Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1728h
if this polytope has a name.
Group : SmallGroup(1728,46356)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 144, 432, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,6}*576a, {12,6}*576e
   4-fold quotients : {12,6}*432g
   6-fold quotients : {6,6}*288a, {6,6}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {12,6}*192a
   12-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c, {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 : {12,2}*48, {4,6}*48a, {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
   108-fold quotients : {4,2}*16
   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)(  5,221)(  6,222)(  7,224)(  8,223)
(  9,225)( 10,226)( 11,228)( 12,227)( 13,241)( 14,242)( 15,244)( 16,243)
( 17,245)( 18,246)( 19,248)( 20,247)( 21,249)( 22,250)( 23,252)( 24,251)
( 25,229)( 26,230)( 27,232)( 28,231)( 29,233)( 30,234)( 31,236)( 32,235)
( 33,237)( 34,238)( 35,240)( 36,239)( 37,289)( 38,290)( 39,292)( 40,291)
( 41,293)( 42,294)( 43,296)( 44,295)( 45,297)( 46,298)( 47,300)( 48,299)
( 49,313)( 50,314)( 51,316)( 52,315)( 53,317)( 54,318)( 55,320)( 56,319)
( 57,321)( 58,322)( 59,324)( 60,323)( 61,301)( 62,302)( 63,304)( 64,303)
( 65,305)( 66,306)( 67,308)( 68,307)( 69,309)( 70,310)( 71,312)( 72,311)
( 73,253)( 74,254)( 75,256)( 76,255)( 77,257)( 78,258)( 79,260)( 80,259)
( 81,261)( 82,262)( 83,264)( 84,263)( 85,277)( 86,278)( 87,280)( 88,279)
( 89,281)( 90,282)( 91,284)( 92,283)( 93,285)( 94,286)( 95,288)( 96,287)
( 97,265)( 98,266)( 99,268)(100,267)(101,269)(102,270)(103,272)(104,271)
(105,273)(106,274)(107,276)(108,275)(109,325)(110,326)(111,328)(112,327)
(113,329)(114,330)(115,332)(116,331)(117,333)(118,334)(119,336)(120,335)
(121,349)(122,350)(123,352)(124,351)(125,353)(126,354)(127,356)(128,355)
(129,357)(130,358)(131,360)(132,359)(133,337)(134,338)(135,340)(136,339)
(137,341)(138,342)(139,344)(140,343)(141,345)(142,346)(143,348)(144,347)
(145,397)(146,398)(147,400)(148,399)(149,401)(150,402)(151,404)(152,403)
(153,405)(154,406)(155,408)(156,407)(157,421)(158,422)(159,424)(160,423)
(161,425)(162,426)(163,428)(164,427)(165,429)(166,430)(167,432)(168,431)
(169,409)(170,410)(171,412)(172,411)(173,413)(174,414)(175,416)(176,415)
(177,417)(178,418)(179,420)(180,419)(181,361)(182,362)(183,364)(184,363)
(185,365)(186,366)(187,368)(188,367)(189,369)(190,370)(191,372)(192,371)
(193,385)(194,386)(195,388)(196,387)(197,389)(198,390)(199,392)(200,391)
(201,393)(202,394)(203,396)(204,395)(205,373)(206,374)(207,376)(208,375)
(209,377)(210,378)(211,380)(212,379)(213,381)(214,382)(215,384)(216,383)
(433,757)(434,758)(435,760)(436,759)(437,761)(438,762)(439,764)(440,763)
(441,765)(442,766)(443,768)(444,767)(445,781)(446,782)(447,784)(448,783)
(449,785)(450,786)(451,788)(452,787)(453,789)(454,790)(455,792)(456,791)
(457,769)(458,770)(459,772)(460,771)(461,773)(462,774)(463,776)(464,775)
(465,777)(466,778)(467,780)(468,779)(469,829)(470,830)(471,832)(472,831)
(473,833)(474,834)(475,836)(476,835)(477,837)(478,838)(479,840)(480,839)
(481,853)(482,854)(483,856)(484,855)(485,857)(486,858)(487,860)(488,859)
(489,861)(490,862)(491,864)(492,863)(493,841)(494,842)(495,844)(496,843)
(497,845)(498,846)(499,848)(500,847)(501,849)(502,850)(503,852)(504,851)
(505,793)(506,794)(507,796)(508,795)(509,797)(510,798)(511,800)(512,799)
(513,801)(514,802)(515,804)(516,803)(517,817)(518,818)(519,820)(520,819)
(521,821)(522,822)(523,824)(524,823)(525,825)(526,826)(527,828)(528,827)
(529,805)(530,806)(531,808)(532,807)(533,809)(534,810)(535,812)(536,811)
(537,813)(538,814)(539,816)(540,815)(541,649)(542,650)(543,652)(544,651)
(545,653)(546,654)(547,656)(548,655)(549,657)(550,658)(551,660)(552,659)
(553,673)(554,674)(555,676)(556,675)(557,677)(558,678)(559,680)(560,679)
(561,681)(562,682)(563,684)(564,683)(565,661)(566,662)(567,664)(568,663)
(569,665)(570,666)(571,668)(572,667)(573,669)(574,670)(575,672)(576,671)
(577,721)(578,722)(579,724)(580,723)(581,725)(582,726)(583,728)(584,727)
(585,729)(586,730)(587,732)(588,731)(589,745)(590,746)(591,748)(592,747)
(593,749)(594,750)(595,752)(596,751)(597,753)(598,754)(599,756)(600,755)
(601,733)(602,734)(603,736)(604,735)(605,737)(606,738)(607,740)(608,739)
(609,741)(610,742)(611,744)(612,743)(613,685)(614,686)(615,688)(616,687)
(617,689)(618,690)(619,692)(620,691)(621,693)(622,694)(623,696)(624,695)
(625,709)(626,710)(627,712)(628,711)(629,713)(630,714)(631,716)(632,715)
(633,717)(634,718)(635,720)(636,719)(637,697)(638,698)(639,700)(640,699)
(641,701)(642,702)(643,704)(644,703)(645,705)(646,706)(647,708)(648,707);;
s1 := (  1,697)(  2,700)(  3,699)(  4,698)(  5,705)(  6,708)(  7,707)(  8,706)
(  9,701)( 10,704)( 11,703)( 12,702)( 13,685)( 14,688)( 15,687)( 16,686)
( 17,693)( 18,696)( 19,695)( 20,694)( 21,689)( 22,692)( 23,691)( 24,690)
( 25,709)( 26,712)( 27,711)( 28,710)( 29,717)( 30,720)( 31,719)( 32,718)
( 33,713)( 34,716)( 35,715)( 36,714)( 37,661)( 38,664)( 39,663)( 40,662)
( 41,669)( 42,672)( 43,671)( 44,670)( 45,665)( 46,668)( 47,667)( 48,666)
( 49,649)( 50,652)( 51,651)( 52,650)( 53,657)( 54,660)( 55,659)( 56,658)
( 57,653)( 58,656)( 59,655)( 60,654)( 61,673)( 62,676)( 63,675)( 64,674)
( 65,681)( 66,684)( 67,683)( 68,682)( 69,677)( 70,680)( 71,679)( 72,678)
( 73,733)( 74,736)( 75,735)( 76,734)( 77,741)( 78,744)( 79,743)( 80,742)
( 81,737)( 82,740)( 83,739)( 84,738)( 85,721)( 86,724)( 87,723)( 88,722)
( 89,729)( 90,732)( 91,731)( 92,730)( 93,725)( 94,728)( 95,727)( 96,726)
( 97,745)( 98,748)( 99,747)(100,746)(101,753)(102,756)(103,755)(104,754)
(105,749)(106,752)(107,751)(108,750)(109,805)(110,808)(111,807)(112,806)
(113,813)(114,816)(115,815)(116,814)(117,809)(118,812)(119,811)(120,810)
(121,793)(122,796)(123,795)(124,794)(125,801)(126,804)(127,803)(128,802)
(129,797)(130,800)(131,799)(132,798)(133,817)(134,820)(135,819)(136,818)
(137,825)(138,828)(139,827)(140,826)(141,821)(142,824)(143,823)(144,822)
(145,769)(146,772)(147,771)(148,770)(149,777)(150,780)(151,779)(152,778)
(153,773)(154,776)(155,775)(156,774)(157,757)(158,760)(159,759)(160,758)
(161,765)(162,768)(163,767)(164,766)(165,761)(166,764)(167,763)(168,762)
(169,781)(170,784)(171,783)(172,782)(173,789)(174,792)(175,791)(176,790)
(177,785)(178,788)(179,787)(180,786)(181,841)(182,844)(183,843)(184,842)
(185,849)(186,852)(187,851)(188,850)(189,845)(190,848)(191,847)(192,846)
(193,829)(194,832)(195,831)(196,830)(197,837)(198,840)(199,839)(200,838)
(201,833)(202,836)(203,835)(204,834)(205,853)(206,856)(207,855)(208,854)
(209,861)(210,864)(211,863)(212,862)(213,857)(214,860)(215,859)(216,858)
(217,481)(218,484)(219,483)(220,482)(221,489)(222,492)(223,491)(224,490)
(225,485)(226,488)(227,487)(228,486)(229,469)(230,472)(231,471)(232,470)
(233,477)(234,480)(235,479)(236,478)(237,473)(238,476)(239,475)(240,474)
(241,493)(242,496)(243,495)(244,494)(245,501)(246,504)(247,503)(248,502)
(249,497)(250,500)(251,499)(252,498)(253,445)(254,448)(255,447)(256,446)
(257,453)(258,456)(259,455)(260,454)(261,449)(262,452)(263,451)(264,450)
(265,433)(266,436)(267,435)(268,434)(269,441)(270,444)(271,443)(272,442)
(273,437)(274,440)(275,439)(276,438)(277,457)(278,460)(279,459)(280,458)
(281,465)(282,468)(283,467)(284,466)(285,461)(286,464)(287,463)(288,462)
(289,517)(290,520)(291,519)(292,518)(293,525)(294,528)(295,527)(296,526)
(297,521)(298,524)(299,523)(300,522)(301,505)(302,508)(303,507)(304,506)
(305,513)(306,516)(307,515)(308,514)(309,509)(310,512)(311,511)(312,510)
(313,529)(314,532)(315,531)(316,530)(317,537)(318,540)(319,539)(320,538)
(321,533)(322,536)(323,535)(324,534)(325,589)(326,592)(327,591)(328,590)
(329,597)(330,600)(331,599)(332,598)(333,593)(334,596)(335,595)(336,594)
(337,577)(338,580)(339,579)(340,578)(341,585)(342,588)(343,587)(344,586)
(345,581)(346,584)(347,583)(348,582)(349,601)(350,604)(351,603)(352,602)
(353,609)(354,612)(355,611)(356,610)(357,605)(358,608)(359,607)(360,606)
(361,553)(362,556)(363,555)(364,554)(365,561)(366,564)(367,563)(368,562)
(369,557)(370,560)(371,559)(372,558)(373,541)(374,544)(375,543)(376,542)
(377,549)(378,552)(379,551)(380,550)(381,545)(382,548)(383,547)(384,546)
(385,565)(386,568)(387,567)(388,566)(389,573)(390,576)(391,575)(392,574)
(393,569)(394,572)(395,571)(396,570)(397,625)(398,628)(399,627)(400,626)
(401,633)(402,636)(403,635)(404,634)(405,629)(406,632)(407,631)(408,630)
(409,613)(410,616)(411,615)(412,614)(413,621)(414,624)(415,623)(416,622)
(417,617)(418,620)(419,619)(420,618)(421,637)(422,640)(423,639)(424,638)
(425,645)(426,648)(427,647)(428,646)(429,641)(430,644)(431,643)(432,642);;
s2 := (  1,654)(  2,653)(  3,655)(  4,656)(  5,650)(  6,649)(  7,651)(  8,652)
(  9,658)( 10,657)( 11,659)( 12,660)( 13,666)( 14,665)( 15,667)( 16,668)
( 17,662)( 18,661)( 19,663)( 20,664)( 21,670)( 22,669)( 23,671)( 24,672)
( 25,678)( 26,677)( 27,679)( 28,680)( 29,674)( 30,673)( 31,675)( 32,676)
( 33,682)( 34,681)( 35,683)( 36,684)( 37,726)( 38,725)( 39,727)( 40,728)
( 41,722)( 42,721)( 43,723)( 44,724)( 45,730)( 46,729)( 47,731)( 48,732)
( 49,738)( 50,737)( 51,739)( 52,740)( 53,734)( 54,733)( 55,735)( 56,736)
( 57,742)( 58,741)( 59,743)( 60,744)( 61,750)( 62,749)( 63,751)( 64,752)
( 65,746)( 66,745)( 67,747)( 68,748)( 69,754)( 70,753)( 71,755)( 72,756)
( 73,690)( 74,689)( 75,691)( 76,692)( 77,686)( 78,685)( 79,687)( 80,688)
( 81,694)( 82,693)( 83,695)( 84,696)( 85,702)( 86,701)( 87,703)( 88,704)
( 89,698)( 90,697)( 91,699)( 92,700)( 93,706)( 94,705)( 95,707)( 96,708)
( 97,714)( 98,713)( 99,715)(100,716)(101,710)(102,709)(103,711)(104,712)
(105,718)(106,717)(107,719)(108,720)(109,762)(110,761)(111,763)(112,764)
(113,758)(114,757)(115,759)(116,760)(117,766)(118,765)(119,767)(120,768)
(121,774)(122,773)(123,775)(124,776)(125,770)(126,769)(127,771)(128,772)
(129,778)(130,777)(131,779)(132,780)(133,786)(134,785)(135,787)(136,788)
(137,782)(138,781)(139,783)(140,784)(141,790)(142,789)(143,791)(144,792)
(145,834)(146,833)(147,835)(148,836)(149,830)(150,829)(151,831)(152,832)
(153,838)(154,837)(155,839)(156,840)(157,846)(158,845)(159,847)(160,848)
(161,842)(162,841)(163,843)(164,844)(165,850)(166,849)(167,851)(168,852)
(169,858)(170,857)(171,859)(172,860)(173,854)(174,853)(175,855)(176,856)
(177,862)(178,861)(179,863)(180,864)(181,798)(182,797)(183,799)(184,800)
(185,794)(186,793)(187,795)(188,796)(189,802)(190,801)(191,803)(192,804)
(193,810)(194,809)(195,811)(196,812)(197,806)(198,805)(199,807)(200,808)
(201,814)(202,813)(203,815)(204,816)(205,822)(206,821)(207,823)(208,824)
(209,818)(210,817)(211,819)(212,820)(213,826)(214,825)(215,827)(216,828)
(217,546)(218,545)(219,547)(220,548)(221,542)(222,541)(223,543)(224,544)
(225,550)(226,549)(227,551)(228,552)(229,558)(230,557)(231,559)(232,560)
(233,554)(234,553)(235,555)(236,556)(237,562)(238,561)(239,563)(240,564)
(241,570)(242,569)(243,571)(244,572)(245,566)(246,565)(247,567)(248,568)
(249,574)(250,573)(251,575)(252,576)(253,618)(254,617)(255,619)(256,620)
(257,614)(258,613)(259,615)(260,616)(261,622)(262,621)(263,623)(264,624)
(265,630)(266,629)(267,631)(268,632)(269,626)(270,625)(271,627)(272,628)
(273,634)(274,633)(275,635)(276,636)(277,642)(278,641)(279,643)(280,644)
(281,638)(282,637)(283,639)(284,640)(285,646)(286,645)(287,647)(288,648)
(289,582)(290,581)(291,583)(292,584)(293,578)(294,577)(295,579)(296,580)
(297,586)(298,585)(299,587)(300,588)(301,594)(302,593)(303,595)(304,596)
(305,590)(306,589)(307,591)(308,592)(309,598)(310,597)(311,599)(312,600)
(313,606)(314,605)(315,607)(316,608)(317,602)(318,601)(319,603)(320,604)
(321,610)(322,609)(323,611)(324,612)(325,438)(326,437)(327,439)(328,440)
(329,434)(330,433)(331,435)(332,436)(333,442)(334,441)(335,443)(336,444)
(337,450)(338,449)(339,451)(340,452)(341,446)(342,445)(343,447)(344,448)
(345,454)(346,453)(347,455)(348,456)(349,462)(350,461)(351,463)(352,464)
(353,458)(354,457)(355,459)(356,460)(357,466)(358,465)(359,467)(360,468)
(361,510)(362,509)(363,511)(364,512)(365,506)(366,505)(367,507)(368,508)
(369,514)(370,513)(371,515)(372,516)(373,522)(374,521)(375,523)(376,524)
(377,518)(378,517)(379,519)(380,520)(381,526)(382,525)(383,527)(384,528)
(385,534)(386,533)(387,535)(388,536)(389,530)(390,529)(391,531)(392,532)
(393,538)(394,537)(395,539)(396,540)(397,474)(398,473)(399,475)(400,476)
(401,470)(402,469)(403,471)(404,472)(405,478)(406,477)(407,479)(408,480)
(409,486)(410,485)(411,487)(412,488)(413,482)(414,481)(415,483)(416,484)
(417,490)(418,489)(419,491)(420,492)(421,498)(422,497)(423,499)(424,500)
(425,494)(426,493)(427,495)(428,496)(429,502)(430,501)(431,503)(432,504);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*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, 
s0*s1*s0*s1*s0*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)!(  1,217)(  2,218)(  3,220)(  4,219)(  5,221)(  6,222)(  7,224)
(  8,223)(  9,225)( 10,226)( 11,228)( 12,227)( 13,241)( 14,242)( 15,244)
( 16,243)( 17,245)( 18,246)( 19,248)( 20,247)( 21,249)( 22,250)( 23,252)
( 24,251)( 25,229)( 26,230)( 27,232)( 28,231)( 29,233)( 30,234)( 31,236)
( 32,235)( 33,237)( 34,238)( 35,240)( 36,239)( 37,289)( 38,290)( 39,292)
( 40,291)( 41,293)( 42,294)( 43,296)( 44,295)( 45,297)( 46,298)( 47,300)
( 48,299)( 49,313)( 50,314)( 51,316)( 52,315)( 53,317)( 54,318)( 55,320)
( 56,319)( 57,321)( 58,322)( 59,324)( 60,323)( 61,301)( 62,302)( 63,304)
( 64,303)( 65,305)( 66,306)( 67,308)( 68,307)( 69,309)( 70,310)( 71,312)
( 72,311)( 73,253)( 74,254)( 75,256)( 76,255)( 77,257)( 78,258)( 79,260)
( 80,259)( 81,261)( 82,262)( 83,264)( 84,263)( 85,277)( 86,278)( 87,280)
( 88,279)( 89,281)( 90,282)( 91,284)( 92,283)( 93,285)( 94,286)( 95,288)
( 96,287)( 97,265)( 98,266)( 99,268)(100,267)(101,269)(102,270)(103,272)
(104,271)(105,273)(106,274)(107,276)(108,275)(109,325)(110,326)(111,328)
(112,327)(113,329)(114,330)(115,332)(116,331)(117,333)(118,334)(119,336)
(120,335)(121,349)(122,350)(123,352)(124,351)(125,353)(126,354)(127,356)
(128,355)(129,357)(130,358)(131,360)(132,359)(133,337)(134,338)(135,340)
(136,339)(137,341)(138,342)(139,344)(140,343)(141,345)(142,346)(143,348)
(144,347)(145,397)(146,398)(147,400)(148,399)(149,401)(150,402)(151,404)
(152,403)(153,405)(154,406)(155,408)(156,407)(157,421)(158,422)(159,424)
(160,423)(161,425)(162,426)(163,428)(164,427)(165,429)(166,430)(167,432)
(168,431)(169,409)(170,410)(171,412)(172,411)(173,413)(174,414)(175,416)
(176,415)(177,417)(178,418)(179,420)(180,419)(181,361)(182,362)(183,364)
(184,363)(185,365)(186,366)(187,368)(188,367)(189,369)(190,370)(191,372)
(192,371)(193,385)(194,386)(195,388)(196,387)(197,389)(198,390)(199,392)
(200,391)(201,393)(202,394)(203,396)(204,395)(205,373)(206,374)(207,376)
(208,375)(209,377)(210,378)(211,380)(212,379)(213,381)(214,382)(215,384)
(216,383)(433,757)(434,758)(435,760)(436,759)(437,761)(438,762)(439,764)
(440,763)(441,765)(442,766)(443,768)(444,767)(445,781)(446,782)(447,784)
(448,783)(449,785)(450,786)(451,788)(452,787)(453,789)(454,790)(455,792)
(456,791)(457,769)(458,770)(459,772)(460,771)(461,773)(462,774)(463,776)
(464,775)(465,777)(466,778)(467,780)(468,779)(469,829)(470,830)(471,832)
(472,831)(473,833)(474,834)(475,836)(476,835)(477,837)(478,838)(479,840)
(480,839)(481,853)(482,854)(483,856)(484,855)(485,857)(486,858)(487,860)
(488,859)(489,861)(490,862)(491,864)(492,863)(493,841)(494,842)(495,844)
(496,843)(497,845)(498,846)(499,848)(500,847)(501,849)(502,850)(503,852)
(504,851)(505,793)(506,794)(507,796)(508,795)(509,797)(510,798)(511,800)
(512,799)(513,801)(514,802)(515,804)(516,803)(517,817)(518,818)(519,820)
(520,819)(521,821)(522,822)(523,824)(524,823)(525,825)(526,826)(527,828)
(528,827)(529,805)(530,806)(531,808)(532,807)(533,809)(534,810)(535,812)
(536,811)(537,813)(538,814)(539,816)(540,815)(541,649)(542,650)(543,652)
(544,651)(545,653)(546,654)(547,656)(548,655)(549,657)(550,658)(551,660)
(552,659)(553,673)(554,674)(555,676)(556,675)(557,677)(558,678)(559,680)
(560,679)(561,681)(562,682)(563,684)(564,683)(565,661)(566,662)(567,664)
(568,663)(569,665)(570,666)(571,668)(572,667)(573,669)(574,670)(575,672)
(576,671)(577,721)(578,722)(579,724)(580,723)(581,725)(582,726)(583,728)
(584,727)(585,729)(586,730)(587,732)(588,731)(589,745)(590,746)(591,748)
(592,747)(593,749)(594,750)(595,752)(596,751)(597,753)(598,754)(599,756)
(600,755)(601,733)(602,734)(603,736)(604,735)(605,737)(606,738)(607,740)
(608,739)(609,741)(610,742)(611,744)(612,743)(613,685)(614,686)(615,688)
(616,687)(617,689)(618,690)(619,692)(620,691)(621,693)(622,694)(623,696)
(624,695)(625,709)(626,710)(627,712)(628,711)(629,713)(630,714)(631,716)
(632,715)(633,717)(634,718)(635,720)(636,719)(637,697)(638,698)(639,700)
(640,699)(641,701)(642,702)(643,704)(644,703)(645,705)(646,706)(647,708)
(648,707);
s1 := Sym(864)!(  1,697)(  2,700)(  3,699)(  4,698)(  5,705)(  6,708)(  7,707)
(  8,706)(  9,701)( 10,704)( 11,703)( 12,702)( 13,685)( 14,688)( 15,687)
( 16,686)( 17,693)( 18,696)( 19,695)( 20,694)( 21,689)( 22,692)( 23,691)
( 24,690)( 25,709)( 26,712)( 27,711)( 28,710)( 29,717)( 30,720)( 31,719)
( 32,718)( 33,713)( 34,716)( 35,715)( 36,714)( 37,661)( 38,664)( 39,663)
( 40,662)( 41,669)( 42,672)( 43,671)( 44,670)( 45,665)( 46,668)( 47,667)
( 48,666)( 49,649)( 50,652)( 51,651)( 52,650)( 53,657)( 54,660)( 55,659)
( 56,658)( 57,653)( 58,656)( 59,655)( 60,654)( 61,673)( 62,676)( 63,675)
( 64,674)( 65,681)( 66,684)( 67,683)( 68,682)( 69,677)( 70,680)( 71,679)
( 72,678)( 73,733)( 74,736)( 75,735)( 76,734)( 77,741)( 78,744)( 79,743)
( 80,742)( 81,737)( 82,740)( 83,739)( 84,738)( 85,721)( 86,724)( 87,723)
( 88,722)( 89,729)( 90,732)( 91,731)( 92,730)( 93,725)( 94,728)( 95,727)
( 96,726)( 97,745)( 98,748)( 99,747)(100,746)(101,753)(102,756)(103,755)
(104,754)(105,749)(106,752)(107,751)(108,750)(109,805)(110,808)(111,807)
(112,806)(113,813)(114,816)(115,815)(116,814)(117,809)(118,812)(119,811)
(120,810)(121,793)(122,796)(123,795)(124,794)(125,801)(126,804)(127,803)
(128,802)(129,797)(130,800)(131,799)(132,798)(133,817)(134,820)(135,819)
(136,818)(137,825)(138,828)(139,827)(140,826)(141,821)(142,824)(143,823)
(144,822)(145,769)(146,772)(147,771)(148,770)(149,777)(150,780)(151,779)
(152,778)(153,773)(154,776)(155,775)(156,774)(157,757)(158,760)(159,759)
(160,758)(161,765)(162,768)(163,767)(164,766)(165,761)(166,764)(167,763)
(168,762)(169,781)(170,784)(171,783)(172,782)(173,789)(174,792)(175,791)
(176,790)(177,785)(178,788)(179,787)(180,786)(181,841)(182,844)(183,843)
(184,842)(185,849)(186,852)(187,851)(188,850)(189,845)(190,848)(191,847)
(192,846)(193,829)(194,832)(195,831)(196,830)(197,837)(198,840)(199,839)
(200,838)(201,833)(202,836)(203,835)(204,834)(205,853)(206,856)(207,855)
(208,854)(209,861)(210,864)(211,863)(212,862)(213,857)(214,860)(215,859)
(216,858)(217,481)(218,484)(219,483)(220,482)(221,489)(222,492)(223,491)
(224,490)(225,485)(226,488)(227,487)(228,486)(229,469)(230,472)(231,471)
(232,470)(233,477)(234,480)(235,479)(236,478)(237,473)(238,476)(239,475)
(240,474)(241,493)(242,496)(243,495)(244,494)(245,501)(246,504)(247,503)
(248,502)(249,497)(250,500)(251,499)(252,498)(253,445)(254,448)(255,447)
(256,446)(257,453)(258,456)(259,455)(260,454)(261,449)(262,452)(263,451)
(264,450)(265,433)(266,436)(267,435)(268,434)(269,441)(270,444)(271,443)
(272,442)(273,437)(274,440)(275,439)(276,438)(277,457)(278,460)(279,459)
(280,458)(281,465)(282,468)(283,467)(284,466)(285,461)(286,464)(287,463)
(288,462)(289,517)(290,520)(291,519)(292,518)(293,525)(294,528)(295,527)
(296,526)(297,521)(298,524)(299,523)(300,522)(301,505)(302,508)(303,507)
(304,506)(305,513)(306,516)(307,515)(308,514)(309,509)(310,512)(311,511)
(312,510)(313,529)(314,532)(315,531)(316,530)(317,537)(318,540)(319,539)
(320,538)(321,533)(322,536)(323,535)(324,534)(325,589)(326,592)(327,591)
(328,590)(329,597)(330,600)(331,599)(332,598)(333,593)(334,596)(335,595)
(336,594)(337,577)(338,580)(339,579)(340,578)(341,585)(342,588)(343,587)
(344,586)(345,581)(346,584)(347,583)(348,582)(349,601)(350,604)(351,603)
(352,602)(353,609)(354,612)(355,611)(356,610)(357,605)(358,608)(359,607)
(360,606)(361,553)(362,556)(363,555)(364,554)(365,561)(366,564)(367,563)
(368,562)(369,557)(370,560)(371,559)(372,558)(373,541)(374,544)(375,543)
(376,542)(377,549)(378,552)(379,551)(380,550)(381,545)(382,548)(383,547)
(384,546)(385,565)(386,568)(387,567)(388,566)(389,573)(390,576)(391,575)
(392,574)(393,569)(394,572)(395,571)(396,570)(397,625)(398,628)(399,627)
(400,626)(401,633)(402,636)(403,635)(404,634)(405,629)(406,632)(407,631)
(408,630)(409,613)(410,616)(411,615)(412,614)(413,621)(414,624)(415,623)
(416,622)(417,617)(418,620)(419,619)(420,618)(421,637)(422,640)(423,639)
(424,638)(425,645)(426,648)(427,647)(428,646)(429,641)(430,644)(431,643)
(432,642);
s2 := Sym(864)!(  1,654)(  2,653)(  3,655)(  4,656)(  5,650)(  6,649)(  7,651)
(  8,652)(  9,658)( 10,657)( 11,659)( 12,660)( 13,666)( 14,665)( 15,667)
( 16,668)( 17,662)( 18,661)( 19,663)( 20,664)( 21,670)( 22,669)( 23,671)
( 24,672)( 25,678)( 26,677)( 27,679)( 28,680)( 29,674)( 30,673)( 31,675)
( 32,676)( 33,682)( 34,681)( 35,683)( 36,684)( 37,726)( 38,725)( 39,727)
( 40,728)( 41,722)( 42,721)( 43,723)( 44,724)( 45,730)( 46,729)( 47,731)
( 48,732)( 49,738)( 50,737)( 51,739)( 52,740)( 53,734)( 54,733)( 55,735)
( 56,736)( 57,742)( 58,741)( 59,743)( 60,744)( 61,750)( 62,749)( 63,751)
( 64,752)( 65,746)( 66,745)( 67,747)( 68,748)( 69,754)( 70,753)( 71,755)
( 72,756)( 73,690)( 74,689)( 75,691)( 76,692)( 77,686)( 78,685)( 79,687)
( 80,688)( 81,694)( 82,693)( 83,695)( 84,696)( 85,702)( 86,701)( 87,703)
( 88,704)( 89,698)( 90,697)( 91,699)( 92,700)( 93,706)( 94,705)( 95,707)
( 96,708)( 97,714)( 98,713)( 99,715)(100,716)(101,710)(102,709)(103,711)
(104,712)(105,718)(106,717)(107,719)(108,720)(109,762)(110,761)(111,763)
(112,764)(113,758)(114,757)(115,759)(116,760)(117,766)(118,765)(119,767)
(120,768)(121,774)(122,773)(123,775)(124,776)(125,770)(126,769)(127,771)
(128,772)(129,778)(130,777)(131,779)(132,780)(133,786)(134,785)(135,787)
(136,788)(137,782)(138,781)(139,783)(140,784)(141,790)(142,789)(143,791)
(144,792)(145,834)(146,833)(147,835)(148,836)(149,830)(150,829)(151,831)
(152,832)(153,838)(154,837)(155,839)(156,840)(157,846)(158,845)(159,847)
(160,848)(161,842)(162,841)(163,843)(164,844)(165,850)(166,849)(167,851)
(168,852)(169,858)(170,857)(171,859)(172,860)(173,854)(174,853)(175,855)
(176,856)(177,862)(178,861)(179,863)(180,864)(181,798)(182,797)(183,799)
(184,800)(185,794)(186,793)(187,795)(188,796)(189,802)(190,801)(191,803)
(192,804)(193,810)(194,809)(195,811)(196,812)(197,806)(198,805)(199,807)
(200,808)(201,814)(202,813)(203,815)(204,816)(205,822)(206,821)(207,823)
(208,824)(209,818)(210,817)(211,819)(212,820)(213,826)(214,825)(215,827)
(216,828)(217,546)(218,545)(219,547)(220,548)(221,542)(222,541)(223,543)
(224,544)(225,550)(226,549)(227,551)(228,552)(229,558)(230,557)(231,559)
(232,560)(233,554)(234,553)(235,555)(236,556)(237,562)(238,561)(239,563)
(240,564)(241,570)(242,569)(243,571)(244,572)(245,566)(246,565)(247,567)
(248,568)(249,574)(250,573)(251,575)(252,576)(253,618)(254,617)(255,619)
(256,620)(257,614)(258,613)(259,615)(260,616)(261,622)(262,621)(263,623)
(264,624)(265,630)(266,629)(267,631)(268,632)(269,626)(270,625)(271,627)
(272,628)(273,634)(274,633)(275,635)(276,636)(277,642)(278,641)(279,643)
(280,644)(281,638)(282,637)(283,639)(284,640)(285,646)(286,645)(287,647)
(288,648)(289,582)(290,581)(291,583)(292,584)(293,578)(294,577)(295,579)
(296,580)(297,586)(298,585)(299,587)(300,588)(301,594)(302,593)(303,595)
(304,596)(305,590)(306,589)(307,591)(308,592)(309,598)(310,597)(311,599)
(312,600)(313,606)(314,605)(315,607)(316,608)(317,602)(318,601)(319,603)
(320,604)(321,610)(322,609)(323,611)(324,612)(325,438)(326,437)(327,439)
(328,440)(329,434)(330,433)(331,435)(332,436)(333,442)(334,441)(335,443)
(336,444)(337,450)(338,449)(339,451)(340,452)(341,446)(342,445)(343,447)
(344,448)(345,454)(346,453)(347,455)(348,456)(349,462)(350,461)(351,463)
(352,464)(353,458)(354,457)(355,459)(356,460)(357,466)(358,465)(359,467)
(360,468)(361,510)(362,509)(363,511)(364,512)(365,506)(366,505)(367,507)
(368,508)(369,514)(370,513)(371,515)(372,516)(373,522)(374,521)(375,523)
(376,524)(377,518)(378,517)(379,519)(380,520)(381,526)(382,525)(383,527)
(384,528)(385,534)(386,533)(387,535)(388,536)(389,530)(390,529)(391,531)
(392,532)(393,538)(394,537)(395,539)(396,540)(397,474)(398,473)(399,475)
(400,476)(401,470)(402,469)(403,471)(404,472)(405,478)(406,477)(407,479)
(408,480)(409,486)(410,485)(411,487)(412,488)(413,482)(414,481)(415,483)
(416,484)(417,490)(418,489)(419,491)(420,492)(421,498)(422,497)(423,499)
(424,500)(425,494)(426,493)(427,495)(428,496)(429,502)(430,501)(431,503)
(432,504);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope