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Polytope of Type {8,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,2}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46634)
Rank : 4
Schlafli Type : {8,6,2}
Number of vertices, edges, etc : 72, 216, 54, 2
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,2}*864b
   3-fold quotients : {8,6,2}*576
   6-fold quotients : {4,6,2}*288
   9-fold quotients : {8,6,2}*192
   12-fold quotients : {4,6,2}*144
   18-fold quotients : {4,6,2}*96a
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,217)(  2,218)(  3,219)(  4,238)(  5,239)(  6,240)(  7,232)(  8,233)
(  9,234)( 10,235)( 11,236)( 12,237)( 13,229)( 14,230)( 15,231)( 16,223)
( 17,224)( 18,225)( 19,226)( 20,227)( 21,228)( 22,220)( 23,221)( 24,222)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,265)( 32,266)
( 33,267)( 34,259)( 35,260)( 36,261)( 37,262)( 38,263)( 39,264)( 40,256)
( 41,257)( 42,258)( 43,250)( 44,251)( 45,252)( 46,253)( 47,254)( 48,255)
( 49,247)( 50,248)( 51,249)( 52,268)( 53,269)( 54,270)( 55,298)( 56,299)
( 57,300)( 58,319)( 59,320)( 60,321)( 61,313)( 62,314)( 63,315)( 64,316)
( 65,317)( 66,318)( 67,310)( 68,311)( 69,312)( 70,304)( 71,305)( 72,306)
( 73,307)( 74,308)( 75,309)( 76,301)( 77,302)( 78,303)( 79,322)( 80,323)
( 81,324)( 82,271)( 83,272)( 84,273)( 85,292)( 86,293)( 87,294)( 88,286)
( 89,287)( 90,288)( 91,289)( 92,290)( 93,291)( 94,283)( 95,284)( 96,285)
( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,274)(104,275)
(105,276)(106,295)(107,296)(108,297)(109,379)(110,380)(111,381)(112,400)
(113,401)(114,402)(115,394)(116,395)(117,396)(118,397)(119,398)(120,399)
(121,391)(122,392)(123,393)(124,385)(125,386)(126,387)(127,388)(128,389)
(129,390)(130,382)(131,383)(132,384)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,427)(140,428)(141,429)(142,421)(143,422)(144,423)
(145,424)(146,425)(147,426)(148,418)(149,419)(150,420)(151,412)(152,413)
(153,414)(154,415)(155,416)(156,417)(157,409)(158,410)(159,411)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,346)(167,347)(168,348)
(169,340)(170,341)(171,342)(172,343)(173,344)(174,345)(175,337)(176,338)
(177,339)(178,331)(179,332)(180,333)(181,334)(182,335)(183,336)(184,328)
(185,329)(186,330)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,373)(194,374)(195,375)(196,367)(197,368)(198,369)(199,370)(200,371)
(201,372)(202,364)(203,365)(204,366)(205,358)(206,359)(207,360)(208,361)
(209,362)(210,363)(211,355)(212,356)(213,357)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,670)(437,671)(438,672)(439,664)(440,665)
(441,666)(442,667)(443,668)(444,669)(445,661)(446,662)(447,663)(448,655)
(449,656)(450,657)(451,658)(452,659)(453,660)(454,652)(455,653)(456,654)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,697)(464,698)
(465,699)(466,691)(467,692)(468,693)(469,694)(470,695)(471,696)(472,688)
(473,689)(474,690)(475,682)(476,683)(477,684)(478,685)(479,686)(480,687)
(481,679)(482,680)(483,681)(484,700)(485,701)(486,702)(487,730)(488,731)
(489,732)(490,751)(491,752)(492,753)(493,745)(494,746)(495,747)(496,748)
(497,749)(498,750)(499,742)(500,743)(501,744)(502,736)(503,737)(504,738)
(505,739)(506,740)(507,741)(508,733)(509,734)(510,735)(511,754)(512,755)
(513,756)(514,703)(515,704)(516,705)(517,724)(518,725)(519,726)(520,718)
(521,719)(522,720)(523,721)(524,722)(525,723)(526,715)(527,716)(528,717)
(529,709)(530,710)(531,711)(532,712)(533,713)(534,714)(535,706)(536,707)
(537,708)(538,727)(539,728)(540,729)(541,811)(542,812)(543,813)(544,832)
(545,833)(546,834)(547,826)(548,827)(549,828)(550,829)(551,830)(552,831)
(553,823)(554,824)(555,825)(556,817)(557,818)(558,819)(559,820)(560,821)
(561,822)(562,814)(563,815)(564,816)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,859)(572,860)(573,861)(574,853)(575,854)(576,855)
(577,856)(578,857)(579,858)(580,850)(581,851)(582,852)(583,844)(584,845)
(585,846)(586,847)(587,848)(588,849)(589,841)(590,842)(591,843)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,778)(599,779)(600,780)
(601,772)(602,773)(603,774)(604,775)(605,776)(606,777)(607,769)(608,770)
(609,771)(610,763)(611,764)(612,765)(613,766)(614,767)(615,768)(616,760)
(617,761)(618,762)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,805)(626,806)(627,807)(628,799)(629,800)(630,801)(631,802)(632,803)
(633,804)(634,796)(635,797)(636,798)(637,790)(638,791)(639,792)(640,793)
(641,794)(642,795)(643,787)(644,788)(645,789)(646,808)(647,809)(648,810);;
s1 := (  1,541)(  2,543)(  3,542)(  4,544)(  5,546)(  6,545)(  7,547)(  8,549)
(  9,548)( 10,565)( 11,567)( 12,566)( 13,559)( 14,561)( 15,560)( 16,562)
( 17,564)( 18,563)( 19,553)( 20,555)( 21,554)( 22,556)( 23,558)( 24,557)
( 25,550)( 26,552)( 27,551)( 28,568)( 29,570)( 30,569)( 31,571)( 32,573)
( 33,572)( 34,574)( 35,576)( 36,575)( 37,592)( 38,594)( 39,593)( 40,586)
( 41,588)( 42,587)( 43,589)( 44,591)( 45,590)( 46,580)( 47,582)( 48,581)
( 49,583)( 50,585)( 51,584)( 52,577)( 53,579)( 54,578)( 55,622)( 56,624)
( 57,623)( 58,625)( 59,627)( 60,626)( 61,628)( 62,630)( 63,629)( 64,646)
( 65,648)( 66,647)( 67,640)( 68,642)( 69,641)( 70,643)( 71,645)( 72,644)
( 73,634)( 74,636)( 75,635)( 76,637)( 77,639)( 78,638)( 79,631)( 80,633)
( 81,632)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)( 88,601)
( 89,603)( 90,602)( 91,619)( 92,621)( 93,620)( 94,613)( 95,615)( 96,614)
( 97,616)( 98,618)( 99,617)(100,607)(101,609)(102,608)(103,610)(104,612)
(105,611)(106,604)(107,606)(108,605)(109,460)(110,462)(111,461)(112,463)
(113,465)(114,464)(115,466)(116,468)(117,467)(118,484)(119,486)(120,485)
(121,478)(122,480)(123,479)(124,481)(125,483)(126,482)(127,472)(128,474)
(129,473)(130,475)(131,477)(132,476)(133,469)(134,471)(135,470)(136,433)
(137,435)(138,434)(139,436)(140,438)(141,437)(142,439)(143,441)(144,440)
(145,457)(146,459)(147,458)(148,451)(149,453)(150,452)(151,454)(152,456)
(153,455)(154,445)(155,447)(156,446)(157,448)(158,450)(159,449)(160,442)
(161,444)(162,443)(163,487)(164,489)(165,488)(166,490)(167,492)(168,491)
(169,493)(170,495)(171,494)(172,511)(173,513)(174,512)(175,505)(176,507)
(177,506)(178,508)(179,510)(180,509)(181,499)(182,501)(183,500)(184,502)
(185,504)(186,503)(187,496)(188,498)(189,497)(190,514)(191,516)(192,515)
(193,517)(194,519)(195,518)(196,520)(197,522)(198,521)(199,538)(200,540)
(201,539)(202,532)(203,534)(204,533)(205,535)(206,537)(207,536)(208,526)
(209,528)(210,527)(211,529)(212,531)(213,530)(214,523)(215,525)(216,524)
(217,784)(218,786)(219,785)(220,787)(221,789)(222,788)(223,790)(224,792)
(225,791)(226,808)(227,810)(228,809)(229,802)(230,804)(231,803)(232,805)
(233,807)(234,806)(235,796)(236,798)(237,797)(238,799)(239,801)(240,800)
(241,793)(242,795)(243,794)(244,757)(245,759)(246,758)(247,760)(248,762)
(249,761)(250,763)(251,765)(252,764)(253,781)(254,783)(255,782)(256,775)
(257,777)(258,776)(259,778)(260,780)(261,779)(262,769)(263,771)(264,770)
(265,772)(266,774)(267,773)(268,766)(269,768)(270,767)(271,811)(272,813)
(273,812)(274,814)(275,816)(276,815)(277,817)(278,819)(279,818)(280,835)
(281,837)(282,836)(283,829)(284,831)(285,830)(286,832)(287,834)(288,833)
(289,823)(290,825)(291,824)(292,826)(293,828)(294,827)(295,820)(296,822)
(297,821)(298,838)(299,840)(300,839)(301,841)(302,843)(303,842)(304,844)
(305,846)(306,845)(307,862)(308,864)(309,863)(310,856)(311,858)(312,857)
(313,859)(314,861)(315,860)(316,850)(317,852)(318,851)(319,853)(320,855)
(321,854)(322,847)(323,849)(324,848)(325,649)(326,651)(327,650)(328,652)
(329,654)(330,653)(331,655)(332,657)(333,656)(334,673)(335,675)(336,674)
(337,667)(338,669)(339,668)(340,670)(341,672)(342,671)(343,661)(344,663)
(345,662)(346,664)(347,666)(348,665)(349,658)(350,660)(351,659)(352,676)
(353,678)(354,677)(355,679)(356,681)(357,680)(358,682)(359,684)(360,683)
(361,700)(362,702)(363,701)(364,694)(365,696)(366,695)(367,697)(368,699)
(369,698)(370,688)(371,690)(372,689)(373,691)(374,693)(375,692)(376,685)
(377,687)(378,686)(379,730)(380,732)(381,731)(382,733)(383,735)(384,734)
(385,736)(386,738)(387,737)(388,754)(389,756)(390,755)(391,748)(392,750)
(393,749)(394,751)(395,753)(396,752)(397,742)(398,744)(399,743)(400,745)
(401,747)(402,746)(403,739)(404,741)(405,740)(406,703)(407,705)(408,704)
(409,706)(410,708)(411,707)(412,709)(413,711)(414,710)(415,727)(416,729)
(417,728)(418,721)(419,723)(420,722)(421,724)(422,726)(423,725)(424,715)
(425,717)(426,716)(427,718)(428,720)(429,719)(430,712)(431,714)(432,713);;
s2 := (  1,500)(  2,499)(  3,501)(  4,497)(  5,496)(  6,498)(  7,503)(  8,502)
(  9,504)( 10,491)( 11,490)( 12,492)( 13,488)( 14,487)( 15,489)( 16,494)
( 17,493)( 18,495)( 19,509)( 20,508)( 21,510)( 22,506)( 23,505)( 24,507)
( 25,512)( 26,511)( 27,513)( 28,527)( 29,526)( 30,528)( 31,524)( 32,523)
( 33,525)( 34,530)( 35,529)( 36,531)( 37,518)( 38,517)( 39,519)( 40,515)
( 41,514)( 42,516)( 43,521)( 44,520)( 45,522)( 46,536)( 47,535)( 48,537)
( 49,533)( 50,532)( 51,534)( 52,539)( 53,538)( 54,540)( 55,473)( 56,472)
( 57,474)( 58,470)( 59,469)( 60,471)( 61,476)( 62,475)( 63,477)( 64,464)
( 65,463)( 66,465)( 67,461)( 68,460)( 69,462)( 70,467)( 71,466)( 72,468)
( 73,482)( 74,481)( 75,483)( 76,479)( 77,478)( 78,480)( 79,485)( 80,484)
( 81,486)( 82,446)( 83,445)( 84,447)( 85,443)( 86,442)( 87,444)( 88,449)
( 89,448)( 90,450)( 91,437)( 92,436)( 93,438)( 94,434)( 95,433)( 96,435)
( 97,440)( 98,439)( 99,441)(100,455)(101,454)(102,456)(103,452)(104,451)
(105,453)(106,458)(107,457)(108,459)(109,635)(110,634)(111,636)(112,632)
(113,631)(114,633)(115,638)(116,637)(117,639)(118,626)(119,625)(120,627)
(121,623)(122,622)(123,624)(124,629)(125,628)(126,630)(127,644)(128,643)
(129,645)(130,641)(131,640)(132,642)(133,647)(134,646)(135,648)(136,608)
(137,607)(138,609)(139,605)(140,604)(141,606)(142,611)(143,610)(144,612)
(145,599)(146,598)(147,600)(148,596)(149,595)(150,597)(151,602)(152,601)
(153,603)(154,617)(155,616)(156,618)(157,614)(158,613)(159,615)(160,620)
(161,619)(162,621)(163,554)(164,553)(165,555)(166,551)(167,550)(168,552)
(169,557)(170,556)(171,558)(172,545)(173,544)(174,546)(175,542)(176,541)
(177,543)(178,548)(179,547)(180,549)(181,563)(182,562)(183,564)(184,560)
(185,559)(186,561)(187,566)(188,565)(189,567)(190,581)(191,580)(192,582)
(193,578)(194,577)(195,579)(196,584)(197,583)(198,585)(199,572)(200,571)
(201,573)(202,569)(203,568)(204,570)(205,575)(206,574)(207,576)(208,590)
(209,589)(210,591)(211,587)(212,586)(213,588)(214,593)(215,592)(216,594)
(217,743)(218,742)(219,744)(220,740)(221,739)(222,741)(223,746)(224,745)
(225,747)(226,734)(227,733)(228,735)(229,731)(230,730)(231,732)(232,737)
(233,736)(234,738)(235,752)(236,751)(237,753)(238,749)(239,748)(240,750)
(241,755)(242,754)(243,756)(244,716)(245,715)(246,717)(247,713)(248,712)
(249,714)(250,719)(251,718)(252,720)(253,707)(254,706)(255,708)(256,704)
(257,703)(258,705)(259,710)(260,709)(261,711)(262,725)(263,724)(264,726)
(265,722)(266,721)(267,723)(268,728)(269,727)(270,729)(271,662)(272,661)
(273,663)(274,659)(275,658)(276,660)(277,665)(278,664)(279,666)(280,653)
(281,652)(282,654)(283,650)(284,649)(285,651)(286,656)(287,655)(288,657)
(289,671)(290,670)(291,672)(292,668)(293,667)(294,669)(295,674)(296,673)
(297,675)(298,689)(299,688)(300,690)(301,686)(302,685)(303,687)(304,692)
(305,691)(306,693)(307,680)(308,679)(309,681)(310,677)(311,676)(312,678)
(313,683)(314,682)(315,684)(316,698)(317,697)(318,699)(319,695)(320,694)
(321,696)(322,701)(323,700)(324,702)(325,824)(326,823)(327,825)(328,821)
(329,820)(330,822)(331,827)(332,826)(333,828)(334,815)(335,814)(336,816)
(337,812)(338,811)(339,813)(340,818)(341,817)(342,819)(343,833)(344,832)
(345,834)(346,830)(347,829)(348,831)(349,836)(350,835)(351,837)(352,851)
(353,850)(354,852)(355,848)(356,847)(357,849)(358,854)(359,853)(360,855)
(361,842)(362,841)(363,843)(364,839)(365,838)(366,840)(367,845)(368,844)
(369,846)(370,860)(371,859)(372,861)(373,857)(374,856)(375,858)(376,863)
(377,862)(378,864)(379,797)(380,796)(381,798)(382,794)(383,793)(384,795)
(385,800)(386,799)(387,801)(388,788)(389,787)(390,789)(391,785)(392,784)
(393,786)(394,791)(395,790)(396,792)(397,806)(398,805)(399,807)(400,803)
(401,802)(402,804)(403,809)(404,808)(405,810)(406,770)(407,769)(408,771)
(409,767)(410,766)(411,768)(412,773)(413,772)(414,774)(415,761)(416,760)
(417,762)(418,758)(419,757)(420,759)(421,764)(422,763)(423,765)(424,779)
(425,778)(426,780)(427,776)(428,775)(429,777)(430,782)(431,781)(432,783);;
s3 := (865,866);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(866)!(  1,217)(  2,218)(  3,219)(  4,238)(  5,239)(  6,240)(  7,232)
(  8,233)(  9,234)( 10,235)( 11,236)( 12,237)( 13,229)( 14,230)( 15,231)
( 16,223)( 17,224)( 18,225)( 19,226)( 20,227)( 21,228)( 22,220)( 23,221)
( 24,222)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,265)
( 32,266)( 33,267)( 34,259)( 35,260)( 36,261)( 37,262)( 38,263)( 39,264)
( 40,256)( 41,257)( 42,258)( 43,250)( 44,251)( 45,252)( 46,253)( 47,254)
( 48,255)( 49,247)( 50,248)( 51,249)( 52,268)( 53,269)( 54,270)( 55,298)
( 56,299)( 57,300)( 58,319)( 59,320)( 60,321)( 61,313)( 62,314)( 63,315)
( 64,316)( 65,317)( 66,318)( 67,310)( 68,311)( 69,312)( 70,304)( 71,305)
( 72,306)( 73,307)( 74,308)( 75,309)( 76,301)( 77,302)( 78,303)( 79,322)
( 80,323)( 81,324)( 82,271)( 83,272)( 84,273)( 85,292)( 86,293)( 87,294)
( 88,286)( 89,287)( 90,288)( 91,289)( 92,290)( 93,291)( 94,283)( 95,284)
( 96,285)( 97,277)( 98,278)( 99,279)(100,280)(101,281)(102,282)(103,274)
(104,275)(105,276)(106,295)(107,296)(108,297)(109,379)(110,380)(111,381)
(112,400)(113,401)(114,402)(115,394)(116,395)(117,396)(118,397)(119,398)
(120,399)(121,391)(122,392)(123,393)(124,385)(125,386)(126,387)(127,388)
(128,389)(129,390)(130,382)(131,383)(132,384)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,427)(140,428)(141,429)(142,421)(143,422)
(144,423)(145,424)(146,425)(147,426)(148,418)(149,419)(150,420)(151,412)
(152,413)(153,414)(154,415)(155,416)(156,417)(157,409)(158,410)(159,411)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,346)(167,347)
(168,348)(169,340)(170,341)(171,342)(172,343)(173,344)(174,345)(175,337)
(176,338)(177,339)(178,331)(179,332)(180,333)(181,334)(182,335)(183,336)
(184,328)(185,329)(186,330)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,373)(194,374)(195,375)(196,367)(197,368)(198,369)(199,370)
(200,371)(201,372)(202,364)(203,365)(204,366)(205,358)(206,359)(207,360)
(208,361)(209,362)(210,363)(211,355)(212,356)(213,357)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,670)(437,671)(438,672)(439,664)
(440,665)(441,666)(442,667)(443,668)(444,669)(445,661)(446,662)(447,663)
(448,655)(449,656)(450,657)(451,658)(452,659)(453,660)(454,652)(455,653)
(456,654)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,697)
(464,698)(465,699)(466,691)(467,692)(468,693)(469,694)(470,695)(471,696)
(472,688)(473,689)(474,690)(475,682)(476,683)(477,684)(478,685)(479,686)
(480,687)(481,679)(482,680)(483,681)(484,700)(485,701)(486,702)(487,730)
(488,731)(489,732)(490,751)(491,752)(492,753)(493,745)(494,746)(495,747)
(496,748)(497,749)(498,750)(499,742)(500,743)(501,744)(502,736)(503,737)
(504,738)(505,739)(506,740)(507,741)(508,733)(509,734)(510,735)(511,754)
(512,755)(513,756)(514,703)(515,704)(516,705)(517,724)(518,725)(519,726)
(520,718)(521,719)(522,720)(523,721)(524,722)(525,723)(526,715)(527,716)
(528,717)(529,709)(530,710)(531,711)(532,712)(533,713)(534,714)(535,706)
(536,707)(537,708)(538,727)(539,728)(540,729)(541,811)(542,812)(543,813)
(544,832)(545,833)(546,834)(547,826)(548,827)(549,828)(550,829)(551,830)
(552,831)(553,823)(554,824)(555,825)(556,817)(557,818)(558,819)(559,820)
(560,821)(561,822)(562,814)(563,815)(564,816)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,859)(572,860)(573,861)(574,853)(575,854)
(576,855)(577,856)(578,857)(579,858)(580,850)(581,851)(582,852)(583,844)
(584,845)(585,846)(586,847)(587,848)(588,849)(589,841)(590,842)(591,843)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,778)(599,779)
(600,780)(601,772)(602,773)(603,774)(604,775)(605,776)(606,777)(607,769)
(608,770)(609,771)(610,763)(611,764)(612,765)(613,766)(614,767)(615,768)
(616,760)(617,761)(618,762)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,805)(626,806)(627,807)(628,799)(629,800)(630,801)(631,802)
(632,803)(633,804)(634,796)(635,797)(636,798)(637,790)(638,791)(639,792)
(640,793)(641,794)(642,795)(643,787)(644,788)(645,789)(646,808)(647,809)
(648,810);
s1 := Sym(866)!(  1,541)(  2,543)(  3,542)(  4,544)(  5,546)(  6,545)(  7,547)
(  8,549)(  9,548)( 10,565)( 11,567)( 12,566)( 13,559)( 14,561)( 15,560)
( 16,562)( 17,564)( 18,563)( 19,553)( 20,555)( 21,554)( 22,556)( 23,558)
( 24,557)( 25,550)( 26,552)( 27,551)( 28,568)( 29,570)( 30,569)( 31,571)
( 32,573)( 33,572)( 34,574)( 35,576)( 36,575)( 37,592)( 38,594)( 39,593)
( 40,586)( 41,588)( 42,587)( 43,589)( 44,591)( 45,590)( 46,580)( 47,582)
( 48,581)( 49,583)( 50,585)( 51,584)( 52,577)( 53,579)( 54,578)( 55,622)
( 56,624)( 57,623)( 58,625)( 59,627)( 60,626)( 61,628)( 62,630)( 63,629)
( 64,646)( 65,648)( 66,647)( 67,640)( 68,642)( 69,641)( 70,643)( 71,645)
( 72,644)( 73,634)( 74,636)( 75,635)( 76,637)( 77,639)( 78,638)( 79,631)
( 80,633)( 81,632)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)
( 88,601)( 89,603)( 90,602)( 91,619)( 92,621)( 93,620)( 94,613)( 95,615)
( 96,614)( 97,616)( 98,618)( 99,617)(100,607)(101,609)(102,608)(103,610)
(104,612)(105,611)(106,604)(107,606)(108,605)(109,460)(110,462)(111,461)
(112,463)(113,465)(114,464)(115,466)(116,468)(117,467)(118,484)(119,486)
(120,485)(121,478)(122,480)(123,479)(124,481)(125,483)(126,482)(127,472)
(128,474)(129,473)(130,475)(131,477)(132,476)(133,469)(134,471)(135,470)
(136,433)(137,435)(138,434)(139,436)(140,438)(141,437)(142,439)(143,441)
(144,440)(145,457)(146,459)(147,458)(148,451)(149,453)(150,452)(151,454)
(152,456)(153,455)(154,445)(155,447)(156,446)(157,448)(158,450)(159,449)
(160,442)(161,444)(162,443)(163,487)(164,489)(165,488)(166,490)(167,492)
(168,491)(169,493)(170,495)(171,494)(172,511)(173,513)(174,512)(175,505)
(176,507)(177,506)(178,508)(179,510)(180,509)(181,499)(182,501)(183,500)
(184,502)(185,504)(186,503)(187,496)(188,498)(189,497)(190,514)(191,516)
(192,515)(193,517)(194,519)(195,518)(196,520)(197,522)(198,521)(199,538)
(200,540)(201,539)(202,532)(203,534)(204,533)(205,535)(206,537)(207,536)
(208,526)(209,528)(210,527)(211,529)(212,531)(213,530)(214,523)(215,525)
(216,524)(217,784)(218,786)(219,785)(220,787)(221,789)(222,788)(223,790)
(224,792)(225,791)(226,808)(227,810)(228,809)(229,802)(230,804)(231,803)
(232,805)(233,807)(234,806)(235,796)(236,798)(237,797)(238,799)(239,801)
(240,800)(241,793)(242,795)(243,794)(244,757)(245,759)(246,758)(247,760)
(248,762)(249,761)(250,763)(251,765)(252,764)(253,781)(254,783)(255,782)
(256,775)(257,777)(258,776)(259,778)(260,780)(261,779)(262,769)(263,771)
(264,770)(265,772)(266,774)(267,773)(268,766)(269,768)(270,767)(271,811)
(272,813)(273,812)(274,814)(275,816)(276,815)(277,817)(278,819)(279,818)
(280,835)(281,837)(282,836)(283,829)(284,831)(285,830)(286,832)(287,834)
(288,833)(289,823)(290,825)(291,824)(292,826)(293,828)(294,827)(295,820)
(296,822)(297,821)(298,838)(299,840)(300,839)(301,841)(302,843)(303,842)
(304,844)(305,846)(306,845)(307,862)(308,864)(309,863)(310,856)(311,858)
(312,857)(313,859)(314,861)(315,860)(316,850)(317,852)(318,851)(319,853)
(320,855)(321,854)(322,847)(323,849)(324,848)(325,649)(326,651)(327,650)
(328,652)(329,654)(330,653)(331,655)(332,657)(333,656)(334,673)(335,675)
(336,674)(337,667)(338,669)(339,668)(340,670)(341,672)(342,671)(343,661)
(344,663)(345,662)(346,664)(347,666)(348,665)(349,658)(350,660)(351,659)
(352,676)(353,678)(354,677)(355,679)(356,681)(357,680)(358,682)(359,684)
(360,683)(361,700)(362,702)(363,701)(364,694)(365,696)(366,695)(367,697)
(368,699)(369,698)(370,688)(371,690)(372,689)(373,691)(374,693)(375,692)
(376,685)(377,687)(378,686)(379,730)(380,732)(381,731)(382,733)(383,735)
(384,734)(385,736)(386,738)(387,737)(388,754)(389,756)(390,755)(391,748)
(392,750)(393,749)(394,751)(395,753)(396,752)(397,742)(398,744)(399,743)
(400,745)(401,747)(402,746)(403,739)(404,741)(405,740)(406,703)(407,705)
(408,704)(409,706)(410,708)(411,707)(412,709)(413,711)(414,710)(415,727)
(416,729)(417,728)(418,721)(419,723)(420,722)(421,724)(422,726)(423,725)
(424,715)(425,717)(426,716)(427,718)(428,720)(429,719)(430,712)(431,714)
(432,713);
s2 := Sym(866)!(  1,500)(  2,499)(  3,501)(  4,497)(  5,496)(  6,498)(  7,503)
(  8,502)(  9,504)( 10,491)( 11,490)( 12,492)( 13,488)( 14,487)( 15,489)
( 16,494)( 17,493)( 18,495)( 19,509)( 20,508)( 21,510)( 22,506)( 23,505)
( 24,507)( 25,512)( 26,511)( 27,513)( 28,527)( 29,526)( 30,528)( 31,524)
( 32,523)( 33,525)( 34,530)( 35,529)( 36,531)( 37,518)( 38,517)( 39,519)
( 40,515)( 41,514)( 42,516)( 43,521)( 44,520)( 45,522)( 46,536)( 47,535)
( 48,537)( 49,533)( 50,532)( 51,534)( 52,539)( 53,538)( 54,540)( 55,473)
( 56,472)( 57,474)( 58,470)( 59,469)( 60,471)( 61,476)( 62,475)( 63,477)
( 64,464)( 65,463)( 66,465)( 67,461)( 68,460)( 69,462)( 70,467)( 71,466)
( 72,468)( 73,482)( 74,481)( 75,483)( 76,479)( 77,478)( 78,480)( 79,485)
( 80,484)( 81,486)( 82,446)( 83,445)( 84,447)( 85,443)( 86,442)( 87,444)
( 88,449)( 89,448)( 90,450)( 91,437)( 92,436)( 93,438)( 94,434)( 95,433)
( 96,435)( 97,440)( 98,439)( 99,441)(100,455)(101,454)(102,456)(103,452)
(104,451)(105,453)(106,458)(107,457)(108,459)(109,635)(110,634)(111,636)
(112,632)(113,631)(114,633)(115,638)(116,637)(117,639)(118,626)(119,625)
(120,627)(121,623)(122,622)(123,624)(124,629)(125,628)(126,630)(127,644)
(128,643)(129,645)(130,641)(131,640)(132,642)(133,647)(134,646)(135,648)
(136,608)(137,607)(138,609)(139,605)(140,604)(141,606)(142,611)(143,610)
(144,612)(145,599)(146,598)(147,600)(148,596)(149,595)(150,597)(151,602)
(152,601)(153,603)(154,617)(155,616)(156,618)(157,614)(158,613)(159,615)
(160,620)(161,619)(162,621)(163,554)(164,553)(165,555)(166,551)(167,550)
(168,552)(169,557)(170,556)(171,558)(172,545)(173,544)(174,546)(175,542)
(176,541)(177,543)(178,548)(179,547)(180,549)(181,563)(182,562)(183,564)
(184,560)(185,559)(186,561)(187,566)(188,565)(189,567)(190,581)(191,580)
(192,582)(193,578)(194,577)(195,579)(196,584)(197,583)(198,585)(199,572)
(200,571)(201,573)(202,569)(203,568)(204,570)(205,575)(206,574)(207,576)
(208,590)(209,589)(210,591)(211,587)(212,586)(213,588)(214,593)(215,592)
(216,594)(217,743)(218,742)(219,744)(220,740)(221,739)(222,741)(223,746)
(224,745)(225,747)(226,734)(227,733)(228,735)(229,731)(230,730)(231,732)
(232,737)(233,736)(234,738)(235,752)(236,751)(237,753)(238,749)(239,748)
(240,750)(241,755)(242,754)(243,756)(244,716)(245,715)(246,717)(247,713)
(248,712)(249,714)(250,719)(251,718)(252,720)(253,707)(254,706)(255,708)
(256,704)(257,703)(258,705)(259,710)(260,709)(261,711)(262,725)(263,724)
(264,726)(265,722)(266,721)(267,723)(268,728)(269,727)(270,729)(271,662)
(272,661)(273,663)(274,659)(275,658)(276,660)(277,665)(278,664)(279,666)
(280,653)(281,652)(282,654)(283,650)(284,649)(285,651)(286,656)(287,655)
(288,657)(289,671)(290,670)(291,672)(292,668)(293,667)(294,669)(295,674)
(296,673)(297,675)(298,689)(299,688)(300,690)(301,686)(302,685)(303,687)
(304,692)(305,691)(306,693)(307,680)(308,679)(309,681)(310,677)(311,676)
(312,678)(313,683)(314,682)(315,684)(316,698)(317,697)(318,699)(319,695)
(320,694)(321,696)(322,701)(323,700)(324,702)(325,824)(326,823)(327,825)
(328,821)(329,820)(330,822)(331,827)(332,826)(333,828)(334,815)(335,814)
(336,816)(337,812)(338,811)(339,813)(340,818)(341,817)(342,819)(343,833)
(344,832)(345,834)(346,830)(347,829)(348,831)(349,836)(350,835)(351,837)
(352,851)(353,850)(354,852)(355,848)(356,847)(357,849)(358,854)(359,853)
(360,855)(361,842)(362,841)(363,843)(364,839)(365,838)(366,840)(367,845)
(368,844)(369,846)(370,860)(371,859)(372,861)(373,857)(374,856)(375,858)
(376,863)(377,862)(378,864)(379,797)(380,796)(381,798)(382,794)(383,793)
(384,795)(385,800)(386,799)(387,801)(388,788)(389,787)(390,789)(391,785)
(392,784)(393,786)(394,791)(395,790)(396,792)(397,806)(398,805)(399,807)
(400,803)(401,802)(402,804)(403,809)(404,808)(405,810)(406,770)(407,769)
(408,771)(409,767)(410,766)(411,768)(412,773)(413,772)(414,774)(415,761)
(416,760)(417,762)(418,758)(419,757)(420,759)(421,764)(422,763)(423,765)
(424,779)(425,778)(426,780)(427,776)(428,775)(429,777)(430,782)(431,781)
(432,783);
s3 := Sym(866)!(865,866);
poly := sub<Sym(866)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1 >; 
 

to this polytope