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Polytope of Type {4,18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,12}*1728a
Also Known As : {{4,18|2},{18,12|2}}. if this polytope has another name.
Group : SmallGroup(1728,14352)
Rank : 4
Schlafli Type : {4,18,12}
Number of vertices, edges, etc : 4, 36, 108, 12
Order of s0s1s2s3 : 36
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 : {2,18,12}*864a, {4,18,6}*864a
   3-fold quotients : {4,18,4}*576a, {4,6,12}*576a
   4-fold quotients : {2,18,6}*432a
   6-fold quotients : {2,18,4}*288a, {4,18,2}*288a, {2,6,12}*288a, {4,6,6}*288a
   9-fold quotients : {4,2,12}*192, {4,6,4}*192a
   12-fold quotients : {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,2,12}*96, {2,6,4}*96a, {4,2,6}*96, {4,6,2}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {4,2,4}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)(  8,224)
(  9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)( 96,312)
( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)(448,664)
(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)(472,688)
(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,712)
(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)(504,720)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)(528,744)
(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)(544,814)
(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)(552,822)
(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)(560,830)
(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)(576,846)
(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)(584,854)
(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)(600,762)
(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)(608,770)
(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)(616,778)
(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)(632,794)
(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)(640,802)
(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)(648,810);;
s1 := (  1,541)(  2,543)(  3,542)(  4,549)(  5,548)(  6,547)(  7,546)(  8,545)
(  9,544)( 10,550)( 11,552)( 12,551)( 13,558)( 14,557)( 15,556)( 16,555)
( 17,554)( 18,553)( 19,559)( 20,561)( 21,560)( 22,567)( 23,566)( 24,565)
( 25,564)( 26,563)( 27,562)( 28,568)( 29,570)( 30,569)( 31,576)( 32,575)
( 33,574)( 34,573)( 35,572)( 36,571)( 37,577)( 38,579)( 39,578)( 40,585)
( 41,584)( 42,583)( 43,582)( 44,581)( 45,580)( 46,586)( 47,588)( 48,587)
( 49,594)( 50,593)( 51,592)( 52,591)( 53,590)( 54,589)( 55,595)( 56,597)
( 57,596)( 58,603)( 59,602)( 60,601)( 61,600)( 62,599)( 63,598)( 64,604)
( 65,606)( 66,605)( 67,612)( 68,611)( 69,610)( 70,609)( 71,608)( 72,607)
( 73,613)( 74,615)( 75,614)( 76,621)( 77,620)( 78,619)( 79,618)( 80,617)
( 81,616)( 82,622)( 83,624)( 84,623)( 85,630)( 86,629)( 87,628)( 88,627)
( 89,626)( 90,625)( 91,631)( 92,633)( 93,632)( 94,639)( 95,638)( 96,637)
( 97,636)( 98,635)( 99,634)(100,640)(101,642)(102,641)(103,648)(104,647)
(105,646)(106,645)(107,644)(108,643)(109,460)(110,462)(111,461)(112,468)
(113,467)(114,466)(115,465)(116,464)(117,463)(118,469)(119,471)(120,470)
(121,477)(122,476)(123,475)(124,474)(125,473)(126,472)(127,478)(128,480)
(129,479)(130,486)(131,485)(132,484)(133,483)(134,482)(135,481)(136,433)
(137,435)(138,434)(139,441)(140,440)(141,439)(142,438)(143,437)(144,436)
(145,442)(146,444)(147,443)(148,450)(149,449)(150,448)(151,447)(152,446)
(153,445)(154,451)(155,453)(156,452)(157,459)(158,458)(159,457)(160,456)
(161,455)(162,454)(163,514)(164,516)(165,515)(166,522)(167,521)(168,520)
(169,519)(170,518)(171,517)(172,523)(173,525)(174,524)(175,531)(176,530)
(177,529)(178,528)(179,527)(180,526)(181,532)(182,534)(183,533)(184,540)
(185,539)(186,538)(187,537)(188,536)(189,535)(190,487)(191,489)(192,488)
(193,495)(194,494)(195,493)(196,492)(197,491)(198,490)(199,496)(200,498)
(201,497)(202,504)(203,503)(204,502)(205,501)(206,500)(207,499)(208,505)
(209,507)(210,506)(211,513)(212,512)(213,511)(214,510)(215,509)(216,508)
(217,757)(218,759)(219,758)(220,765)(221,764)(222,763)(223,762)(224,761)
(225,760)(226,766)(227,768)(228,767)(229,774)(230,773)(231,772)(232,771)
(233,770)(234,769)(235,775)(236,777)(237,776)(238,783)(239,782)(240,781)
(241,780)(242,779)(243,778)(244,784)(245,786)(246,785)(247,792)(248,791)
(249,790)(250,789)(251,788)(252,787)(253,793)(254,795)(255,794)(256,801)
(257,800)(258,799)(259,798)(260,797)(261,796)(262,802)(263,804)(264,803)
(265,810)(266,809)(267,808)(268,807)(269,806)(270,805)(271,811)(272,813)
(273,812)(274,819)(275,818)(276,817)(277,816)(278,815)(279,814)(280,820)
(281,822)(282,821)(283,828)(284,827)(285,826)(286,825)(287,824)(288,823)
(289,829)(290,831)(291,830)(292,837)(293,836)(294,835)(295,834)(296,833)
(297,832)(298,838)(299,840)(300,839)(301,846)(302,845)(303,844)(304,843)
(305,842)(306,841)(307,847)(308,849)(309,848)(310,855)(311,854)(312,853)
(313,852)(314,851)(315,850)(316,856)(317,858)(318,857)(319,864)(320,863)
(321,862)(322,861)(323,860)(324,859)(325,676)(326,678)(327,677)(328,684)
(329,683)(330,682)(331,681)(332,680)(333,679)(334,685)(335,687)(336,686)
(337,693)(338,692)(339,691)(340,690)(341,689)(342,688)(343,694)(344,696)
(345,695)(346,702)(347,701)(348,700)(349,699)(350,698)(351,697)(352,649)
(353,651)(354,650)(355,657)(356,656)(357,655)(358,654)(359,653)(360,652)
(361,658)(362,660)(363,659)(364,666)(365,665)(366,664)(367,663)(368,662)
(369,661)(370,667)(371,669)(372,668)(373,675)(374,674)(375,673)(376,672)
(377,671)(378,670)(379,730)(380,732)(381,731)(382,738)(383,737)(384,736)
(385,735)(386,734)(387,733)(388,739)(389,741)(390,740)(391,747)(392,746)
(393,745)(394,744)(395,743)(396,742)(397,748)(398,750)(399,749)(400,756)
(401,755)(402,754)(403,753)(404,752)(405,751)(406,703)(407,705)(408,704)
(409,711)(410,710)(411,709)(412,708)(413,707)(414,706)(415,712)(416,714)
(417,713)(418,720)(419,719)(420,718)(421,717)(422,716)(423,715)(424,721)
(425,723)(426,722)(427,729)(428,728)(429,727)(430,726)(431,725)(432,724);;
s2 := (  1,436)(  2,438)(  3,437)(  4,433)(  5,435)(  6,434)(  7,441)(  8,440)
(  9,439)( 10,454)( 11,456)( 12,455)( 13,451)( 14,453)( 15,452)( 16,459)
( 17,458)( 18,457)( 19,445)( 20,447)( 21,446)( 22,442)( 23,444)( 24,443)
( 25,450)( 26,449)( 27,448)( 28,463)( 29,465)( 30,464)( 31,460)( 32,462)
( 33,461)( 34,468)( 35,467)( 36,466)( 37,481)( 38,483)( 39,482)( 40,478)
( 41,480)( 42,479)( 43,486)( 44,485)( 45,484)( 46,472)( 47,474)( 48,473)
( 49,469)( 50,471)( 51,470)( 52,477)( 53,476)( 54,475)( 55,490)( 56,492)
( 57,491)( 58,487)( 59,489)( 60,488)( 61,495)( 62,494)( 63,493)( 64,508)
( 65,510)( 66,509)( 67,505)( 68,507)( 69,506)( 70,513)( 71,512)( 72,511)
( 73,499)( 74,501)( 75,500)( 76,496)( 77,498)( 78,497)( 79,504)( 80,503)
( 81,502)( 82,517)( 83,519)( 84,518)( 85,514)( 86,516)( 87,515)( 88,522)
( 89,521)( 90,520)( 91,535)( 92,537)( 93,536)( 94,532)( 95,534)( 96,533)
( 97,540)( 98,539)( 99,538)(100,526)(101,528)(102,527)(103,523)(104,525)
(105,524)(106,531)(107,530)(108,529)(109,571)(110,573)(111,572)(112,568)
(113,570)(114,569)(115,576)(116,575)(117,574)(118,589)(119,591)(120,590)
(121,586)(122,588)(123,587)(124,594)(125,593)(126,592)(127,580)(128,582)
(129,581)(130,577)(131,579)(132,578)(133,585)(134,584)(135,583)(136,544)
(137,546)(138,545)(139,541)(140,543)(141,542)(142,549)(143,548)(144,547)
(145,562)(146,564)(147,563)(148,559)(149,561)(150,560)(151,567)(152,566)
(153,565)(154,553)(155,555)(156,554)(157,550)(158,552)(159,551)(160,558)
(161,557)(162,556)(163,625)(164,627)(165,626)(166,622)(167,624)(168,623)
(169,630)(170,629)(171,628)(172,643)(173,645)(174,644)(175,640)(176,642)
(177,641)(178,648)(179,647)(180,646)(181,634)(182,636)(183,635)(184,631)
(185,633)(186,632)(187,639)(188,638)(189,637)(190,598)(191,600)(192,599)
(193,595)(194,597)(195,596)(196,603)(197,602)(198,601)(199,616)(200,618)
(201,617)(202,613)(203,615)(204,614)(205,621)(206,620)(207,619)(208,607)
(209,609)(210,608)(211,604)(212,606)(213,605)(214,612)(215,611)(216,610)
(217,652)(218,654)(219,653)(220,649)(221,651)(222,650)(223,657)(224,656)
(225,655)(226,670)(227,672)(228,671)(229,667)(230,669)(231,668)(232,675)
(233,674)(234,673)(235,661)(236,663)(237,662)(238,658)(239,660)(240,659)
(241,666)(242,665)(243,664)(244,679)(245,681)(246,680)(247,676)(248,678)
(249,677)(250,684)(251,683)(252,682)(253,697)(254,699)(255,698)(256,694)
(257,696)(258,695)(259,702)(260,701)(261,700)(262,688)(263,690)(264,689)
(265,685)(266,687)(267,686)(268,693)(269,692)(270,691)(271,706)(272,708)
(273,707)(274,703)(275,705)(276,704)(277,711)(278,710)(279,709)(280,724)
(281,726)(282,725)(283,721)(284,723)(285,722)(286,729)(287,728)(288,727)
(289,715)(290,717)(291,716)(292,712)(293,714)(294,713)(295,720)(296,719)
(297,718)(298,733)(299,735)(300,734)(301,730)(302,732)(303,731)(304,738)
(305,737)(306,736)(307,751)(308,753)(309,752)(310,748)(311,750)(312,749)
(313,756)(314,755)(315,754)(316,742)(317,744)(318,743)(319,739)(320,741)
(321,740)(322,747)(323,746)(324,745)(325,787)(326,789)(327,788)(328,784)
(329,786)(330,785)(331,792)(332,791)(333,790)(334,805)(335,807)(336,806)
(337,802)(338,804)(339,803)(340,810)(341,809)(342,808)(343,796)(344,798)
(345,797)(346,793)(347,795)(348,794)(349,801)(350,800)(351,799)(352,760)
(353,762)(354,761)(355,757)(356,759)(357,758)(358,765)(359,764)(360,763)
(361,778)(362,780)(363,779)(364,775)(365,777)(366,776)(367,783)(368,782)
(369,781)(370,769)(371,771)(372,770)(373,766)(374,768)(375,767)(376,774)
(377,773)(378,772)(379,841)(380,843)(381,842)(382,838)(383,840)(384,839)
(385,846)(386,845)(387,844)(388,859)(389,861)(390,860)(391,856)(392,858)
(393,857)(394,864)(395,863)(396,862)(397,850)(398,852)(399,851)(400,847)
(401,849)(402,848)(403,855)(404,854)(405,853)(406,814)(407,816)(408,815)
(409,811)(410,813)(411,812)(412,819)(413,818)(414,817)(415,832)(416,834)
(417,833)(418,829)(419,831)(420,830)(421,837)(422,836)(423,835)(424,823)
(425,825)(426,824)(427,820)(428,822)(429,821)(430,828)(431,827)(432,826);;
s3 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)(111,147)(112,148)
(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)(119,137)(120,138)
(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(163,199)
(164,200)(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)(171,207)
(172,190)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)(179,197)
(180,198)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)(187,214)
(188,215)(189,216)(217,226)(218,227)(219,228)(220,229)(221,230)(222,231)
(223,232)(224,233)(225,234)(244,253)(245,254)(246,255)(247,256)(248,257)
(249,258)(250,259)(251,260)(252,261)(271,280)(272,281)(273,282)(274,283)
(275,284)(276,285)(277,286)(278,287)(279,288)(298,307)(299,308)(300,309)
(301,310)(302,311)(303,312)(304,313)(305,314)(306,315)(325,361)(326,362)
(327,363)(328,364)(329,365)(330,366)(331,367)(332,368)(333,369)(334,352)
(335,353)(336,354)(337,355)(338,356)(339,357)(340,358)(341,359)(342,360)
(343,370)(344,371)(345,372)(346,373)(347,374)(348,375)(349,376)(350,377)
(351,378)(379,415)(380,416)(381,417)(382,418)(383,419)(384,420)(385,421)
(386,422)(387,423)(388,406)(389,407)(390,408)(391,409)(392,410)(393,411)
(394,412)(395,413)(396,414)(397,424)(398,425)(399,426)(400,427)(401,428)
(402,429)(403,430)(404,431)(405,432)(433,469)(434,470)(435,471)(436,472)
(437,473)(438,474)(439,475)(440,476)(441,477)(442,460)(443,461)(444,462)
(445,463)(446,464)(447,465)(448,466)(449,467)(450,468)(451,478)(452,479)
(453,480)(454,481)(455,482)(456,483)(457,484)(458,485)(459,486)(487,523)
(488,524)(489,525)(490,526)(491,527)(492,528)(493,529)(494,530)(495,531)
(496,514)(497,515)(498,516)(499,517)(500,518)(501,519)(502,520)(503,521)
(504,522)(505,532)(506,533)(507,534)(508,535)(509,536)(510,537)(511,538)
(512,539)(513,540)(541,550)(542,551)(543,552)(544,553)(545,554)(546,555)
(547,556)(548,557)(549,558)(568,577)(569,578)(570,579)(571,580)(572,581)
(573,582)(574,583)(575,584)(576,585)(595,604)(596,605)(597,606)(598,607)
(599,608)(600,609)(601,610)(602,611)(603,612)(622,631)(623,632)(624,633)
(625,634)(626,635)(627,636)(628,637)(629,638)(630,639)(649,685)(650,686)
(651,687)(652,688)(653,689)(654,690)(655,691)(656,692)(657,693)(658,676)
(659,677)(660,678)(661,679)(662,680)(663,681)(664,682)(665,683)(666,684)
(667,694)(668,695)(669,696)(670,697)(671,698)(672,699)(673,700)(674,701)
(675,702)(703,739)(704,740)(705,741)(706,742)(707,743)(708,744)(709,745)
(710,746)(711,747)(712,730)(713,731)(714,732)(715,733)(716,734)(717,735)
(718,736)(719,737)(720,738)(721,748)(722,749)(723,750)(724,751)(725,752)
(726,753)(727,754)(728,755)(729,756)(757,766)(758,767)(759,768)(760,769)
(761,770)(762,771)(763,772)(764,773)(765,774)(784,793)(785,794)(786,795)
(787,796)(788,797)(789,798)(790,799)(791,800)(792,801)(811,820)(812,821)
(813,822)(814,823)(815,824)(816,825)(817,826)(818,827)(819,828)(838,847)
(839,848)(840,849)(841,850)(842,851)(843,852)(844,853)(845,854)(846,855);;
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*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)
(  8,224)(  9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)
( 96,312)( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)
(448,664)(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)
(472,688)(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,712)(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)
(504,720)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)
(528,744)(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)
(544,814)(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)
(552,822)(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)
(560,830)(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)
(576,846)(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)
(584,854)(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)
(600,762)(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)
(608,770)(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)
(616,778)(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)
(632,794)(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)
(640,802)(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)
(648,810);
s1 := Sym(864)!(  1,541)(  2,543)(  3,542)(  4,549)(  5,548)(  6,547)(  7,546)
(  8,545)(  9,544)( 10,550)( 11,552)( 12,551)( 13,558)( 14,557)( 15,556)
( 16,555)( 17,554)( 18,553)( 19,559)( 20,561)( 21,560)( 22,567)( 23,566)
( 24,565)( 25,564)( 26,563)( 27,562)( 28,568)( 29,570)( 30,569)( 31,576)
( 32,575)( 33,574)( 34,573)( 35,572)( 36,571)( 37,577)( 38,579)( 39,578)
( 40,585)( 41,584)( 42,583)( 43,582)( 44,581)( 45,580)( 46,586)( 47,588)
( 48,587)( 49,594)( 50,593)( 51,592)( 52,591)( 53,590)( 54,589)( 55,595)
( 56,597)( 57,596)( 58,603)( 59,602)( 60,601)( 61,600)( 62,599)( 63,598)
( 64,604)( 65,606)( 66,605)( 67,612)( 68,611)( 69,610)( 70,609)( 71,608)
( 72,607)( 73,613)( 74,615)( 75,614)( 76,621)( 77,620)( 78,619)( 79,618)
( 80,617)( 81,616)( 82,622)( 83,624)( 84,623)( 85,630)( 86,629)( 87,628)
( 88,627)( 89,626)( 90,625)( 91,631)( 92,633)( 93,632)( 94,639)( 95,638)
( 96,637)( 97,636)( 98,635)( 99,634)(100,640)(101,642)(102,641)(103,648)
(104,647)(105,646)(106,645)(107,644)(108,643)(109,460)(110,462)(111,461)
(112,468)(113,467)(114,466)(115,465)(116,464)(117,463)(118,469)(119,471)
(120,470)(121,477)(122,476)(123,475)(124,474)(125,473)(126,472)(127,478)
(128,480)(129,479)(130,486)(131,485)(132,484)(133,483)(134,482)(135,481)
(136,433)(137,435)(138,434)(139,441)(140,440)(141,439)(142,438)(143,437)
(144,436)(145,442)(146,444)(147,443)(148,450)(149,449)(150,448)(151,447)
(152,446)(153,445)(154,451)(155,453)(156,452)(157,459)(158,458)(159,457)
(160,456)(161,455)(162,454)(163,514)(164,516)(165,515)(166,522)(167,521)
(168,520)(169,519)(170,518)(171,517)(172,523)(173,525)(174,524)(175,531)
(176,530)(177,529)(178,528)(179,527)(180,526)(181,532)(182,534)(183,533)
(184,540)(185,539)(186,538)(187,537)(188,536)(189,535)(190,487)(191,489)
(192,488)(193,495)(194,494)(195,493)(196,492)(197,491)(198,490)(199,496)
(200,498)(201,497)(202,504)(203,503)(204,502)(205,501)(206,500)(207,499)
(208,505)(209,507)(210,506)(211,513)(212,512)(213,511)(214,510)(215,509)
(216,508)(217,757)(218,759)(219,758)(220,765)(221,764)(222,763)(223,762)
(224,761)(225,760)(226,766)(227,768)(228,767)(229,774)(230,773)(231,772)
(232,771)(233,770)(234,769)(235,775)(236,777)(237,776)(238,783)(239,782)
(240,781)(241,780)(242,779)(243,778)(244,784)(245,786)(246,785)(247,792)
(248,791)(249,790)(250,789)(251,788)(252,787)(253,793)(254,795)(255,794)
(256,801)(257,800)(258,799)(259,798)(260,797)(261,796)(262,802)(263,804)
(264,803)(265,810)(266,809)(267,808)(268,807)(269,806)(270,805)(271,811)
(272,813)(273,812)(274,819)(275,818)(276,817)(277,816)(278,815)(279,814)
(280,820)(281,822)(282,821)(283,828)(284,827)(285,826)(286,825)(287,824)
(288,823)(289,829)(290,831)(291,830)(292,837)(293,836)(294,835)(295,834)
(296,833)(297,832)(298,838)(299,840)(300,839)(301,846)(302,845)(303,844)
(304,843)(305,842)(306,841)(307,847)(308,849)(309,848)(310,855)(311,854)
(312,853)(313,852)(314,851)(315,850)(316,856)(317,858)(318,857)(319,864)
(320,863)(321,862)(322,861)(323,860)(324,859)(325,676)(326,678)(327,677)
(328,684)(329,683)(330,682)(331,681)(332,680)(333,679)(334,685)(335,687)
(336,686)(337,693)(338,692)(339,691)(340,690)(341,689)(342,688)(343,694)
(344,696)(345,695)(346,702)(347,701)(348,700)(349,699)(350,698)(351,697)
(352,649)(353,651)(354,650)(355,657)(356,656)(357,655)(358,654)(359,653)
(360,652)(361,658)(362,660)(363,659)(364,666)(365,665)(366,664)(367,663)
(368,662)(369,661)(370,667)(371,669)(372,668)(373,675)(374,674)(375,673)
(376,672)(377,671)(378,670)(379,730)(380,732)(381,731)(382,738)(383,737)
(384,736)(385,735)(386,734)(387,733)(388,739)(389,741)(390,740)(391,747)
(392,746)(393,745)(394,744)(395,743)(396,742)(397,748)(398,750)(399,749)
(400,756)(401,755)(402,754)(403,753)(404,752)(405,751)(406,703)(407,705)
(408,704)(409,711)(410,710)(411,709)(412,708)(413,707)(414,706)(415,712)
(416,714)(417,713)(418,720)(419,719)(420,718)(421,717)(422,716)(423,715)
(424,721)(425,723)(426,722)(427,729)(428,728)(429,727)(430,726)(431,725)
(432,724);
s2 := Sym(864)!(  1,436)(  2,438)(  3,437)(  4,433)(  5,435)(  6,434)(  7,441)
(  8,440)(  9,439)( 10,454)( 11,456)( 12,455)( 13,451)( 14,453)( 15,452)
( 16,459)( 17,458)( 18,457)( 19,445)( 20,447)( 21,446)( 22,442)( 23,444)
( 24,443)( 25,450)( 26,449)( 27,448)( 28,463)( 29,465)( 30,464)( 31,460)
( 32,462)( 33,461)( 34,468)( 35,467)( 36,466)( 37,481)( 38,483)( 39,482)
( 40,478)( 41,480)( 42,479)( 43,486)( 44,485)( 45,484)( 46,472)( 47,474)
( 48,473)( 49,469)( 50,471)( 51,470)( 52,477)( 53,476)( 54,475)( 55,490)
( 56,492)( 57,491)( 58,487)( 59,489)( 60,488)( 61,495)( 62,494)( 63,493)
( 64,508)( 65,510)( 66,509)( 67,505)( 68,507)( 69,506)( 70,513)( 71,512)
( 72,511)( 73,499)( 74,501)( 75,500)( 76,496)( 77,498)( 78,497)( 79,504)
( 80,503)( 81,502)( 82,517)( 83,519)( 84,518)( 85,514)( 86,516)( 87,515)
( 88,522)( 89,521)( 90,520)( 91,535)( 92,537)( 93,536)( 94,532)( 95,534)
( 96,533)( 97,540)( 98,539)( 99,538)(100,526)(101,528)(102,527)(103,523)
(104,525)(105,524)(106,531)(107,530)(108,529)(109,571)(110,573)(111,572)
(112,568)(113,570)(114,569)(115,576)(116,575)(117,574)(118,589)(119,591)
(120,590)(121,586)(122,588)(123,587)(124,594)(125,593)(126,592)(127,580)
(128,582)(129,581)(130,577)(131,579)(132,578)(133,585)(134,584)(135,583)
(136,544)(137,546)(138,545)(139,541)(140,543)(141,542)(142,549)(143,548)
(144,547)(145,562)(146,564)(147,563)(148,559)(149,561)(150,560)(151,567)
(152,566)(153,565)(154,553)(155,555)(156,554)(157,550)(158,552)(159,551)
(160,558)(161,557)(162,556)(163,625)(164,627)(165,626)(166,622)(167,624)
(168,623)(169,630)(170,629)(171,628)(172,643)(173,645)(174,644)(175,640)
(176,642)(177,641)(178,648)(179,647)(180,646)(181,634)(182,636)(183,635)
(184,631)(185,633)(186,632)(187,639)(188,638)(189,637)(190,598)(191,600)
(192,599)(193,595)(194,597)(195,596)(196,603)(197,602)(198,601)(199,616)
(200,618)(201,617)(202,613)(203,615)(204,614)(205,621)(206,620)(207,619)
(208,607)(209,609)(210,608)(211,604)(212,606)(213,605)(214,612)(215,611)
(216,610)(217,652)(218,654)(219,653)(220,649)(221,651)(222,650)(223,657)
(224,656)(225,655)(226,670)(227,672)(228,671)(229,667)(230,669)(231,668)
(232,675)(233,674)(234,673)(235,661)(236,663)(237,662)(238,658)(239,660)
(240,659)(241,666)(242,665)(243,664)(244,679)(245,681)(246,680)(247,676)
(248,678)(249,677)(250,684)(251,683)(252,682)(253,697)(254,699)(255,698)
(256,694)(257,696)(258,695)(259,702)(260,701)(261,700)(262,688)(263,690)
(264,689)(265,685)(266,687)(267,686)(268,693)(269,692)(270,691)(271,706)
(272,708)(273,707)(274,703)(275,705)(276,704)(277,711)(278,710)(279,709)
(280,724)(281,726)(282,725)(283,721)(284,723)(285,722)(286,729)(287,728)
(288,727)(289,715)(290,717)(291,716)(292,712)(293,714)(294,713)(295,720)
(296,719)(297,718)(298,733)(299,735)(300,734)(301,730)(302,732)(303,731)
(304,738)(305,737)(306,736)(307,751)(308,753)(309,752)(310,748)(311,750)
(312,749)(313,756)(314,755)(315,754)(316,742)(317,744)(318,743)(319,739)
(320,741)(321,740)(322,747)(323,746)(324,745)(325,787)(326,789)(327,788)
(328,784)(329,786)(330,785)(331,792)(332,791)(333,790)(334,805)(335,807)
(336,806)(337,802)(338,804)(339,803)(340,810)(341,809)(342,808)(343,796)
(344,798)(345,797)(346,793)(347,795)(348,794)(349,801)(350,800)(351,799)
(352,760)(353,762)(354,761)(355,757)(356,759)(357,758)(358,765)(359,764)
(360,763)(361,778)(362,780)(363,779)(364,775)(365,777)(366,776)(367,783)
(368,782)(369,781)(370,769)(371,771)(372,770)(373,766)(374,768)(375,767)
(376,774)(377,773)(378,772)(379,841)(380,843)(381,842)(382,838)(383,840)
(384,839)(385,846)(386,845)(387,844)(388,859)(389,861)(390,860)(391,856)
(392,858)(393,857)(394,864)(395,863)(396,862)(397,850)(398,852)(399,851)
(400,847)(401,849)(402,848)(403,855)(404,854)(405,853)(406,814)(407,816)
(408,815)(409,811)(410,813)(411,812)(412,819)(413,818)(414,817)(415,832)
(416,834)(417,833)(418,829)(419,831)(420,830)(421,837)(422,836)(423,835)
(424,823)(425,825)(426,824)(427,820)(428,822)(429,821)(430,828)(431,827)
(432,826);
s3 := Sym(864)!(  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)
(  8, 17)(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)(111,147)
(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)(119,137)
(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,154)
(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)
(163,199)(164,200)(165,201)(166,202)(167,203)(168,204)(169,205)(170,206)
(171,207)(172,190)(173,191)(174,192)(175,193)(176,194)(177,195)(178,196)
(179,197)(180,198)(181,208)(182,209)(183,210)(184,211)(185,212)(186,213)
(187,214)(188,215)(189,216)(217,226)(218,227)(219,228)(220,229)(221,230)
(222,231)(223,232)(224,233)(225,234)(244,253)(245,254)(246,255)(247,256)
(248,257)(249,258)(250,259)(251,260)(252,261)(271,280)(272,281)(273,282)
(274,283)(275,284)(276,285)(277,286)(278,287)(279,288)(298,307)(299,308)
(300,309)(301,310)(302,311)(303,312)(304,313)(305,314)(306,315)(325,361)
(326,362)(327,363)(328,364)(329,365)(330,366)(331,367)(332,368)(333,369)
(334,352)(335,353)(336,354)(337,355)(338,356)(339,357)(340,358)(341,359)
(342,360)(343,370)(344,371)(345,372)(346,373)(347,374)(348,375)(349,376)
(350,377)(351,378)(379,415)(380,416)(381,417)(382,418)(383,419)(384,420)
(385,421)(386,422)(387,423)(388,406)(389,407)(390,408)(391,409)(392,410)
(393,411)(394,412)(395,413)(396,414)(397,424)(398,425)(399,426)(400,427)
(401,428)(402,429)(403,430)(404,431)(405,432)(433,469)(434,470)(435,471)
(436,472)(437,473)(438,474)(439,475)(440,476)(441,477)(442,460)(443,461)
(444,462)(445,463)(446,464)(447,465)(448,466)(449,467)(450,468)(451,478)
(452,479)(453,480)(454,481)(455,482)(456,483)(457,484)(458,485)(459,486)
(487,523)(488,524)(489,525)(490,526)(491,527)(492,528)(493,529)(494,530)
(495,531)(496,514)(497,515)(498,516)(499,517)(500,518)(501,519)(502,520)
(503,521)(504,522)(505,532)(506,533)(507,534)(508,535)(509,536)(510,537)
(511,538)(512,539)(513,540)(541,550)(542,551)(543,552)(544,553)(545,554)
(546,555)(547,556)(548,557)(549,558)(568,577)(569,578)(570,579)(571,580)
(572,581)(573,582)(574,583)(575,584)(576,585)(595,604)(596,605)(597,606)
(598,607)(599,608)(600,609)(601,610)(602,611)(603,612)(622,631)(623,632)
(624,633)(625,634)(626,635)(627,636)(628,637)(629,638)(630,639)(649,685)
(650,686)(651,687)(652,688)(653,689)(654,690)(655,691)(656,692)(657,693)
(658,676)(659,677)(660,678)(661,679)(662,680)(663,681)(664,682)(665,683)
(666,684)(667,694)(668,695)(669,696)(670,697)(671,698)(672,699)(673,700)
(674,701)(675,702)(703,739)(704,740)(705,741)(706,742)(707,743)(708,744)
(709,745)(710,746)(711,747)(712,730)(713,731)(714,732)(715,733)(716,734)
(717,735)(718,736)(719,737)(720,738)(721,748)(722,749)(723,750)(724,751)
(725,752)(726,753)(727,754)(728,755)(729,756)(757,766)(758,767)(759,768)
(760,769)(761,770)(762,771)(763,772)(764,773)(765,774)(784,793)(785,794)
(786,795)(787,796)(788,797)(789,798)(790,799)(791,800)(792,801)(811,820)
(812,821)(813,822)(814,823)(815,824)(816,825)(817,826)(818,827)(819,828)
(838,847)(839,848)(840,849)(841,850)(842,851)(843,852)(844,853)(845,854)
(846,855);
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*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope