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Polytope of Type {12,18,4}

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