152,758)(153,765)(154,768)(155,767)(156,766)(157,773)(158,776)(159,775)
(160,774)(161,769)(162,772)(163,771)(164,770)(165,777)(166,780)(167,779)
(168,778)(169,785)(170,788)(171,787)(172,786)(173,781)(174,784)(175,783)
(176,782)(177,789)(178,792)(179,791)(180,790)(181,833)(182,836)(183,835)
(184,834)(185,829)(186,832)(187,831)(188,830)(189,837)(190,840)(191,839)
(192,838)(193,845)(194,848)(195,847)(196,846)(197,841)(198,844)(199,843)
(200,842)(201,849)(202,852)(203,851)(204,850)(205,857)(206,860)(207,859)
(208,858)(209,853)(210,856)(211,855)(212,854)(213,861)(214,864)(215,863)
(216,862)(217,581)(218,584)(219,583)(220,582)(221,577)(222,580)(223,579)
(224,578)(225,585)(226,588)(227,587)(228,586)(229,593)(230,596)(231,595)
(232,594)(233,589)(234,592)(235,591)(236,590)(237,597)(238,600)(239,599)
(240,598)(241,605)(242,608)(243,607)(244,606)(245,601)(246,604)(247,603)
(248,602)(249,609)(250,612)(251,611)(252,610)(253,545)(254,548)(255,547)
(256,546)(257,541)(258,544)(259,543)(260,542)(261,549)(262,552)(263,551)
(264,550)(265,557)(266,560)(267,559)(268,558)(269,553)(270,556)(271,555)
(272,554)(273,561)(274,564)(275,563)(276,562)(277,569)(278,572)(279,571)
(280,570)(281,565)(282,568)(283,567)(284,566)(285,573)(286,576)(287,575)
(288,574)(289,617)(290,620)(291,619)(292,618)(293,613)(294,616)(295,615)
(296,614)(297,621)(298,624)(299,623)(300,622)(301,629)(302,632)(303,631)
(304,630)(305,625)(306,628)(307,627)(308,626)(309,633)(310,636)(311,635)
(312,634)(313,641)(314,644)(315,643)(316,642)(317,637)(318,640)(319,639)
(320,638)(321,645)(322,648)(323,647)(324,646)(325,473)(326,476)(327,475)
(328,474)(329,469)(330,472)(331,471)(332,470)(333,477)(334,480)(335,479)
(336,478)(337,485)(338,488)(339,487)(340,486)(341,481)(342,484)(343,483)
(344,482)(345,489)(346,492)(347,491)(348,490)(349,497)(350,500)(351,499)
(352,498)(353,493)(354,496)(355,495)(356,494)(357,501)(358,504)(359,503)
(360,502)(361,437)(362,440)(363,439)(364,438)(365,433)(366,436)(367,435)
(368,434)(369,441)(370,444)(371,443)(372,442)(373,449)(374,452)(375,451)
(376,450)(377,445)(378,448)(379,447)(380,446)(381,453)(382,456)(383,455)
(384,454)(385,461)(386,464)(387,463)(388,462)(389,457)(390,460)(391,459)
(392,458)(393,465)(394,468)(395,467)(396,466)(397,509)(398,512)(399,511)
(400,510)(401,505)(402,508)(403,507)(404,506)(405,513)(406,516)(407,515)
(408,514)(409,521)(410,524)(411,523)(412,522)(413,517)(414,520)(415,519)
(416,518)(417,525)(418,528)(419,527)(420,526)(421,533)(422,536)(423,535)
(424,534)(425,529)(426,532)(427,531)(428,530)(429,537)(430,540)(431,539)
(432,538);
s2 := Sym(864)!(  1,218)(  2,217)(  3,219)(  4,220)(  5,222)(  6,221)(  7,223)
(  8,224)(  9,226)( 10,225)( 11,227)( 12,228)( 13,242)( 14,241)( 15,243)
( 16,244)( 17,246)( 18,245)( 19,247)( 20,248)( 21,250)( 22,249)( 23,251)
( 24,252)( 25,230)( 26,229)( 27,231)( 28,232)( 29,234)( 30,233)( 31,235)
( 32,236)( 33,238)( 34,237)( 35,239)( 36,240)( 37,290)( 38,289)( 39,291)
( 40,292)( 41,294)( 42,293)( 43,295)( 44,296)( 45,298)( 46,297)( 47,299)
( 48,300)( 49,314)( 50,313)( 51,315)( 52,316)( 53,318)( 54,317)( 55,319)
( 56,320)( 57,322)( 58,321)( 59,323)( 60,324)( 61,302)( 62,301)( 63,303)
( 64,304)( 65,306)( 66,305)( 67,307)( 68,308)( 69,310)( 70,309)( 71,311)
( 72,312)( 73,254)( 74,253)( 75,255)( 76,256)( 77,258)( 78,257)( 79,259)
( 80,260)( 81,262)( 82,261)( 83,263)( 84,264)( 85,278)( 86,277)( 87,279)
( 88,280)( 89,282)( 90,281)( 91,283)( 92,284)( 93,286)( 94,285)( 95,287)
( 96,288)( 97,266)( 98,265)( 99,267)(100,268)(101,270)(102,269)(103,271)
(104,272)(105,274)(106,273)(107,275)(108,276)(109,326)(110,325)(111,327)
(112,328)(113,330)(114,329)(115,331)(116,332)(117,334)(118,333)(119,335)
(120,336)(121,350)(122,349)(123,351)(124,352)(125,354)(126,353)(127,355)
(128,356)(129,358)(130,357)(131,359)(132,360)(133,338)(134,337)(135,339)
(136,340)(137,342)(138,341)(139,343)(140,344)(141,346)(142,345)(143,347)
(144,348)(145,398)(146,397)(147,399)(148,400)(149,402)(150,401)(151,403)
(152,404)(153,406)(154,405)(155,407)(156,408)(157,422)(158,421)(159,423)
(160,424)(161,426)(162,425)(163,427)(164,428)(165,430)(166,429)(167,431)
(168,432)(169,410)(170,409)(171,411)(172,412)(173,414)(174,413)(175,415)
(176,416)(177,418)(178,417)(179,419)(180,420)(181,362)(182,361)(183,363)
(184,364)(185,366)(186,365)(187,367)(188,368)(189,370)(190,369)(191,371)
(192,372)(193,386)(194,385)(195,387)(196,388)(197,390)(198,389)(199,391)
(200,392)(201,394)(202,393)(203,395)(204,396)(205,374)(206,373)(207,375)
(208,376)(209,378)(210,377)(211,379)(212,380)(213,382)(214,381)(215,383)
(216,384)(433,758)(434,757)(435,759)(436,760)(437,762)(438,761)(439,763)
(440,764)(441,766)(442,765)(443,767)(444,768)(445,782)(446,781)(447,783)
(448,784)(449,786)(450,785)(451,787)(452,788)(453,790)(454,789)(455,791)
(456,792)(457,770)(458,769)(459,771)(460,772)(461,774)(462,773)(463,775)
(464,776)(465,778)(466,777)(467,779)(468,780)(469,830)(470,829)(471,831)
(472,832)(473,834)(474,833)(475,835)(476,836)(477,838)(478,837)(479,839)
(480,840)(481,854)(482,853)(483,855)(484,856)(485,858)(486,857)(487,859)
(488,860)(489,862)(490,861)(491,863)(492,864)(493,842)(494,841)(495,843)
(496,844)(497,846)(498,845)(499,847)(500,848)(501,850)(502,849)(503,851)
(504,852)(505,794)(506,793)(507,795)(508,796)(509,798)(510,797)(511,799)
(512,800)(513,802)(514,801)(515,803)(516,804)(517,818)(518,817)(519,819)
(520,820)(521,822)(522,821)(523,823)(524,824)(525,826)(526,825)(527,827)
(528,828)(529,806)(530,805)(531,807)(532,808)(533,810)(534,809)(535,811)
(536,812)(537,814)(538,813)(539,815)(540,816)(541,650)(542,649)(543,651)
(544,652)(545,654)(546,653)(547,655)(548,656)(549,658)(550,657)(551,659)
(552,660)(553,674)(554,673)(555,675)(556,676)(557,678)(558,677)(559,679)
(560,680)(561,682)(562,681)(563,683)(564,684)(565,662)(566,661)(567,663)
(568,664)(569,666)(570,665)(571,667)(572,668)(573,670)(574,669)(575,671)
(576,672)(577,722)(578,721)(579,723)(580,724)(581,726)(582,725)(583,727)
(584,728)(585,730)(586,729)(587,731)(588,732)(589,746)(590,745)(591,747)
(592,748)(593,750)(594,749)(595,751)(596,752)(597,754)(598,753)(599,755)
(600,756)(601,734)(602,733)(603,735)(604,736)(605,738)(606,737)(607,739)
(608,740)(609,742)(610,741)(611,743)(612,744)(613,686)(614,685)(615,687)
(616,688)(617,690)(618,689)(619,691)(620,692)(621,694)(622,693)(623,695)
(624,696)(625,710)(626,709)(627,711)(628,712)(629,714)(630,713)(631,715)
(632,716)(633,718)(634,717)(635,719)(636,720)(637,698)(638,697)(639,699)
(640,700)(641,702)(642,701)(643,703)(644,704)(645,706)(646,705)(647,707)
(648,708);
s3 := Sym(864)!(  1, 13)(  2, 14)(  3, 15)(  4, 16)(  5, 17)(  6, 18)(  7, 19)
(  8, 20)(  9, 21)( 10, 22)( 11, 23)( 12, 24)( 37, 49)( 38, 50)( 39, 51)
( 40, 52)( 41, 53)( 42, 54)( 43, 55)( 44, 56)( 45, 57)( 46, 58)( 47, 59)
( 48, 60)( 73, 85)( 74, 86)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 91)
( 80, 92)( 81, 93)( 82, 94)( 83, 95)( 84, 96)(109,121)(110,122)(111,123)
(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,131)
(120,132)(145,157)(146,158)(147,159)(148,160)(149,161)(150,162)(151,163)
(152,164)(153,165)(154,166)(155,167)(156,168)(181,193)(182,194)(183,195)
(184,196)(185,197)(186,198)(187,199)(188,200)(189,201)(190,202)(191,203)
(192,204)(217,337)(218,338)(219,339)(220,340)(221,341)(222,342)(223,343)
(224,344)(225,345)(226,346)(227,347)(228,348)(229,325)(230,326)(231,327)
(232,328)(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)
(240,336)(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)(247,355)
(248,356)(249,357)(250,358)(251,359)(252,360)(253,373)(254,374)(255,375)
(256,376)(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)
(264,384)(265,361)(266,362)(267,363)(268,364)(269,365)(270,366)(271,367)
(272,368)(273,369)(274,370)(275,371)(276,372)(277,385)(278,386)(279,387)
(280,388)(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)
(288,396)(289,409)(290,410)(291,411)(292,412)(293,413)(294,414)(295,415)
(296,416)(297,417)(298,418)(299,419)(300,420)(301,397)(302,398)(303,399)
(304,400)(305,401)(306,402)(307,403)(308,404)(309,405)(310,406)(311,407)
(312,408)(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)
(320,428)(321,429)(322,430)(323,431)(324,432)(433,553)(434,554)(435,555)
(436,556)(437,557)(438,558)(439,559)(440,560)(441,561)(442,562)(443,563)
(444,564)(445,541)(446,542)(447,543)(448,544)(449,545)(450,546)(451,547)
(452,548)(453,549)(454,550)(455,551)(456,552)(457,565)(458,566)(459,567)
(460,568)(461,569)(462,570)(463,571)(464,572)(465,573)(466,574)(467,575)
(468,576)(469,589)(470,590)(471,591)(472,592)(473,593)(474,594)(475,595)
(476,596)(477,597)(478,598)(479,599)(480,600)(481,577)(482,578)(483,579)
(484,580)(485,581)(486,582)(487,583)(488,584)(489,585)(490,586)(491,587)
(492,588)(493,601)(494,602)(495,603)(496,604)(497,605)(498,606)(499,607)
(500,608)(501,609)(502,610)(503,611)(504,612)(505,625)(506,626)(507,627)
(508,628)(509,629)(510,630)(511,631)(512,632)(513,633)(514,634)(515,635)
(516,636)(517,613)(518,614)(519,615)(520,616)(521,617)(522,618)(523,619)
(524,620)(525,621)(526,622)(527,623)(528,624)(529,637)(530,638)(531,639)
(532,640)(533,641)(534,642)(535,643)(536,644)(537,645)(538,646)(539,647)
(540,648)(649,661)(650,662)(651,663)(652,664)(653,665)(654,666)(655,667)
(656,668)(657,669)(658,670)(659,671)(660,672)(685,697)(686,698)(687,699)
(688,700)(689,701)(690,702)(691,703)(692,704)(693,705)(694,706)(695,707)
(696,708)(721,733)(722,734)(723,735)(724,736)(725,737)(726,738)(727,739)
(728,740)(729,741)(730,742)(731,743)(732,744)(757,769)(758,770)(759,771)
(760,772)(761,773)(762,774)(763,775)(764,776)(765,777)(766,778)(767,779)
(768,780)(793,805)(794,806)(795,807)(796,808)(797,809)(798,810)(799,811)
(800,812)(801,813)(802,814)(803,815)(804,816)(829,841)(830,842)(831,843)
(832,844)(833,845)(834,846)(835,847)(836,848)(837,849)(838,850)(839,851)
(840,852);
poly := sub<Sym(864)|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, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope