Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1728g
if this polytope has a name.
Group : SmallGroup(1728,46303)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 144, 432, 72
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
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}*576c, {12,6}*576d
   6-fold quotients : {12,3}*288, {6,6}*288a, {6,6}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {12,6}*192b
   12-fold quotients : {3,6}*144, {6,3}*144
   18-fold quotients : {12,3}*96, {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)
(433,649)(434,650)(435,652)(436,651)(437,655)(438,656)(439,653)(440,654)
(441,657)(442,658)(443,660)(444,659)(445,663)(446,664)(447,661)(448,662)
(449,665)(450,666)(451,668)(452,667)(453,671)(454,672)(455,669)(456,670)
(457,697)(458,698)(459,700)(460,699)(461,703)(462,704)(463,701)(464,702)
(465,705)(466,706)(467,708)(468,707)(469,711)(470,712)(471,709)(472,710)
(473,713)(474,714)(475,716)(476,715)(477,719)(478,720)(479,717)(480,718)
(481,673)(482,674)(483,676)(484,675)(485,679)(486,680)(487,677)(488,678)
(489,681)(490,682)(491,684)(492,683)(493,687)(494,688)(495,685)(496,686)
(497,689)(498,690)(499,692)(500,691)(501,695)(502,696)(503,693)(504,694)
(505,793)(506,794)(507,796)(508,795)(509,799)(510,800)(511,797)(512,798)
(513,801)(514,802)(515,804)(516,803)(517,807)(518,808)(519,805)(520,806)
(521,809)(522,810)(523,812)(524,811)(525,815)(526,816)(527,813)(528,814)
(529,841)(530,842)(531,844)(532,843)(533,847)(534,848)(535,845)(536,846)
(537,849)(538,850)(539,852)(540,851)(541,855)(542,856)(543,853)(544,854)
(545,857)(546,858)(547,860)(548,859)(549,863)(550,864)(551,861)(552,862)
(553,817)(554,818)(555,820)(556,819)(557,823)(558,824)(559,821)(560,822)
(561,825)(562,826)(563,828)(564,827)(565,831)(566,832)(567,829)(568,830)
(569,833)(570,834)(571,836)(572,835)(573,839)(574,840)(575,837)(576,838)
(577,721)(578,722)(579,724)(580,723)(581,727)(582,728)(583,725)(584,726)
(585,729)(586,730)(587,732)(588,731)(589,735)(590,736)(591,733)(592,734)
(593,737)(594,738)(595,740)(596,739)(597,743)(598,744)(599,741)(600,742)
(601,769)(602,770)(603,772)(604,771)(605,775)(606,776)(607,773)(608,774)
(609,777)(610,778)(611,780)(612,779)(613,783)(614,784)(615,781)(616,782)
(617,785)(618,786)(619,788)(620,787)(621,791)(622,792)(623,789)(624,790)
(625,745)(626,746)(627,748)(628,747)(629,751)(630,752)(631,749)(632,750)
(633,753)(634,754)(635,756)(636,755)(637,759)(638,760)(639,757)(640,758)
(641,761)(642,762)(643,764)(644,763)(645,767)(646,768)(647,765)(648,766);;
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)
(217,444)(218,443)(219,442)(220,441)(221,445)(222,446)(223,448)(224,447)
(225,436)(226,435)(227,434)(228,433)(229,437)(230,438)(231,440)(232,439)
(233,452)(234,451)(235,450)(236,449)(237,453)(238,454)(239,456)(240,455)
(241,468)(242,467)(243,466)(244,465)(245,469)(246,470)(247,472)(248,471)
(249,460)(250,459)(251,458)(252,457)(253,461)(254,462)(255,464)(256,463)
(257,476)(258,475)(259,474)(260,473)(261,477)(262,478)(263,480)(264,479)
(265,492)(266,491)(267,490)(268,489)(269,493)(270,494)(271,496)(272,495)
(273,484)(274,483)(275,482)(276,481)(277,485)(278,486)(279,488)(280,487)
(281,500)(282,499)(283,498)(284,497)(285,501)(286,502)(287,504)(288,503)
(289,588)(290,587)(291,586)(292,585)(293,589)(294,590)(295,592)(296,591)
(297,580)(298,579)(299,578)(300,577)(301,581)(302,582)(303,584)(304,583)
(305,596)(306,595)(307,594)(308,593)(309,597)(310,598)(311,600)(312,599)
(313,612)(314,611)(315,610)(316,609)(317,613)(318,614)(319,616)(320,615)
(321,604)(322,603)(323,602)(324,601)(325,605)(326,606)(327,608)(328,607)
(329,620)(330,619)(331,618)(332,617)(333,621)(334,622)(335,624)(336,623)
(337,636)(338,635)(339,634)(340,633)(341,637)(342,638)(343,640)(344,639)
(345,628)(346,627)(347,626)(348,625)(349,629)(350,630)(351,632)(352,631)
(353,644)(354,643)(355,642)(356,641)(357,645)(358,646)(359,648)(360,647)
(361,516)(362,515)(363,514)(364,513)(365,517)(366,518)(367,520)(368,519)
(369,508)(370,507)(371,506)(372,505)(373,509)(374,510)(375,512)(376,511)
(377,524)(378,523)(379,522)(380,521)(381,525)(382,526)(383,528)(384,527)
(385,540)(386,539)(387,538)(388,537)(389,541)(390,542)(391,544)(392,543)
(393,532)(394,531)(395,530)(396,529)(397,533)(398,534)(399,536)(400,535)
(401,548)(402,547)(403,546)(404,545)(405,549)(406,550)(407,552)(408,551)
(409,564)(410,563)(411,562)(412,561)(413,565)(414,566)(415,568)(416,567)
(417,556)(418,555)(419,554)(420,553)(421,557)(422,558)(423,560)(424,559)
(425,572)(426,571)(427,570)(428,569)(429,573)(430,574)(431,576)(432,575);;
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*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*s0*s1*s0*s2*s1*s2*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)(433,649)(434,650)(435,652)(436,651)(437,655)(438,656)(439,653)
(440,654)(441,657)(442,658)(443,660)(444,659)(445,663)(446,664)(447,661)
(448,662)(449,665)(450,666)(451,668)(452,667)(453,671)(454,672)(455,669)
(456,670)(457,697)(458,698)(459,700)(460,699)(461,703)(462,704)(463,701)
(464,702)(465,705)(466,706)(467,708)(468,707)(469,711)(470,712)(471,709)
(472,710)(473,713)(474,714)(475,716)(476,715)(477,719)(478,720)(479,717)
(480,718)(481,673)(482,674)(483,676)(484,675)(485,679)(486,680)(487,677)
(488,678)(489,681)(490,682)(491,684)(492,683)(493,687)(494,688)(495,685)
(496,686)(497,689)(498,690)(499,692)(500,691)(501,695)(502,696)(503,693)
(504,694)(505,793)(506,794)(507,796)(508,795)(509,799)(510,800)(511,797)
(512,798)(513,801)(514,802)(515,804)(516,803)(517,807)(518,808)(519,805)
(520,806)(521,809)(522,810)(523,812)(524,811)(525,815)(526,816)(527,813)
(528,814)(529,841)(530,842)(531,844)(532,843)(533,847)(534,848)(535,845)
(536,846)(537,849)(538,850)(539,852)(540,851)(541,855)(542,856)(543,853)
(544,854)(545,857)(546,858)(547,860)(548,859)(549,863)(550,864)(551,861)
(552,862)(553,817)(554,818)(555,820)(556,819)(557,823)(558,824)(559,821)
(560,822)(561,825)(562,826)(563,828)(564,827)(565,831)(566,832)(567,829)
(568,830)(569,833)(570,834)(571,836)(572,835)(573,839)(574,840)(575,837)
(576,838)(577,721)(578,722)(579,724)(580,723)(581,727)(582,728)(583,725)
(584,726)(585,729)(586,730)(587,732)(588,731)(589,735)(590,736)(591,733)
(592,734)(593,737)(594,738)(595,740)(596,739)(597,743)(598,744)(599,741)
(600,742)(601,769)(602,770)(603,772)(604,771)(605,775)(606,776)(607,773)
(608,774)(609,777)(610,778)(611,780)(612,779)(613,783)(614,784)(615,781)
(616,782)(617,785)(618,786)(619,788)(620,787)(621,791)(622,792)(623,789)
(624,790)(625,745)(626,746)(627,748)(628,747)(629,751)(630,752)(631,749)
(632,750)(633,753)(634,754)(635,756)(636,755)(637,759)(638,760)(639,757)
(640,758)(641,761)(642,762)(643,764)(644,763)(645,767)(646,768)(647,765)
(648,766);
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)(217,444)(218,443)(219,442)(220,441)(221,445)(222,446)(223,448)
(224,447)(225,436)(226,435)(227,434)(228,433)(229,437)(230,438)(231,440)
(232,439)(233,452)(234,451)(235,450)(236,449)(237,453)(238,454)(239,456)
(240,455)(241,468)(242,467)(243,466)(244,465)(245,469)(246,470)(247,472)
(248,471)(249,460)(250,459)(251,458)(252,457)(253,461)(254,462)(255,464)
(256,463)(257,476)(258,475)(259,474)(260,473)(261,477)(262,478)(263,480)
(264,479)(265,492)(266,491)(267,490)(268,489)(269,493)(270,494)(271,496)
(272,495)(273,484)(274,483)(275,482)(276,481)(277,485)(278,486)(279,488)
(280,487)(281,500)(282,499)(283,498)(284,497)(285,501)(286,502)(287,504)
(288,503)(289,588)(290,587)(291,586)(292,585)(293,589)(294,590)(295,592)
(296,591)(297,580)(298,579)(299,578)(300,577)(301,581)(302,582)(303,584)
(304,583)(305,596)(306,595)(307,594)(308,593)(309,597)(310,598)(311,600)
(312,599)(313,612)(314,611)(315,610)(316,609)(317,613)(318,614)(319,616)
(320,615)(321,604)(322,603)(323,602)(324,601)(325,605)(326,606)(327,608)
(328,607)(329,620)(330,619)(331,618)(332,617)(333,621)(334,622)(335,624)
(336,623)(337,636)(338,635)(339,634)(340,633)(341,637)(342,638)(343,640)
(344,639)(345,628)(346,627)(347,626)(348,625)(349,629)(350,630)(351,632)
(352,631)(353,644)(354,643)(355,642)(356,641)(357,645)(358,646)(359,648)
(360,647)(361,516)(362,515)(363,514)(364,513)(365,517)(366,518)(367,520)
(368,519)(369,508)(370,507)(371,506)(372,505)(373,509)(374,510)(375,512)
(376,511)(377,524)(378,523)(379,522)(380,521)(381,525)(382,526)(383,528)
(384,527)(385,540)(386,539)(387,538)(388,537)(389,541)(390,542)(391,544)
(392,543)(393,532)(394,531)(395,530)(396,529)(397,533)(398,534)(399,536)
(400,535)(401,548)(402,547)(403,546)(404,545)(405,549)(406,550)(407,552)
(408,551)(409,564)(410,563)(411,562)(412,561)(413,565)(414,566)(415,568)
(416,567)(417,556)(418,555)(419,554)(420,553)(421,557)(422,558)(423,560)
(424,559)(425,572)(426,571)(427,570)(428,569)(429,573)(430,574)(431,576)
(432,575);
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*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*s0*s1*s0*s2*s1*s2*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