Polytope of Type {24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6}*1728g
if this polytope has a name.
Group : SmallGroup(1728,46313)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 144, 432, 36
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 24
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 : {12,6}*864c
   3-fold quotients : {24,6}*576c, {24,6}*576d
   6-fold quotients : {12,6}*288a, {12,6}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {8,6}*192c
   12-fold quotients : {12,3}*144, {12,6}*144d
   18-fold quotients : {4,6}*96
   24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {3,6}*36, {6,3}*36
   72-fold quotients : {4,3}*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 := (  1,437)(  2,438)(  3,440)(  4,439)(  5,434)(  6,433)(  7,435)(  8,436)
(  9,453)( 10,454)( 11,456)( 12,455)( 13,450)( 14,449)( 15,451)( 16,452)
( 17,445)( 18,446)( 19,448)( 20,447)( 21,442)( 22,441)( 23,443)( 24,444)
( 25,485)( 26,486)( 27,488)( 28,487)( 29,482)( 30,481)( 31,483)( 32,484)
( 33,501)( 34,502)( 35,504)( 36,503)( 37,498)( 38,497)( 39,499)( 40,500)
( 41,493)( 42,494)( 43,496)( 44,495)( 45,490)( 46,489)( 47,491)( 48,492)
( 49,461)( 50,462)( 51,464)( 52,463)( 53,458)( 54,457)( 55,459)( 56,460)
( 57,477)( 58,478)( 59,480)( 60,479)( 61,474)( 62,473)( 63,475)( 64,476)
( 65,469)( 66,470)( 67,472)( 68,471)( 69,466)( 70,465)( 71,467)( 72,468)
( 73,509)( 74,510)( 75,512)( 76,511)( 77,506)( 78,505)( 79,507)( 80,508)
( 81,525)( 82,526)( 83,528)( 84,527)( 85,522)( 86,521)( 87,523)( 88,524)
( 89,517)( 90,518)( 91,520)( 92,519)( 93,514)( 94,513)( 95,515)( 96,516)
( 97,557)( 98,558)( 99,560)(100,559)(101,554)(102,553)(103,555)(104,556)
(105,573)(106,574)(107,576)(108,575)(109,570)(110,569)(111,571)(112,572)
(113,565)(114,566)(115,568)(116,567)(117,562)(118,561)(119,563)(120,564)
(121,533)(122,534)(123,536)(124,535)(125,530)(126,529)(127,531)(128,532)
(129,549)(130,550)(131,552)(132,551)(133,546)(134,545)(135,547)(136,548)
(137,541)(138,542)(139,544)(140,543)(141,538)(142,537)(143,539)(144,540)
(145,581)(146,582)(147,584)(148,583)(149,578)(150,577)(151,579)(152,580)
(153,597)(154,598)(155,600)(156,599)(157,594)(158,593)(159,595)(160,596)
(161,589)(162,590)(163,592)(164,591)(165,586)(166,585)(167,587)(168,588)
(169,629)(170,630)(171,632)(172,631)(173,626)(174,625)(175,627)(176,628)
(177,645)(178,646)(179,648)(180,647)(181,642)(182,641)(183,643)(184,644)
(185,637)(186,638)(187,640)(188,639)(189,634)(190,633)(191,635)(192,636)
(193,605)(194,606)(195,608)(196,607)(197,602)(198,601)(199,603)(200,604)
(201,621)(202,622)(203,624)(204,623)(205,618)(206,617)(207,619)(208,620)
(209,613)(210,614)(211,616)(212,615)(213,610)(214,609)(215,611)(216,612)
(217,653)(218,654)(219,656)(220,655)(221,650)(222,649)(223,651)(224,652)
(225,669)(226,670)(227,672)(228,671)(229,666)(230,665)(231,667)(232,668)
(233,661)(234,662)(235,664)(236,663)(237,658)(238,657)(239,659)(240,660)
(241,701)(242,702)(243,704)(244,703)(245,698)(246,697)(247,699)(248,700)
(249,717)(250,718)(251,720)(252,719)(253,714)(254,713)(255,715)(256,716)
(257,709)(258,710)(259,712)(260,711)(261,706)(262,705)(263,707)(264,708)
(265,677)(266,678)(267,680)(268,679)(269,674)(270,673)(271,675)(272,676)
(273,693)(274,694)(275,696)(276,695)(277,690)(278,689)(279,691)(280,692)
(281,685)(282,686)(283,688)(284,687)(285,682)(286,681)(287,683)(288,684)
(289,725)(290,726)(291,728)(292,727)(293,722)(294,721)(295,723)(296,724)
(297,741)(298,742)(299,744)(300,743)(301,738)(302,737)(303,739)(304,740)
(305,733)(306,734)(307,736)(308,735)(309,730)(310,729)(311,731)(312,732)
(313,773)(314,774)(315,776)(316,775)(317,770)(318,769)(319,771)(320,772)
(321,789)(322,790)(323,792)(324,791)(325,786)(326,785)(327,787)(328,788)
(329,781)(330,782)(331,784)(332,783)(333,778)(334,777)(335,779)(336,780)
(337,749)(338,750)(339,752)(340,751)(341,746)(342,745)(343,747)(344,748)
(345,765)(346,766)(347,768)(348,767)(349,762)(350,761)(351,763)(352,764)
(353,757)(354,758)(355,760)(356,759)(357,754)(358,753)(359,755)(360,756)
(361,797)(362,798)(363,800)(364,799)(365,794)(366,793)(367,795)(368,796)
(369,813)(370,814)(371,816)(372,815)(373,810)(374,809)(375,811)(376,812)
(377,805)(378,806)(379,808)(380,807)(381,802)(382,801)(383,803)(384,804)
(385,845)(386,846)(387,848)(388,847)(389,842)(390,841)(391,843)(392,844)
(393,861)(394,862)(395,864)(396,863)(397,858)(398,857)(399,859)(400,860)
(401,853)(402,854)(403,856)(404,855)(405,850)(406,849)(407,851)(408,852)
(409,821)(410,822)(411,824)(412,823)(413,818)(414,817)(415,819)(416,820)
(417,837)(418,838)(419,840)(420,839)(421,834)(422,833)(423,835)(424,836)
(425,829)(426,830)(427,832)(428,831)(429,826)(430,825)(431,827)(432,828);;
s1 := (  1,681)(  2,682)(  3,684)(  4,683)(  5,687)(  6,688)(  7,685)(  8,686)
(  9,673)( 10,674)( 11,676)( 12,675)( 13,679)( 14,680)( 15,677)( 16,678)
( 17,689)( 18,690)( 19,692)( 20,691)( 21,695)( 22,696)( 23,693)( 24,694)
( 25,657)( 26,658)( 27,660)( 28,659)( 29,663)( 30,664)( 31,661)( 32,662)
( 33,649)( 34,650)( 35,652)( 36,651)( 37,655)( 38,656)( 39,653)( 40,654)
( 41,665)( 42,666)( 43,668)( 44,667)( 45,671)( 46,672)( 47,669)( 48,670)
( 49,705)( 50,706)( 51,708)( 52,707)( 53,711)( 54,712)( 55,709)( 56,710)
( 57,697)( 58,698)( 59,700)( 60,699)( 61,703)( 62,704)( 63,701)( 64,702)
( 65,713)( 66,714)( 67,716)( 68,715)( 69,719)( 70,720)( 71,717)( 72,718)
( 73,825)( 74,826)( 75,828)( 76,827)( 77,831)( 78,832)( 79,829)( 80,830)
( 81,817)( 82,818)( 83,820)( 84,819)( 85,823)( 86,824)( 87,821)( 88,822)
( 89,833)( 90,834)( 91,836)( 92,835)( 93,839)( 94,840)( 95,837)( 96,838)
( 97,801)( 98,802)( 99,804)(100,803)(101,807)(102,808)(103,805)(104,806)
(105,793)(106,794)(107,796)(108,795)(109,799)(110,800)(111,797)(112,798)
(113,809)(114,810)(115,812)(116,811)(117,815)(118,816)(119,813)(120,814)
(121,849)(122,850)(123,852)(124,851)(125,855)(126,856)(127,853)(128,854)
(129,841)(130,842)(131,844)(132,843)(133,847)(134,848)(135,845)(136,846)
(137,857)(138,858)(139,860)(140,859)(141,863)(142,864)(143,861)(144,862)
(145,753)(146,754)(147,756)(148,755)(149,759)(150,760)(151,757)(152,758)
(153,745)(154,746)(155,748)(156,747)(157,751)(158,752)(159,749)(160,750)
(161,761)(162,762)(163,764)(164,763)(165,767)(166,768)(167,765)(168,766)
(169,729)(170,730)(171,732)(172,731)(173,735)(174,736)(175,733)(176,734)
(177,721)(178,722)(179,724)(180,723)(181,727)(182,728)(183,725)(184,726)
(185,737)(186,738)(187,740)(188,739)(189,743)(190,744)(191,741)(192,742)
(193,777)(194,778)(195,780)(196,779)(197,783)(198,784)(199,781)(200,782)
(201,769)(202,770)(203,772)(204,771)(205,775)(206,776)(207,773)(208,774)
(209,785)(210,786)(211,788)(212,787)(213,791)(214,792)(215,789)(216,790)
(217,466)(218,465)(219,467)(220,468)(221,472)(222,471)(223,470)(224,469)
(225,458)(226,457)(227,459)(228,460)(229,464)(230,463)(231,462)(232,461)
(233,474)(234,473)(235,475)(236,476)(237,480)(238,479)(239,478)(240,477)
(241,442)(242,441)(243,443)(244,444)(245,448)(246,447)(247,446)(248,445)
(249,434)(250,433)(251,435)(252,436)(253,440)(254,439)(255,438)(256,437)
(257,450)(258,449)(259,451)(260,452)(261,456)(262,455)(263,454)(264,453)
(265,490)(266,489)(267,491)(268,492)(269,496)(270,495)(271,494)(272,493)
(273,482)(274,481)(275,483)(276,484)(277,488)(278,487)(279,486)(280,485)
(281,498)(282,497)(283,499)(284,500)(285,504)(286,503)(287,502)(288,501)
(289,610)(290,609)(291,611)(292,612)(293,616)(294,615)(295,614)(296,613)
(297,602)(298,601)(299,603)(300,604)(301,608)(302,607)(303,606)(304,605)
(305,618)(306,617)(307,619)(308,620)(309,624)(310,623)(311,622)(312,621)
(313,586)(314,585)(315,587)(316,588)(317,592)(318,591)(319,590)(320,589)
(321,578)(322,577)(323,579)(324,580)(325,584)(326,583)(327,582)(328,581)
(329,594)(330,593)(331,595)(332,596)(333,600)(334,599)(335,598)(336,597)
(337,634)(338,633)(339,635)(340,636)(341,640)(342,639)(343,638)(344,637)
(345,626)(346,625)(347,627)(348,628)(349,632)(350,631)(351,630)(352,629)
(353,642)(354,641)(355,643)(356,644)(357,648)(358,647)(359,646)(360,645)
(361,538)(362,537)(363,539)(364,540)(365,544)(366,543)(367,542)(368,541)
(369,530)(370,529)(371,531)(372,532)(373,536)(374,535)(375,534)(376,533)
(377,546)(378,545)(379,547)(380,548)(381,552)(382,551)(383,550)(384,549)
(385,514)(386,513)(387,515)(388,516)(389,520)(390,519)(391,518)(392,517)
(393,506)(394,505)(395,507)(396,508)(397,512)(398,511)(399,510)(400,509)
(401,522)(402,521)(403,523)(404,524)(405,528)(406,527)(407,526)(408,525)
(409,562)(410,561)(411,563)(412,564)(413,568)(414,567)(415,566)(416,565)
(417,554)(418,553)(419,555)(420,556)(421,560)(422,559)(423,558)(424,557)
(425,570)(426,569)(427,571)(428,572)(429,576)(430,575)(431,574)(432,573);;
s2 := (  1,289)(  2,290)(  3,295)(  4,296)(  5,294)(  6,293)(  7,291)(  8,292)
(  9,297)( 10,298)( 11,303)( 12,304)( 13,302)( 14,301)( 15,299)( 16,300)
( 17,305)( 18,306)( 19,311)( 20,312)( 21,310)( 22,309)( 23,307)( 24,308)
( 25,337)( 26,338)( 27,343)( 28,344)( 29,342)( 30,341)( 31,339)( 32,340)
( 33,345)( 34,346)( 35,351)( 36,352)( 37,350)( 38,349)( 39,347)( 40,348)
( 41,353)( 42,354)( 43,359)( 44,360)( 45,358)( 46,357)( 47,355)( 48,356)
( 49,313)( 50,314)( 51,319)( 52,320)( 53,318)( 54,317)( 55,315)( 56,316)
( 57,321)( 58,322)( 59,327)( 60,328)( 61,326)( 62,325)( 63,323)( 64,324)
( 65,329)( 66,330)( 67,335)( 68,336)( 69,334)( 70,333)( 71,331)( 72,332)
( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)( 80,220)
( 81,225)( 82,226)( 83,231)( 84,232)( 85,230)( 86,229)( 87,227)( 88,228)
( 89,233)( 90,234)( 91,239)( 92,240)( 93,238)( 94,237)( 95,235)( 96,236)
( 97,265)( 98,266)( 99,271)(100,272)(101,270)(102,269)(103,267)(104,268)
(105,273)(106,274)(107,279)(108,280)(109,278)(110,277)(111,275)(112,276)
(113,281)(114,282)(115,287)(116,288)(117,286)(118,285)(119,283)(120,284)
(121,241)(122,242)(123,247)(124,248)(125,246)(126,245)(127,243)(128,244)
(129,249)(130,250)(131,255)(132,256)(133,254)(134,253)(135,251)(136,252)
(137,257)(138,258)(139,263)(140,264)(141,262)(142,261)(143,259)(144,260)
(145,361)(146,362)(147,367)(148,368)(149,366)(150,365)(151,363)(152,364)
(153,369)(154,370)(155,375)(156,376)(157,374)(158,373)(159,371)(160,372)
(161,377)(162,378)(163,383)(164,384)(165,382)(166,381)(167,379)(168,380)
(169,409)(170,410)(171,415)(172,416)(173,414)(174,413)(175,411)(176,412)
(177,417)(178,418)(179,423)(180,424)(181,422)(182,421)(183,419)(184,420)
(185,425)(186,426)(187,431)(188,432)(189,430)(190,429)(191,427)(192,428)
(193,385)(194,386)(195,391)(196,392)(197,390)(198,389)(199,387)(200,388)
(201,393)(202,394)(203,399)(204,400)(205,398)(206,397)(207,395)(208,396)
(209,401)(210,402)(211,407)(212,408)(213,406)(214,405)(215,403)(216,404)
(433,722)(434,721)(435,728)(436,727)(437,725)(438,726)(439,724)(440,723)
(441,730)(442,729)(443,736)(444,735)(445,733)(446,734)(447,732)(448,731)
(449,738)(450,737)(451,744)(452,743)(453,741)(454,742)(455,740)(456,739)
(457,770)(458,769)(459,776)(460,775)(461,773)(462,774)(463,772)(464,771)
(465,778)(466,777)(467,784)(468,783)(469,781)(470,782)(471,780)(472,779)
(473,786)(474,785)(475,792)(476,791)(477,789)(478,790)(479,788)(480,787)
(481,746)(482,745)(483,752)(484,751)(485,749)(486,750)(487,748)(488,747)
(489,754)(490,753)(491,760)(492,759)(493,757)(494,758)(495,756)(496,755)
(497,762)(498,761)(499,768)(500,767)(501,765)(502,766)(503,764)(504,763)
(505,650)(506,649)(507,656)(508,655)(509,653)(510,654)(511,652)(512,651)
(513,658)(514,657)(515,664)(516,663)(517,661)(518,662)(519,660)(520,659)
(521,666)(522,665)(523,672)(524,671)(525,669)(526,670)(527,668)(528,667)
(529,698)(530,697)(531,704)(532,703)(533,701)(534,702)(535,700)(536,699)
(537,706)(538,705)(539,712)(540,711)(541,709)(542,710)(543,708)(544,707)
(545,714)(546,713)(547,720)(548,719)(549,717)(550,718)(551,716)(552,715)
(553,674)(554,673)(555,680)(556,679)(557,677)(558,678)(559,676)(560,675)
(561,682)(562,681)(563,688)(564,687)(565,685)(566,686)(567,684)(568,683)
(569,690)(570,689)(571,696)(572,695)(573,693)(574,694)(575,692)(576,691)
(577,794)(578,793)(579,800)(580,799)(581,797)(582,798)(583,796)(584,795)
(585,802)(586,801)(587,808)(588,807)(589,805)(590,806)(591,804)(592,803)
(593,810)(594,809)(595,816)(596,815)(597,813)(598,814)(599,812)(600,811)
(601,842)(602,841)(603,848)(604,847)(605,845)(606,846)(607,844)(608,843)
(609,850)(610,849)(611,856)(612,855)(613,853)(614,854)(615,852)(616,851)
(617,858)(618,857)(619,864)(620,863)(621,861)(622,862)(623,860)(624,859)
(625,818)(626,817)(627,824)(628,823)(629,821)(630,822)(631,820)(632,819)
(633,826)(634,825)(635,832)(636,831)(637,829)(638,830)(639,828)(640,827)
(641,834)(642,833)(643,840)(644,839)(645,837)(646,838)(647,836)(648,835);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope