Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*1728f
if this polytope has a name.
Group : SmallGroup(1728,46313)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 144, 432, 144
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*864c
   3-fold quotients : {6,6}*576a, {6,6}*576b
   6-fold quotients : {6,6}*288a, {6,6}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {6,6}*192b
   12-fold quotients : {3,6}*144, {6,3}*144
   18-fold quotients : {6,6}*96
   24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   36-fold quotients : {3,6}*48, {6,3}*48
   48-fold quotients : {3,6}*36, {6,3}*36
   72-fold quotients : {3,3}*24, {2,6}*24, {6,2}*24
   144-fold quotients : {2,3}*12, {3,2}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope