Polytope of Type {8,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,54}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11367)
Rank : 3
Schlafli Type : {8,54}
Number of vertices, edges, etc : 16, 432, 108
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {8,27}*864, {4,54}*864
   3-fold quotients : {8,18}*576b
   4-fold quotients : {4,27}*432, {4,54}*432b, {4,54}*432c
   6-fold quotients : {8,9}*288, {4,18}*288
   8-fold quotients : {4,27}*216, {2,54}*216
   9-fold quotients : {8,6}*192b
   12-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
   16-fold quotients : {2,27}*108
   18-fold quotients : {8,3}*96, {4,6}*96
   24-fold quotients : {4,9}*72, {2,18}*72
   36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {2,9}*36
   72-fold quotients : {4,3}*24, {2,6}*24
   144-fold quotients : {2,3}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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*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*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