Polytope of Type {8,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,6}*1728f
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 4
Schlafli Type : {8,6,6}
Number of vertices, edges, etc : 24, 72, 54, 6
Order of s0s1s2s3 : 24
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 : {4,6,6}*864j
   3-fold quotients : {8,6,2}*576
   6-fold quotients : {4,6,2}*288
   9-fold quotients : {8,2,6}*192
   12-fold quotients : {4,6,2}*144
   18-fold quotients : {8,2,3}*96, {4,2,6}*96
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)
( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)
( 48, 51)( 49, 52)( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)
( 61, 85)( 62, 83)( 63, 90)( 64, 91)( 65, 98)( 66, 96)( 67, 97)( 68, 95)
( 69, 93)( 70, 94)( 71, 92)( 72, 99)( 73,100)( 74,107)( 75,105)( 76,106)
( 77,104)( 78,102)( 79,103)( 80,101)( 81,108)(109,136)(110,143)(111,141)
(112,142)(113,140)(114,138)(115,139)(116,137)(117,144)(118,145)(119,152)
(120,150)(121,151)(122,149)(123,147)(124,148)(125,146)(126,153)(127,154)
(128,161)(129,159)(130,160)(131,158)(132,156)(133,157)(134,155)(135,162)
(164,170)(165,168)(166,169)(173,179)(174,177)(175,178)(182,188)(183,186)
(184,187)(191,197)(192,195)(193,196)(200,206)(201,204)(202,205)(209,215)
(210,213)(211,214)(218,224)(219,222)(220,223)(227,233)(228,231)(229,232)
(236,242)(237,240)(238,241)(245,251)(246,249)(247,250)(254,260)(255,258)
(256,259)(263,269)(264,267)(265,268)(271,298)(272,305)(273,303)(274,304)
(275,302)(276,300)(277,301)(278,299)(279,306)(280,307)(281,314)(282,312)
(283,313)(284,311)(285,309)(286,310)(287,308)(288,315)(289,316)(290,323)
(291,321)(292,322)(293,320)(294,318)(295,319)(296,317)(297,324)(325,352)
(326,359)(327,357)(328,358)(329,356)(330,354)(331,355)(332,353)(333,360)
(334,361)(335,368)(336,366)(337,367)(338,365)(339,363)(340,364)(341,362)
(342,369)(343,370)(344,377)(345,375)(346,376)(347,374)(348,372)(349,373)
(350,371)(351,378)(380,386)(381,384)(382,385)(389,395)(390,393)(391,394)
(398,404)(399,402)(400,403)(407,413)(408,411)(409,412)(416,422)(417,420)
(418,421)(425,431)(426,429)(427,430)(433,487)(434,494)(435,492)(436,493)
(437,491)(438,489)(439,490)(440,488)(441,495)(442,496)(443,503)(444,501)
(445,502)(446,500)(447,498)(448,499)(449,497)(450,504)(451,505)(452,512)
(453,510)(454,511)(455,509)(456,507)(457,508)(458,506)(459,513)(460,514)
(461,521)(462,519)(463,520)(464,518)(465,516)(466,517)(467,515)(468,522)
(469,523)(470,530)(471,528)(472,529)(473,527)(474,525)(475,526)(476,524)
(477,531)(478,532)(479,539)(480,537)(481,538)(482,536)(483,534)(484,535)
(485,533)(486,540)(541,622)(542,629)(543,627)(544,628)(545,626)(546,624)
(547,625)(548,623)(549,630)(550,631)(551,638)(552,636)(553,637)(554,635)
(555,633)(556,634)(557,632)(558,639)(559,640)(560,647)(561,645)(562,646)
(563,644)(564,642)(565,643)(566,641)(567,648)(568,595)(569,602)(570,600)
(571,601)(572,599)(573,597)(574,598)(575,596)(576,603)(577,604)(578,611)
(579,609)(580,610)(581,608)(582,606)(583,607)(584,605)(585,612)(586,613)
(587,620)(588,618)(589,619)(590,617)(591,615)(592,616)(593,614)(594,621)
(649,703)(650,710)(651,708)(652,709)(653,707)(654,705)(655,706)(656,704)
(657,711)(658,712)(659,719)(660,717)(661,718)(662,716)(663,714)(664,715)
(665,713)(666,720)(667,721)(668,728)(669,726)(670,727)(671,725)(672,723)
(673,724)(674,722)(675,729)(676,730)(677,737)(678,735)(679,736)(680,734)
(681,732)(682,733)(683,731)(684,738)(685,739)(686,746)(687,744)(688,745)
(689,743)(690,741)(691,742)(692,740)(693,747)(694,748)(695,755)(696,753)
(697,754)(698,752)(699,750)(700,751)(701,749)(702,756)(757,838)(758,845)
(759,843)(760,844)(761,842)(762,840)(763,841)(764,839)(765,846)(766,847)
(767,854)(768,852)(769,853)(770,851)(771,849)(772,850)(773,848)(774,855)
(775,856)(776,863)(777,861)(778,862)(779,860)(780,858)(781,859)(782,857)
(783,864)(784,811)(785,818)(786,816)(787,817)(788,815)(789,813)(790,814)
(791,812)(792,819)(793,820)(794,827)(795,825)(796,826)(797,824)(798,822)
(799,823)(800,821)(801,828)(802,829)(803,836)(804,834)(805,835)(806,833)
(807,831)(808,832)(809,830)(810,837);;
s1 := (  1,433)(  2,434)(  3,435)(  4,441)(  5,439)(  6,440)(  7,437)(  8,438)
(  9,436)( 10,442)( 11,443)( 12,444)( 13,450)( 14,448)( 15,449)( 16,446)
( 17,447)( 18,445)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)( 24,458)
( 25,455)( 26,456)( 27,454)( 28,460)( 29,461)( 30,462)( 31,468)( 32,466)
( 33,467)( 34,464)( 35,465)( 36,463)( 37,469)( 38,470)( 39,471)( 40,477)
( 41,475)( 42,476)( 43,473)( 44,474)( 45,472)( 46,478)( 47,479)( 48,480)
( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,514)( 56,515)
( 57,516)( 58,522)( 59,520)( 60,521)( 61,518)( 62,519)( 63,517)( 64,523)
( 65,524)( 66,525)( 67,531)( 68,529)( 69,530)( 70,527)( 71,528)( 72,526)
( 73,532)( 74,533)( 75,534)( 76,540)( 77,538)( 78,539)( 79,536)( 80,537)
( 81,535)( 82,487)( 83,488)( 84,489)( 85,495)( 86,493)( 87,494)( 88,491)
( 89,492)( 90,490)( 91,496)( 92,497)( 93,498)( 94,504)( 95,502)( 96,503)
( 97,500)( 98,501)( 99,499)(100,505)(101,506)(102,507)(103,513)(104,511)
(105,512)(106,509)(107,510)(108,508)(109,568)(110,569)(111,570)(112,576)
(113,574)(114,575)(115,572)(116,573)(117,571)(118,577)(119,578)(120,579)
(121,585)(122,583)(123,584)(124,581)(125,582)(126,580)(127,586)(128,587)
(129,588)(130,594)(131,592)(132,593)(133,590)(134,591)(135,589)(136,541)
(137,542)(138,543)(139,549)(140,547)(141,548)(142,545)(143,546)(144,544)
(145,550)(146,551)(147,552)(148,558)(149,556)(150,557)(151,554)(152,555)
(153,553)(154,559)(155,560)(156,561)(157,567)(158,565)(159,566)(160,563)
(161,564)(162,562)(163,595)(164,596)(165,597)(166,603)(167,601)(168,602)
(169,599)(170,600)(171,598)(172,604)(173,605)(174,606)(175,612)(176,610)
(177,611)(178,608)(179,609)(180,607)(181,613)(182,614)(183,615)(184,621)
(185,619)(186,620)(187,617)(188,618)(189,616)(190,622)(191,623)(192,624)
(193,630)(194,628)(195,629)(196,626)(197,627)(198,625)(199,631)(200,632)
(201,633)(202,639)(203,637)(204,638)(205,635)(206,636)(207,634)(208,640)
(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)(216,643)
(217,649)(218,650)(219,651)(220,657)(221,655)(222,656)(223,653)(224,654)
(225,652)(226,658)(227,659)(228,660)(229,666)(230,664)(231,665)(232,662)
(233,663)(234,661)(235,667)(236,668)(237,669)(238,675)(239,673)(240,674)
(241,671)(242,672)(243,670)(244,676)(245,677)(246,678)(247,684)(248,682)
(249,683)(250,680)(251,681)(252,679)(253,685)(254,686)(255,687)(256,693)
(257,691)(258,692)(259,689)(260,690)(261,688)(262,694)(263,695)(264,696)
(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,730)(272,731)
(273,732)(274,738)(275,736)(276,737)(277,734)(278,735)(279,733)(280,739)
(281,740)(282,741)(283,747)(284,745)(285,746)(286,743)(287,744)(288,742)
(289,748)(290,749)(291,750)(292,756)(293,754)(294,755)(295,752)(296,753)
(297,751)(298,703)(299,704)(300,705)(301,711)(302,709)(303,710)(304,707)
(305,708)(306,706)(307,712)(308,713)(309,714)(310,720)(311,718)(312,719)
(313,716)(314,717)(315,715)(316,721)(317,722)(318,723)(319,729)(320,727)
(321,728)(322,725)(323,726)(324,724)(325,784)(326,785)(327,786)(328,792)
(329,790)(330,791)(331,788)(332,789)(333,787)(334,793)(335,794)(336,795)
(337,801)(338,799)(339,800)(340,797)(341,798)(342,796)(343,802)(344,803)
(345,804)(346,810)(347,808)(348,809)(349,806)(350,807)(351,805)(352,757)
(353,758)(354,759)(355,765)(356,763)(357,764)(358,761)(359,762)(360,760)
(361,766)(362,767)(363,768)(364,774)(365,772)(366,773)(367,770)(368,771)
(369,769)(370,775)(371,776)(372,777)(373,783)(374,781)(375,782)(376,779)
(377,780)(378,778)(379,811)(380,812)(381,813)(382,819)(383,817)(384,818)
(385,815)(386,816)(387,814)(388,820)(389,821)(390,822)(391,828)(392,826)
(393,827)(394,824)(395,825)(396,823)(397,829)(398,830)(399,831)(400,837)
(401,835)(402,836)(403,833)(404,834)(405,832)(406,838)(407,839)(408,840)
(409,846)(410,844)(411,845)(412,842)(413,843)(414,841)(415,847)(416,848)
(417,849)(418,855)(419,853)(420,854)(421,851)(422,852)(423,850)(424,856)
(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)(432,859);;
s2 := (  1,383)(  2,382)(  3,384)(  4,380)(  5,379)(  6,381)(  7,386)(  8,385)
(  9,387)( 10,401)( 11,400)( 12,402)( 13,398)( 14,397)( 15,399)( 16,404)
( 17,403)( 18,405)( 19,392)( 20,391)( 21,393)( 22,389)( 23,388)( 24,390)
( 25,395)( 26,394)( 27,396)( 28,410)( 29,409)( 30,411)( 31,407)( 32,406)
( 33,408)( 34,413)( 35,412)( 36,414)( 37,428)( 38,427)( 39,429)( 40,425)
( 41,424)( 42,426)( 43,431)( 44,430)( 45,432)( 46,419)( 47,418)( 48,420)
( 49,416)( 50,415)( 51,417)( 52,422)( 53,421)( 54,423)( 55,356)( 56,355)
( 57,357)( 58,353)( 59,352)( 60,354)( 61,359)( 62,358)( 63,360)( 64,374)
( 65,373)( 66,375)( 67,371)( 68,370)( 69,372)( 70,377)( 71,376)( 72,378)
( 73,365)( 74,364)( 75,366)( 76,362)( 77,361)( 78,363)( 79,368)( 80,367)
( 81,369)( 82,329)( 83,328)( 84,330)( 85,326)( 86,325)( 87,327)( 88,332)
( 89,331)( 90,333)( 91,347)( 92,346)( 93,348)( 94,344)( 95,343)( 96,345)
( 97,350)( 98,349)( 99,351)(100,338)(101,337)(102,339)(103,335)(104,334)
(105,336)(106,341)(107,340)(108,342)(109,302)(110,301)(111,303)(112,299)
(113,298)(114,300)(115,305)(116,304)(117,306)(118,320)(119,319)(120,321)
(121,317)(122,316)(123,318)(124,323)(125,322)(126,324)(127,311)(128,310)
(129,312)(130,308)(131,307)(132,309)(133,314)(134,313)(135,315)(136,275)
(137,274)(138,276)(139,272)(140,271)(141,273)(142,278)(143,277)(144,279)
(145,293)(146,292)(147,294)(148,290)(149,289)(150,291)(151,296)(152,295)
(153,297)(154,284)(155,283)(156,285)(157,281)(158,280)(159,282)(160,287)
(161,286)(162,288)(163,221)(164,220)(165,222)(166,218)(167,217)(168,219)
(169,224)(170,223)(171,225)(172,239)(173,238)(174,240)(175,236)(176,235)
(177,237)(178,242)(179,241)(180,243)(181,230)(182,229)(183,231)(184,227)
(185,226)(186,228)(187,233)(188,232)(189,234)(190,248)(191,247)(192,249)
(193,245)(194,244)(195,246)(196,251)(197,250)(198,252)(199,266)(200,265)
(201,267)(202,263)(203,262)(204,264)(205,269)(206,268)(207,270)(208,257)
(209,256)(210,258)(211,254)(212,253)(213,255)(214,260)(215,259)(216,261)
(433,815)(434,814)(435,816)(436,812)(437,811)(438,813)(439,818)(440,817)
(441,819)(442,833)(443,832)(444,834)(445,830)(446,829)(447,831)(448,836)
(449,835)(450,837)(451,824)(452,823)(453,825)(454,821)(455,820)(456,822)
(457,827)(458,826)(459,828)(460,842)(461,841)(462,843)(463,839)(464,838)
(465,840)(466,845)(467,844)(468,846)(469,860)(470,859)(471,861)(472,857)
(473,856)(474,858)(475,863)(476,862)(477,864)(478,851)(479,850)(480,852)
(481,848)(482,847)(483,849)(484,854)(485,853)(486,855)(487,788)(488,787)
(489,789)(490,785)(491,784)(492,786)(493,791)(494,790)(495,792)(496,806)
(497,805)(498,807)(499,803)(500,802)(501,804)(502,809)(503,808)(504,810)
(505,797)(506,796)(507,798)(508,794)(509,793)(510,795)(511,800)(512,799)
(513,801)(514,761)(515,760)(516,762)(517,758)(518,757)(519,759)(520,764)
(521,763)(522,765)(523,779)(524,778)(525,780)(526,776)(527,775)(528,777)
(529,782)(530,781)(531,783)(532,770)(533,769)(534,771)(535,767)(536,766)
(537,768)(538,773)(539,772)(540,774)(541,734)(542,733)(543,735)(544,731)
(545,730)(546,732)(547,737)(548,736)(549,738)(550,752)(551,751)(552,753)
(553,749)(554,748)(555,750)(556,755)(557,754)(558,756)(559,743)(560,742)
(561,744)(562,740)(563,739)(564,741)(565,746)(566,745)(567,747)(568,707)
(569,706)(570,708)(571,704)(572,703)(573,705)(574,710)(575,709)(576,711)
(577,725)(578,724)(579,726)(580,722)(581,721)(582,723)(583,728)(584,727)
(585,729)(586,716)(587,715)(588,717)(589,713)(590,712)(591,714)(592,719)
(593,718)(594,720)(595,653)(596,652)(597,654)(598,650)(599,649)(600,651)
(601,656)(602,655)(603,657)(604,671)(605,670)(606,672)(607,668)(608,667)
(609,669)(610,674)(611,673)(612,675)(613,662)(614,661)(615,663)(616,659)
(617,658)(618,660)(619,665)(620,664)(621,666)(622,680)(623,679)(624,681)
(625,677)(626,676)(627,678)(628,683)(629,682)(630,684)(631,698)(632,697)
(633,699)(634,695)(635,694)(636,696)(637,701)(638,700)(639,702)(640,689)
(641,688)(642,690)(643,686)(644,685)(645,687)(646,692)(647,691)(648,693);;
s3 := (  1,226)(  2,227)(  3,228)(  4,229)(  5,230)(  6,231)(  7,232)(  8,233)
(  9,234)( 10,217)( 11,218)( 12,219)( 13,220)( 14,221)( 15,222)( 16,223)
( 17,224)( 18,225)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,253)( 29,254)( 30,255)( 31,256)( 32,257)
( 33,258)( 34,259)( 35,260)( 36,261)( 37,244)( 38,245)( 39,246)( 40,247)
( 41,248)( 42,249)( 43,250)( 44,251)( 45,252)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,280)( 56,281)
( 57,282)( 58,283)( 59,284)( 60,285)( 61,286)( 62,287)( 63,288)( 64,271)
( 65,272)( 66,273)( 67,274)( 68,275)( 69,276)( 70,277)( 71,278)( 72,279)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)( 88,313)
( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)( 96,303)
( 97,304)( 98,305)( 99,306)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,334)(110,335)(111,336)(112,337)
(113,338)(114,339)(115,340)(116,341)(117,342)(118,325)(119,326)(120,327)
(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,343)(128,344)
(129,345)(130,346)(131,347)(132,348)(133,349)(134,350)(135,351)(136,361)
(137,362)(138,363)(139,364)(140,365)(141,366)(142,367)(143,368)(144,369)
(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)(152,359)
(153,360)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)(160,376)
(161,377)(162,378)(163,388)(164,389)(165,390)(166,391)(167,392)(168,393)
(169,394)(170,395)(171,396)(172,379)(173,380)(174,381)(175,382)(176,383)
(177,384)(178,385)(179,386)(180,387)(181,397)(182,398)(183,399)(184,400)
(185,401)(186,402)(187,403)(188,404)(189,405)(190,415)(191,416)(192,417)
(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)(200,407)
(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)(208,424)
(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)(216,432)
(433,658)(434,659)(435,660)(436,661)(437,662)(438,663)(439,664)(440,665)
(441,666)(442,649)(443,650)(444,651)(445,652)(446,653)(447,654)(448,655)
(449,656)(450,657)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,685)(461,686)(462,687)(463,688)(464,689)
(465,690)(466,691)(467,692)(468,693)(469,676)(470,677)(471,678)(472,679)
(473,680)(474,681)(475,682)(476,683)(477,684)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,712)(488,713)
(489,714)(490,715)(491,716)(492,717)(493,718)(494,719)(495,720)(496,703)
(497,704)(498,705)(499,706)(500,707)(501,708)(502,709)(503,710)(504,711)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,739)(515,740)(516,741)(517,742)(518,743)(519,744)(520,745)
(521,746)(522,747)(523,730)(524,731)(525,732)(526,733)(527,734)(528,735)
(529,736)(530,737)(531,738)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,766)(542,767)(543,768)(544,769)
(545,770)(546,771)(547,772)(548,773)(549,774)(550,757)(551,758)(552,759)
(553,760)(554,761)(555,762)(556,763)(557,764)(558,765)(559,775)(560,776)
(561,777)(562,778)(563,779)(564,780)(565,781)(566,782)(567,783)(568,793)
(569,794)(570,795)(571,796)(572,797)(573,798)(574,799)(575,800)(576,801)
(577,784)(578,785)(579,786)(580,787)(581,788)(582,789)(583,790)(584,791)
(585,792)(586,802)(587,803)(588,804)(589,805)(590,806)(591,807)(592,808)
(593,809)(594,810)(595,820)(596,821)(597,822)(598,823)(599,824)(600,825)
(601,826)(602,827)(603,828)(604,811)(605,812)(606,813)(607,814)(608,815)
(609,816)(610,817)(611,818)(612,819)(613,829)(614,830)(615,831)(616,832)
(617,833)(618,834)(619,835)(620,836)(621,837)(622,847)(623,848)(624,849)
(625,850)(626,851)(627,852)(628,853)(629,854)(630,855)(631,838)(632,839)
(633,840)(634,841)(635,842)(636,843)(637,844)(638,845)(639,846)(640,856)
(641,857)(642,858)(643,859)(644,860)(645,861)(646,862)(647,863)(648,864);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(864)!(  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)
( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)
( 47, 53)( 48, 51)( 49, 52)( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)
( 60, 84)( 61, 85)( 62, 83)( 63, 90)( 64, 91)( 65, 98)( 66, 96)( 67, 97)
( 68, 95)( 69, 93)( 70, 94)( 71, 92)( 72, 99)( 73,100)( 74,107)( 75,105)
( 76,106)( 77,104)( 78,102)( 79,103)( 80,101)( 81,108)(109,136)(110,143)
(111,141)(112,142)(113,140)(114,138)(115,139)(116,137)(117,144)(118,145)
(119,152)(120,150)(121,151)(122,149)(123,147)(124,148)(125,146)(126,153)
(127,154)(128,161)(129,159)(130,160)(131,158)(132,156)(133,157)(134,155)
(135,162)(164,170)(165,168)(166,169)(173,179)(174,177)(175,178)(182,188)
(183,186)(184,187)(191,197)(192,195)(193,196)(200,206)(201,204)(202,205)
(209,215)(210,213)(211,214)(218,224)(219,222)(220,223)(227,233)(228,231)
(229,232)(236,242)(237,240)(238,241)(245,251)(246,249)(247,250)(254,260)
(255,258)(256,259)(263,269)(264,267)(265,268)(271,298)(272,305)(273,303)
(274,304)(275,302)(276,300)(277,301)(278,299)(279,306)(280,307)(281,314)
(282,312)(283,313)(284,311)(285,309)(286,310)(287,308)(288,315)(289,316)
(290,323)(291,321)(292,322)(293,320)(294,318)(295,319)(296,317)(297,324)
(325,352)(326,359)(327,357)(328,358)(329,356)(330,354)(331,355)(332,353)
(333,360)(334,361)(335,368)(336,366)(337,367)(338,365)(339,363)(340,364)
(341,362)(342,369)(343,370)(344,377)(345,375)(346,376)(347,374)(348,372)
(349,373)(350,371)(351,378)(380,386)(381,384)(382,385)(389,395)(390,393)
(391,394)(398,404)(399,402)(400,403)(407,413)(408,411)(409,412)(416,422)
(417,420)(418,421)(425,431)(426,429)(427,430)(433,487)(434,494)(435,492)
(436,493)(437,491)(438,489)(439,490)(440,488)(441,495)(442,496)(443,503)
(444,501)(445,502)(446,500)(447,498)(448,499)(449,497)(450,504)(451,505)
(452,512)(453,510)(454,511)(455,509)(456,507)(457,508)(458,506)(459,513)
(460,514)(461,521)(462,519)(463,520)(464,518)(465,516)(466,517)(467,515)
(468,522)(469,523)(470,530)(471,528)(472,529)(473,527)(474,525)(475,526)
(476,524)(477,531)(478,532)(479,539)(480,537)(481,538)(482,536)(483,534)
(484,535)(485,533)(486,540)(541,622)(542,629)(543,627)(544,628)(545,626)
(546,624)(547,625)(548,623)(549,630)(550,631)(551,638)(552,636)(553,637)
(554,635)(555,633)(556,634)(557,632)(558,639)(559,640)(560,647)(561,645)
(562,646)(563,644)(564,642)(565,643)(566,641)(567,648)(568,595)(569,602)
(570,600)(571,601)(572,599)(573,597)(574,598)(575,596)(576,603)(577,604)
(578,611)(579,609)(580,610)(581,608)(582,606)(583,607)(584,605)(585,612)
(586,613)(587,620)(588,618)(589,619)(590,617)(591,615)(592,616)(593,614)
(594,621)(649,703)(650,710)(651,708)(652,709)(653,707)(654,705)(655,706)
(656,704)(657,711)(658,712)(659,719)(660,717)(661,718)(662,716)(663,714)
(664,715)(665,713)(666,720)(667,721)(668,728)(669,726)(670,727)(671,725)
(672,723)(673,724)(674,722)(675,729)(676,730)(677,737)(678,735)(679,736)
(680,734)(681,732)(682,733)(683,731)(684,738)(685,739)(686,746)(687,744)
(688,745)(689,743)(690,741)(691,742)(692,740)(693,747)(694,748)(695,755)
(696,753)(697,754)(698,752)(699,750)(700,751)(701,749)(702,756)(757,838)
(758,845)(759,843)(760,844)(761,842)(762,840)(763,841)(764,839)(765,846)
(766,847)(767,854)(768,852)(769,853)(770,851)(771,849)(772,850)(773,848)
(774,855)(775,856)(776,863)(777,861)(778,862)(779,860)(780,858)(781,859)
(782,857)(783,864)(784,811)(785,818)(786,816)(787,817)(788,815)(789,813)
(790,814)(791,812)(792,819)(793,820)(794,827)(795,825)(796,826)(797,824)
(798,822)(799,823)(800,821)(801,828)(802,829)(803,836)(804,834)(805,835)
(806,833)(807,831)(808,832)(809,830)(810,837);
s1 := Sym(864)!(  1,433)(  2,434)(  3,435)(  4,441)(  5,439)(  6,440)(  7,437)
(  8,438)(  9,436)( 10,442)( 11,443)( 12,444)( 13,450)( 14,448)( 15,449)
( 16,446)( 17,447)( 18,445)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)
( 24,458)( 25,455)( 26,456)( 27,454)( 28,460)( 29,461)( 30,462)( 31,468)
( 32,466)( 33,467)( 34,464)( 35,465)( 36,463)( 37,469)( 38,470)( 39,471)
( 40,477)( 41,475)( 42,476)( 43,473)( 44,474)( 45,472)( 46,478)( 47,479)
( 48,480)( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,514)
( 56,515)( 57,516)( 58,522)( 59,520)( 60,521)( 61,518)( 62,519)( 63,517)
( 64,523)( 65,524)( 66,525)( 67,531)( 68,529)( 69,530)( 70,527)( 71,528)
( 72,526)( 73,532)( 74,533)( 75,534)( 76,540)( 77,538)( 78,539)( 79,536)
( 80,537)( 81,535)( 82,487)( 83,488)( 84,489)( 85,495)( 86,493)( 87,494)
( 88,491)( 89,492)( 90,490)( 91,496)( 92,497)( 93,498)( 94,504)( 95,502)
( 96,503)( 97,500)( 98,501)( 99,499)(100,505)(101,506)(102,507)(103,513)
(104,511)(105,512)(106,509)(107,510)(108,508)(109,568)(110,569)(111,570)
(112,576)(113,574)(114,575)(115,572)(116,573)(117,571)(118,577)(119,578)
(120,579)(121,585)(122,583)(123,584)(124,581)(125,582)(126,580)(127,586)
(128,587)(129,588)(130,594)(131,592)(132,593)(133,590)(134,591)(135,589)
(136,541)(137,542)(138,543)(139,549)(140,547)(141,548)(142,545)(143,546)
(144,544)(145,550)(146,551)(147,552)(148,558)(149,556)(150,557)(151,554)
(152,555)(153,553)(154,559)(155,560)(156,561)(157,567)(158,565)(159,566)
(160,563)(161,564)(162,562)(163,595)(164,596)(165,597)(166,603)(167,601)
(168,602)(169,599)(170,600)(171,598)(172,604)(173,605)(174,606)(175,612)
(176,610)(177,611)(178,608)(179,609)(180,607)(181,613)(182,614)(183,615)
(184,621)(185,619)(186,620)(187,617)(188,618)(189,616)(190,622)(191,623)
(192,624)(193,630)(194,628)(195,629)(196,626)(197,627)(198,625)(199,631)
(200,632)(201,633)(202,639)(203,637)(204,638)(205,635)(206,636)(207,634)
(208,640)(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)
(216,643)(217,649)(218,650)(219,651)(220,657)(221,655)(222,656)(223,653)
(224,654)(225,652)(226,658)(227,659)(228,660)(229,666)(230,664)(231,665)
(232,662)(233,663)(234,661)(235,667)(236,668)(237,669)(238,675)(239,673)
(240,674)(241,671)(242,672)(243,670)(244,676)(245,677)(246,678)(247,684)
(248,682)(249,683)(250,680)(251,681)(252,679)(253,685)(254,686)(255,687)
(256,693)(257,691)(258,692)(259,689)(260,690)(261,688)(262,694)(263,695)
(264,696)(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,730)
(272,731)(273,732)(274,738)(275,736)(276,737)(277,734)(278,735)(279,733)
(280,739)(281,740)(282,741)(283,747)(284,745)(285,746)(286,743)(287,744)
(288,742)(289,748)(290,749)(291,750)(292,756)(293,754)(294,755)(295,752)
(296,753)(297,751)(298,703)(299,704)(300,705)(301,711)(302,709)(303,710)
(304,707)(305,708)(306,706)(307,712)(308,713)(309,714)(310,720)(311,718)
(312,719)(313,716)(314,717)(315,715)(316,721)(317,722)(318,723)(319,729)
(320,727)(321,728)(322,725)(323,726)(324,724)(325,784)(326,785)(327,786)
(328,792)(329,790)(330,791)(331,788)(332,789)(333,787)(334,793)(335,794)
(336,795)(337,801)(338,799)(339,800)(340,797)(341,798)(342,796)(343,802)
(344,803)(345,804)(346,810)(347,808)(348,809)(349,806)(350,807)(351,805)
(352,757)(353,758)(354,759)(355,765)(356,763)(357,764)(358,761)(359,762)
(360,760)(361,766)(362,767)(363,768)(364,774)(365,772)(366,773)(367,770)
(368,771)(369,769)(370,775)(371,776)(372,777)(373,783)(374,781)(375,782)
(376,779)(377,780)(378,778)(379,811)(380,812)(381,813)(382,819)(383,817)
(384,818)(385,815)(386,816)(387,814)(388,820)(389,821)(390,822)(391,828)
(392,826)(393,827)(394,824)(395,825)(396,823)(397,829)(398,830)(399,831)
(400,837)(401,835)(402,836)(403,833)(404,834)(405,832)(406,838)(407,839)
(408,840)(409,846)(410,844)(411,845)(412,842)(413,843)(414,841)(415,847)
(416,848)(417,849)(418,855)(419,853)(420,854)(421,851)(422,852)(423,850)
(424,856)(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)
(432,859);
s2 := Sym(864)!(  1,383)(  2,382)(  3,384)(  4,380)(  5,379)(  6,381)(  7,386)
(  8,385)(  9,387)( 10,401)( 11,400)( 12,402)( 13,398)( 14,397)( 15,399)
( 16,404)( 17,403)( 18,405)( 19,392)( 20,391)( 21,393)( 22,389)( 23,388)
( 24,390)( 25,395)( 26,394)( 27,396)( 28,410)( 29,409)( 30,411)( 31,407)
( 32,406)( 33,408)( 34,413)( 35,412)( 36,414)( 37,428)( 38,427)( 39,429)
( 40,425)( 41,424)( 42,426)( 43,431)( 44,430)( 45,432)( 46,419)( 47,418)
( 48,420)( 49,416)( 50,415)( 51,417)( 52,422)( 53,421)( 54,423)( 55,356)
( 56,355)( 57,357)( 58,353)( 59,352)( 60,354)( 61,359)( 62,358)( 63,360)
( 64,374)( 65,373)( 66,375)( 67,371)( 68,370)( 69,372)( 70,377)( 71,376)
( 72,378)( 73,365)( 74,364)( 75,366)( 76,362)( 77,361)( 78,363)( 79,368)
( 80,367)( 81,369)( 82,329)( 83,328)( 84,330)( 85,326)( 86,325)( 87,327)
( 88,332)( 89,331)( 90,333)( 91,347)( 92,346)( 93,348)( 94,344)( 95,343)
( 96,345)( 97,350)( 98,349)( 99,351)(100,338)(101,337)(102,339)(103,335)
(104,334)(105,336)(106,341)(107,340)(108,342)(109,302)(110,301)(111,303)
(112,299)(113,298)(114,300)(115,305)(116,304)(117,306)(118,320)(119,319)
(120,321)(121,317)(122,316)(123,318)(124,323)(125,322)(126,324)(127,311)
(128,310)(129,312)(130,308)(131,307)(132,309)(133,314)(134,313)(135,315)
(136,275)(137,274)(138,276)(139,272)(140,271)(141,273)(142,278)(143,277)
(144,279)(145,293)(146,292)(147,294)(148,290)(149,289)(150,291)(151,296)
(152,295)(153,297)(154,284)(155,283)(156,285)(157,281)(158,280)(159,282)
(160,287)(161,286)(162,288)(163,221)(164,220)(165,222)(166,218)(167,217)
(168,219)(169,224)(170,223)(171,225)(172,239)(173,238)(174,240)(175,236)
(176,235)(177,237)(178,242)(179,241)(180,243)(181,230)(182,229)(183,231)
(184,227)(185,226)(186,228)(187,233)(188,232)(189,234)(190,248)(191,247)
(192,249)(193,245)(194,244)(195,246)(196,251)(197,250)(198,252)(199,266)
(200,265)(201,267)(202,263)(203,262)(204,264)(205,269)(206,268)(207,270)
(208,257)(209,256)(210,258)(211,254)(212,253)(213,255)(214,260)(215,259)
(216,261)(433,815)(434,814)(435,816)(436,812)(437,811)(438,813)(439,818)
(440,817)(441,819)(442,833)(443,832)(444,834)(445,830)(446,829)(447,831)
(448,836)(449,835)(450,837)(451,824)(452,823)(453,825)(454,821)(455,820)
(456,822)(457,827)(458,826)(459,828)(460,842)(461,841)(462,843)(463,839)
(464,838)(465,840)(466,845)(467,844)(468,846)(469,860)(470,859)(471,861)
(472,857)(473,856)(474,858)(475,863)(476,862)(477,864)(478,851)(479,850)
(480,852)(481,848)(482,847)(483,849)(484,854)(485,853)(486,855)(487,788)
(488,787)(489,789)(490,785)(491,784)(492,786)(493,791)(494,790)(495,792)
(496,806)(497,805)(498,807)(499,803)(500,802)(501,804)(502,809)(503,808)
(504,810)(505,797)(506,796)(507,798)(508,794)(509,793)(510,795)(511,800)
(512,799)(513,801)(514,761)(515,760)(516,762)(517,758)(518,757)(519,759)
(520,764)(521,763)(522,765)(523,779)(524,778)(525,780)(526,776)(527,775)
(528,777)(529,782)(530,781)(531,783)(532,770)(533,769)(534,771)(535,767)
(536,766)(537,768)(538,773)(539,772)(540,774)(541,734)(542,733)(543,735)
(544,731)(545,730)(546,732)(547,737)(548,736)(549,738)(550,752)(551,751)
(552,753)(553,749)(554,748)(555,750)(556,755)(557,754)(558,756)(559,743)
(560,742)(561,744)(562,740)(563,739)(564,741)(565,746)(566,745)(567,747)
(568,707)(569,706)(570,708)(571,704)(572,703)(573,705)(574,710)(575,709)
(576,711)(577,725)(578,724)(579,726)(580,722)(581,721)(582,723)(583,728)
(584,727)(585,729)(586,716)(587,715)(588,717)(589,713)(590,712)(591,714)
(592,719)(593,718)(594,720)(595,653)(596,652)(597,654)(598,650)(599,649)
(600,651)(601,656)(602,655)(603,657)(604,671)(605,670)(606,672)(607,668)
(608,667)(609,669)(610,674)(611,673)(612,675)(613,662)(614,661)(615,663)
(616,659)(617,658)(618,660)(619,665)(620,664)(621,666)(622,680)(623,679)
(624,681)(625,677)(626,676)(627,678)(628,683)(629,682)(630,684)(631,698)
(632,697)(633,699)(634,695)(635,694)(636,696)(637,701)(638,700)(639,702)
(640,689)(641,688)(642,690)(643,686)(644,685)(645,687)(646,692)(647,691)
(648,693);
s3 := Sym(864)!(  1,226)(  2,227)(  3,228)(  4,229)(  5,230)(  6,231)(  7,232)
(  8,233)(  9,234)( 10,217)( 11,218)( 12,219)( 13,220)( 14,221)( 15,222)
( 16,223)( 17,224)( 18,225)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,253)( 29,254)( 30,255)( 31,256)
( 32,257)( 33,258)( 34,259)( 35,260)( 36,261)( 37,244)( 38,245)( 39,246)
( 40,247)( 41,248)( 42,249)( 43,250)( 44,251)( 45,252)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,280)
( 56,281)( 57,282)( 58,283)( 59,284)( 60,285)( 61,286)( 62,287)( 63,288)
( 64,271)( 65,272)( 66,273)( 67,274)( 68,275)( 69,276)( 70,277)( 71,278)
( 72,279)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)
( 88,313)( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)
( 96,303)( 97,304)( 98,305)( 99,306)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,334)(110,335)(111,336)
(112,337)(113,338)(114,339)(115,340)(116,341)(117,342)(118,325)(119,326)
(120,327)(121,328)(122,329)(123,330)(124,331)(125,332)(126,333)(127,343)
(128,344)(129,345)(130,346)(131,347)(132,348)(133,349)(134,350)(135,351)
(136,361)(137,362)(138,363)(139,364)(140,365)(141,366)(142,367)(143,368)
(144,369)(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)
(152,359)(153,360)(154,370)(155,371)(156,372)(157,373)(158,374)(159,375)
(160,376)(161,377)(162,378)(163,388)(164,389)(165,390)(166,391)(167,392)
(168,393)(169,394)(170,395)(171,396)(172,379)(173,380)(174,381)(175,382)
(176,383)(177,384)(178,385)(179,386)(180,387)(181,397)(182,398)(183,399)
(184,400)(185,401)(186,402)(187,403)(188,404)(189,405)(190,415)(191,416)
(192,417)(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)
(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)
(208,424)(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)
(216,432)(433,658)(434,659)(435,660)(436,661)(437,662)(438,663)(439,664)
(440,665)(441,666)(442,649)(443,650)(444,651)(445,652)(446,653)(447,654)
(448,655)(449,656)(450,657)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,685)(461,686)(462,687)(463,688)
(464,689)(465,690)(466,691)(467,692)(468,693)(469,676)(470,677)(471,678)
(472,679)(473,680)(474,681)(475,682)(476,683)(477,684)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,712)
(488,713)(489,714)(490,715)(491,716)(492,717)(493,718)(494,719)(495,720)
(496,703)(497,704)(498,705)(499,706)(500,707)(501,708)(502,709)(503,710)
(504,711)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,739)(515,740)(516,741)(517,742)(518,743)(519,744)
(520,745)(521,746)(522,747)(523,730)(524,731)(525,732)(526,733)(527,734)
(528,735)(529,736)(530,737)(531,738)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,766)(542,767)(543,768)
(544,769)(545,770)(546,771)(547,772)(548,773)(549,774)(550,757)(551,758)
(552,759)(553,760)(554,761)(555,762)(556,763)(557,764)(558,765)(559,775)
(560,776)(561,777)(562,778)(563,779)(564,780)(565,781)(566,782)(567,783)
(568,793)(569,794)(570,795)(571,796)(572,797)(573,798)(574,799)(575,800)
(576,801)(577,784)(578,785)(579,786)(580,787)(581,788)(582,789)(583,790)
(584,791)(585,792)(586,802)(587,803)(588,804)(589,805)(590,806)(591,807)
(592,808)(593,809)(594,810)(595,820)(596,821)(597,822)(598,823)(599,824)
(600,825)(601,826)(602,827)(603,828)(604,811)(605,812)(606,813)(607,814)
(608,815)(609,816)(610,817)(611,818)(612,819)(613,829)(614,830)(615,831)
(616,832)(617,833)(618,834)(619,835)(620,836)(621,837)(622,847)(623,848)
(624,849)(625,850)(626,851)(627,852)(628,853)(629,854)(630,855)(631,838)
(632,839)(633,840)(634,841)(635,842)(636,843)(637,844)(638,845)(639,846)
(640,856)(641,857)(642,858)(643,859)(644,860)(645,861)(646,862)(647,863)
(648,864);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope