Polytope of Type {3,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,8}*864b
if this polytope has a name.
Group : SmallGroup(864,4094)
Rank : 4
Schlafli Type : {3,6,8}
Number of vertices, edges, etc : 3, 27, 72, 24
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,8,2} of size 1728
Vertex Figure Of :
{2,3,6,8} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,4}*432b
9-fold quotients : {3,2,8}*96
18-fold quotients : {3,2,4}*48
36-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,16}*1728b, {6,6,8}*1728g
Permutation Representation (GAP) :
s0 := ( 1,487)( 2,489)( 3,488)( 4,493)( 5,495)( 6,494)( 7,490)( 8,492)
( 9,491)( 10,505)( 11,507)( 12,506)( 13,511)( 14,513)( 15,512)( 16,508)
( 17,510)( 18,509)( 19,496)( 20,498)( 21,497)( 22,502)( 23,504)( 24,503)
( 25,499)( 26,501)( 27,500)( 28,514)( 29,516)( 30,515)( 31,520)( 32,522)
( 33,521)( 34,517)( 35,519)( 36,518)( 37,532)( 38,534)( 39,533)( 40,538)
( 41,540)( 42,539)( 43,535)( 44,537)( 45,536)( 46,523)( 47,525)( 48,524)
( 49,529)( 50,531)( 51,530)( 52,526)( 53,528)( 54,527)( 55,460)( 56,462)
( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)( 64,478)
( 65,480)( 66,479)( 67,484)( 68,486)( 69,485)( 70,481)( 71,483)( 72,482)
( 73,469)( 74,471)( 75,470)( 76,475)( 77,477)( 78,476)( 79,472)( 80,474)
( 81,473)( 82,433)( 83,435)( 84,434)( 85,439)( 86,441)( 87,440)( 88,436)
( 89,438)( 90,437)( 91,451)( 92,453)( 93,452)( 94,457)( 95,459)( 96,458)
( 97,454)( 98,456)( 99,455)(100,442)(101,444)(102,443)(103,448)(104,450)
(105,449)(106,445)(107,447)(108,446)(109,622)(110,624)(111,623)(112,628)
(113,630)(114,629)(115,625)(116,627)(117,626)(118,640)(119,642)(120,641)
(121,646)(122,648)(123,647)(124,643)(125,645)(126,644)(127,631)(128,633)
(129,632)(130,637)(131,639)(132,638)(133,634)(134,636)(135,635)(136,595)
(137,597)(138,596)(139,601)(140,603)(141,602)(142,598)(143,600)(144,599)
(145,613)(146,615)(147,614)(148,619)(149,621)(150,620)(151,616)(152,618)
(153,617)(154,604)(155,606)(156,605)(157,610)(158,612)(159,611)(160,607)
(161,609)(162,608)(163,541)(164,543)(165,542)(166,547)(167,549)(168,548)
(169,544)(170,546)(171,545)(172,559)(173,561)(174,560)(175,565)(176,567)
(177,566)(178,562)(179,564)(180,563)(181,550)(182,552)(183,551)(184,556)
(185,558)(186,557)(187,553)(188,555)(189,554)(190,568)(191,570)(192,569)
(193,574)(194,576)(195,575)(196,571)(197,573)(198,572)(199,586)(200,588)
(201,587)(202,592)(203,594)(204,593)(205,589)(206,591)(207,590)(208,577)
(209,579)(210,578)(211,583)(212,585)(213,584)(214,580)(215,582)(216,581)
(217,730)(218,732)(219,731)(220,736)(221,738)(222,737)(223,733)(224,735)
(225,734)(226,748)(227,750)(228,749)(229,754)(230,756)(231,755)(232,751)
(233,753)(234,752)(235,739)(236,741)(237,740)(238,745)(239,747)(240,746)
(241,742)(242,744)(243,743)(244,703)(245,705)(246,704)(247,709)(248,711)
(249,710)(250,706)(251,708)(252,707)(253,721)(254,723)(255,722)(256,727)
(257,729)(258,728)(259,724)(260,726)(261,725)(262,712)(263,714)(264,713)
(265,718)(266,720)(267,719)(268,715)(269,717)(270,716)(271,649)(272,651)
(273,650)(274,655)(275,657)(276,656)(277,652)(278,654)(279,653)(280,667)
(281,669)(282,668)(283,673)(284,675)(285,674)(286,670)(287,672)(288,671)
(289,658)(290,660)(291,659)(292,664)(293,666)(294,665)(295,661)(296,663)
(297,662)(298,676)(299,678)(300,677)(301,682)(302,684)(303,683)(304,679)
(305,681)(306,680)(307,694)(308,696)(309,695)(310,700)(311,702)(312,701)
(313,697)(314,699)(315,698)(316,685)(317,687)(318,686)(319,691)(320,693)
(321,692)(322,688)(323,690)(324,689)(325,811)(326,813)(327,812)(328,817)
(329,819)(330,818)(331,814)(332,816)(333,815)(334,829)(335,831)(336,830)
(337,835)(338,837)(339,836)(340,832)(341,834)(342,833)(343,820)(344,822)
(345,821)(346,826)(347,828)(348,827)(349,823)(350,825)(351,824)(352,838)
(353,840)(354,839)(355,844)(356,846)(357,845)(358,841)(359,843)(360,842)
(361,856)(362,858)(363,857)(364,862)(365,864)(366,863)(367,859)(368,861)
(369,860)(370,847)(371,849)(372,848)(373,853)(374,855)(375,854)(376,850)
(377,852)(378,851)(379,784)(380,786)(381,785)(382,790)(383,792)(384,791)
(385,787)(386,789)(387,788)(388,802)(389,804)(390,803)(391,808)(392,810)
(393,809)(394,805)(395,807)(396,806)(397,793)(398,795)(399,794)(400,799)
(401,801)(402,800)(403,796)(404,798)(405,797)(406,757)(407,759)(408,758)
(409,763)(410,765)(411,764)(412,760)(413,762)(414,761)(415,775)(416,777)
(417,776)(418,781)(419,783)(420,782)(421,778)(422,780)(423,779)(424,766)
(425,768)(426,767)(427,772)(428,774)(429,773)(430,769)(431,771)(432,770);;
s1 := ( 1,497)( 2,496)( 3,498)( 4,503)( 5,502)( 6,504)( 7,500)( 8,499)
( 9,501)( 10,488)( 11,487)( 12,489)( 13,494)( 14,493)( 15,495)( 16,491)
( 17,490)( 18,492)( 19,506)( 20,505)( 21,507)( 22,512)( 23,511)( 24,513)
( 25,509)( 26,508)( 27,510)( 28,524)( 29,523)( 30,525)( 31,530)( 32,529)
( 33,531)( 34,527)( 35,526)( 36,528)( 37,515)( 38,514)( 39,516)( 40,521)
( 41,520)( 42,522)( 43,518)( 44,517)( 45,519)( 46,533)( 47,532)( 48,534)
( 49,539)( 50,538)( 51,540)( 52,536)( 53,535)( 54,537)( 55,470)( 56,469)
( 57,471)( 58,476)( 59,475)( 60,477)( 61,473)( 62,472)( 63,474)( 64,461)
( 65,460)( 66,462)( 67,467)( 68,466)( 69,468)( 70,464)( 71,463)( 72,465)
( 73,479)( 74,478)( 75,480)( 76,485)( 77,484)( 78,486)( 79,482)( 80,481)
( 81,483)( 82,443)( 83,442)( 84,444)( 85,449)( 86,448)( 87,450)( 88,446)
( 89,445)( 90,447)( 91,434)( 92,433)( 93,435)( 94,440)( 95,439)( 96,441)
( 97,437)( 98,436)( 99,438)(100,452)(101,451)(102,453)(103,458)(104,457)
(105,459)(106,455)(107,454)(108,456)(109,632)(110,631)(111,633)(112,638)
(113,637)(114,639)(115,635)(116,634)(117,636)(118,623)(119,622)(120,624)
(121,629)(122,628)(123,630)(124,626)(125,625)(126,627)(127,641)(128,640)
(129,642)(130,647)(131,646)(132,648)(133,644)(134,643)(135,645)(136,605)
(137,604)(138,606)(139,611)(140,610)(141,612)(142,608)(143,607)(144,609)
(145,596)(146,595)(147,597)(148,602)(149,601)(150,603)(151,599)(152,598)
(153,600)(154,614)(155,613)(156,615)(157,620)(158,619)(159,621)(160,617)
(161,616)(162,618)(163,551)(164,550)(165,552)(166,557)(167,556)(168,558)
(169,554)(170,553)(171,555)(172,542)(173,541)(174,543)(175,548)(176,547)
(177,549)(178,545)(179,544)(180,546)(181,560)(182,559)(183,561)(184,566)
(185,565)(186,567)(187,563)(188,562)(189,564)(190,578)(191,577)(192,579)
(193,584)(194,583)(195,585)(196,581)(197,580)(198,582)(199,569)(200,568)
(201,570)(202,575)(203,574)(204,576)(205,572)(206,571)(207,573)(208,587)
(209,586)(210,588)(211,593)(212,592)(213,594)(214,590)(215,589)(216,591)
(217,740)(218,739)(219,741)(220,746)(221,745)(222,747)(223,743)(224,742)
(225,744)(226,731)(227,730)(228,732)(229,737)(230,736)(231,738)(232,734)
(233,733)(234,735)(235,749)(236,748)(237,750)(238,755)(239,754)(240,756)
(241,752)(242,751)(243,753)(244,713)(245,712)(246,714)(247,719)(248,718)
(249,720)(250,716)(251,715)(252,717)(253,704)(254,703)(255,705)(256,710)
(257,709)(258,711)(259,707)(260,706)(261,708)(262,722)(263,721)(264,723)
(265,728)(266,727)(267,729)(268,725)(269,724)(270,726)(271,659)(272,658)
(273,660)(274,665)(275,664)(276,666)(277,662)(278,661)(279,663)(280,650)
(281,649)(282,651)(283,656)(284,655)(285,657)(286,653)(287,652)(288,654)
(289,668)(290,667)(291,669)(292,674)(293,673)(294,675)(295,671)(296,670)
(297,672)(298,686)(299,685)(300,687)(301,692)(302,691)(303,693)(304,689)
(305,688)(306,690)(307,677)(308,676)(309,678)(310,683)(311,682)(312,684)
(313,680)(314,679)(315,681)(316,695)(317,694)(318,696)(319,701)(320,700)
(321,702)(322,698)(323,697)(324,699)(325,821)(326,820)(327,822)(328,827)
(329,826)(330,828)(331,824)(332,823)(333,825)(334,812)(335,811)(336,813)
(337,818)(338,817)(339,819)(340,815)(341,814)(342,816)(343,830)(344,829)
(345,831)(346,836)(347,835)(348,837)(349,833)(350,832)(351,834)(352,848)
(353,847)(354,849)(355,854)(356,853)(357,855)(358,851)(359,850)(360,852)
(361,839)(362,838)(363,840)(364,845)(365,844)(366,846)(367,842)(368,841)
(369,843)(370,857)(371,856)(372,858)(373,863)(374,862)(375,864)(376,860)
(377,859)(378,861)(379,794)(380,793)(381,795)(382,800)(383,799)(384,801)
(385,797)(386,796)(387,798)(388,785)(389,784)(390,786)(391,791)(392,790)
(393,792)(394,788)(395,787)(396,789)(397,803)(398,802)(399,804)(400,809)
(401,808)(402,810)(403,806)(404,805)(405,807)(406,767)(407,766)(408,768)
(409,773)(410,772)(411,774)(412,770)(413,769)(414,771)(415,758)(416,757)
(417,759)(418,764)(419,763)(420,765)(421,761)(422,760)(423,762)(424,776)
(425,775)(426,777)(427,782)(428,781)(429,783)(430,779)(431,778)(432,780);;
s2 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,133)( 11,134)( 12,135)( 13,127)( 14,128)( 15,129)( 16,130)
( 17,131)( 18,132)( 19,121)( 20,122)( 21,123)( 22,124)( 23,125)( 24,126)
( 25,118)( 26,119)( 27,120)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,160)( 38,161)( 39,162)( 40,154)
( 41,155)( 42,156)( 43,157)( 44,158)( 45,159)( 46,148)( 47,149)( 48,150)
( 49,151)( 50,152)( 51,153)( 52,145)( 53,146)( 54,147)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,214)
( 65,215)( 66,216)( 67,208)( 68,209)( 69,210)( 70,211)( 71,212)( 72,213)
( 73,202)( 74,203)( 75,204)( 76,205)( 77,206)( 78,207)( 79,199)( 80,200)
( 81,201)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,187)( 92,188)( 93,189)( 94,181)( 95,182)( 96,183)
( 97,184)( 98,185)( 99,186)(100,175)(101,176)(102,177)(103,178)(104,179)
(105,180)(106,172)(107,173)(108,174)(217,325)(218,326)(219,327)(220,328)
(221,329)(222,330)(223,331)(224,332)(225,333)(226,349)(227,350)(228,351)
(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,337)(236,338)
(237,339)(238,340)(239,341)(240,342)(241,334)(242,335)(243,336)(244,352)
(245,353)(246,354)(247,355)(248,356)(249,357)(250,358)(251,359)(252,360)
(253,376)(254,377)(255,378)(256,370)(257,371)(258,372)(259,373)(260,374)
(261,375)(262,364)(263,365)(264,366)(265,367)(266,368)(267,369)(268,361)
(269,362)(270,363)(271,406)(272,407)(273,408)(274,409)(275,410)(276,411)
(277,412)(278,413)(279,414)(280,430)(281,431)(282,432)(283,424)(284,425)
(285,426)(286,427)(287,428)(288,429)(289,418)(290,419)(291,420)(292,421)
(293,422)(294,423)(295,415)(296,416)(297,417)(298,379)(299,380)(300,381)
(301,382)(302,383)(303,384)(304,385)(305,386)(306,387)(307,403)(308,404)
(309,405)(310,397)(311,398)(312,399)(313,400)(314,401)(315,402)(316,391)
(317,392)(318,393)(319,394)(320,395)(321,396)(322,388)(323,389)(324,390)
(433,541)(434,542)(435,543)(436,544)(437,545)(438,546)(439,547)(440,548)
(441,549)(442,565)(443,566)(444,567)(445,559)(446,560)(447,561)(448,562)
(449,563)(450,564)(451,553)(452,554)(453,555)(454,556)(455,557)(456,558)
(457,550)(458,551)(459,552)(460,568)(461,569)(462,570)(463,571)(464,572)
(465,573)(466,574)(467,575)(468,576)(469,592)(470,593)(471,594)(472,586)
(473,587)(474,588)(475,589)(476,590)(477,591)(478,580)(479,581)(480,582)
(481,583)(482,584)(483,585)(484,577)(485,578)(486,579)(487,622)(488,623)
(489,624)(490,625)(491,626)(492,627)(493,628)(494,629)(495,630)(496,646)
(497,647)(498,648)(499,640)(500,641)(501,642)(502,643)(503,644)(504,645)
(505,634)(506,635)(507,636)(508,637)(509,638)(510,639)(511,631)(512,632)
(513,633)(514,595)(515,596)(516,597)(517,598)(518,599)(519,600)(520,601)
(521,602)(522,603)(523,619)(524,620)(525,621)(526,613)(527,614)(528,615)
(529,616)(530,617)(531,618)(532,607)(533,608)(534,609)(535,610)(536,611)
(537,612)(538,604)(539,605)(540,606)(649,757)(650,758)(651,759)(652,760)
(653,761)(654,762)(655,763)(656,764)(657,765)(658,781)(659,782)(660,783)
(661,775)(662,776)(663,777)(664,778)(665,779)(666,780)(667,769)(668,770)
(669,771)(670,772)(671,773)(672,774)(673,766)(674,767)(675,768)(676,784)
(677,785)(678,786)(679,787)(680,788)(681,789)(682,790)(683,791)(684,792)
(685,808)(686,809)(687,810)(688,802)(689,803)(690,804)(691,805)(692,806)
(693,807)(694,796)(695,797)(696,798)(697,799)(698,800)(699,801)(700,793)
(701,794)(702,795)(703,838)(704,839)(705,840)(706,841)(707,842)(708,843)
(709,844)(710,845)(711,846)(712,862)(713,863)(714,864)(715,856)(716,857)
(717,858)(718,859)(719,860)(720,861)(721,850)(722,851)(723,852)(724,853)
(725,854)(726,855)(727,847)(728,848)(729,849)(730,811)(731,812)(732,813)
(733,814)(734,815)(735,816)(736,817)(737,818)(738,819)(739,835)(740,836)
(741,837)(742,829)(743,830)(744,831)(745,832)(746,833)(747,834)(748,823)
(749,824)(750,825)(751,826)(752,827)(753,828)(754,820)(755,821)(756,822);;
s3 := ( 1,271)( 2,272)( 3,273)( 4,286)( 5,287)( 6,288)( 7,292)( 8,293)
( 9,294)( 10,280)( 11,281)( 12,282)( 13,295)( 14,296)( 15,297)( 16,274)
( 17,275)( 18,276)( 19,289)( 20,290)( 21,291)( 22,277)( 23,278)( 24,279)
( 25,283)( 26,284)( 27,285)( 28,298)( 29,299)( 30,300)( 31,313)( 32,314)
( 33,315)( 34,319)( 35,320)( 36,321)( 37,307)( 38,308)( 39,309)( 40,322)
( 41,323)( 42,324)( 43,301)( 44,302)( 45,303)( 46,316)( 47,317)( 48,318)
( 49,304)( 50,305)( 51,306)( 52,310)( 53,311)( 54,312)( 55,217)( 56,218)
( 57,219)( 58,232)( 59,233)( 60,234)( 61,238)( 62,239)( 63,240)( 64,226)
( 65,227)( 66,228)( 67,241)( 68,242)( 69,243)( 70,220)( 71,221)( 72,222)
( 73,235)( 74,236)( 75,237)( 76,223)( 77,224)( 78,225)( 79,229)( 80,230)
( 81,231)( 82,244)( 83,245)( 84,246)( 85,259)( 86,260)( 87,261)( 88,265)
( 89,266)( 90,267)( 91,253)( 92,254)( 93,255)( 94,268)( 95,269)( 96,270)
( 97,247)( 98,248)( 99,249)(100,262)(101,263)(102,264)(103,250)(104,251)
(105,252)(106,256)(107,257)(108,258)(109,352)(110,353)(111,354)(112,367)
(113,368)(114,369)(115,373)(116,374)(117,375)(118,361)(119,362)(120,363)
(121,376)(122,377)(123,378)(124,355)(125,356)(126,357)(127,370)(128,371)
(129,372)(130,358)(131,359)(132,360)(133,364)(134,365)(135,366)(136,325)
(137,326)(138,327)(139,340)(140,341)(141,342)(142,346)(143,347)(144,348)
(145,334)(146,335)(147,336)(148,349)(149,350)(150,351)(151,328)(152,329)
(153,330)(154,343)(155,344)(156,345)(157,331)(158,332)(159,333)(160,337)
(161,338)(162,339)(163,379)(164,380)(165,381)(166,394)(167,395)(168,396)
(169,400)(170,401)(171,402)(172,388)(173,389)(174,390)(175,403)(176,404)
(177,405)(178,382)(179,383)(180,384)(181,397)(182,398)(183,399)(184,385)
(185,386)(186,387)(187,391)(188,392)(189,393)(190,406)(191,407)(192,408)
(193,421)(194,422)(195,423)(196,427)(197,428)(198,429)(199,415)(200,416)
(201,417)(202,430)(203,431)(204,432)(205,409)(206,410)(207,411)(208,424)
(209,425)(210,426)(211,412)(212,413)(213,414)(214,418)(215,419)(216,420)
(433,703)(434,704)(435,705)(436,718)(437,719)(438,720)(439,724)(440,725)
(441,726)(442,712)(443,713)(444,714)(445,727)(446,728)(447,729)(448,706)
(449,707)(450,708)(451,721)(452,722)(453,723)(454,709)(455,710)(456,711)
(457,715)(458,716)(459,717)(460,730)(461,731)(462,732)(463,745)(464,746)
(465,747)(466,751)(467,752)(468,753)(469,739)(470,740)(471,741)(472,754)
(473,755)(474,756)(475,733)(476,734)(477,735)(478,748)(479,749)(480,750)
(481,736)(482,737)(483,738)(484,742)(485,743)(486,744)(487,649)(488,650)
(489,651)(490,664)(491,665)(492,666)(493,670)(494,671)(495,672)(496,658)
(497,659)(498,660)(499,673)(500,674)(501,675)(502,652)(503,653)(504,654)
(505,667)(506,668)(507,669)(508,655)(509,656)(510,657)(511,661)(512,662)
(513,663)(514,676)(515,677)(516,678)(517,691)(518,692)(519,693)(520,697)
(521,698)(522,699)(523,685)(524,686)(525,687)(526,700)(527,701)(528,702)
(529,679)(530,680)(531,681)(532,694)(533,695)(534,696)(535,682)(536,683)
(537,684)(538,688)(539,689)(540,690)(541,784)(542,785)(543,786)(544,799)
(545,800)(546,801)(547,805)(548,806)(549,807)(550,793)(551,794)(552,795)
(553,808)(554,809)(555,810)(556,787)(557,788)(558,789)(559,802)(560,803)
(561,804)(562,790)(563,791)(564,792)(565,796)(566,797)(567,798)(568,757)
(569,758)(570,759)(571,772)(572,773)(573,774)(574,778)(575,779)(576,780)
(577,766)(578,767)(579,768)(580,781)(581,782)(582,783)(583,760)(584,761)
(585,762)(586,775)(587,776)(588,777)(589,763)(590,764)(591,765)(592,769)
(593,770)(594,771)(595,811)(596,812)(597,813)(598,826)(599,827)(600,828)
(601,832)(602,833)(603,834)(604,820)(605,821)(606,822)(607,835)(608,836)
(609,837)(610,814)(611,815)(612,816)(613,829)(614,830)(615,831)(616,817)
(617,818)(618,819)(619,823)(620,824)(621,825)(622,838)(623,839)(624,840)
(625,853)(626,854)(627,855)(628,859)(629,860)(630,861)(631,847)(632,848)
(633,849)(634,862)(635,863)(636,864)(637,841)(638,842)(639,843)(640,856)
(641,857)(642,858)(643,844)(644,845)(645,846)(646,850)(647,851)(648,852);;
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, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(864)!( 1,487)( 2,489)( 3,488)( 4,493)( 5,495)( 6,494)( 7,490)
( 8,492)( 9,491)( 10,505)( 11,507)( 12,506)( 13,511)( 14,513)( 15,512)
( 16,508)( 17,510)( 18,509)( 19,496)( 20,498)( 21,497)( 22,502)( 23,504)
( 24,503)( 25,499)( 26,501)( 27,500)( 28,514)( 29,516)( 30,515)( 31,520)
( 32,522)( 33,521)( 34,517)( 35,519)( 36,518)( 37,532)( 38,534)( 39,533)
( 40,538)( 41,540)( 42,539)( 43,535)( 44,537)( 45,536)( 46,523)( 47,525)
( 48,524)( 49,529)( 50,531)( 51,530)( 52,526)( 53,528)( 54,527)( 55,460)
( 56,462)( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)
( 64,478)( 65,480)( 66,479)( 67,484)( 68,486)( 69,485)( 70,481)( 71,483)
( 72,482)( 73,469)( 74,471)( 75,470)( 76,475)( 77,477)( 78,476)( 79,472)
( 80,474)( 81,473)( 82,433)( 83,435)( 84,434)( 85,439)( 86,441)( 87,440)
( 88,436)( 89,438)( 90,437)( 91,451)( 92,453)( 93,452)( 94,457)( 95,459)
( 96,458)( 97,454)( 98,456)( 99,455)(100,442)(101,444)(102,443)(103,448)
(104,450)(105,449)(106,445)(107,447)(108,446)(109,622)(110,624)(111,623)
(112,628)(113,630)(114,629)(115,625)(116,627)(117,626)(118,640)(119,642)
(120,641)(121,646)(122,648)(123,647)(124,643)(125,645)(126,644)(127,631)
(128,633)(129,632)(130,637)(131,639)(132,638)(133,634)(134,636)(135,635)
(136,595)(137,597)(138,596)(139,601)(140,603)(141,602)(142,598)(143,600)
(144,599)(145,613)(146,615)(147,614)(148,619)(149,621)(150,620)(151,616)
(152,618)(153,617)(154,604)(155,606)(156,605)(157,610)(158,612)(159,611)
(160,607)(161,609)(162,608)(163,541)(164,543)(165,542)(166,547)(167,549)
(168,548)(169,544)(170,546)(171,545)(172,559)(173,561)(174,560)(175,565)
(176,567)(177,566)(178,562)(179,564)(180,563)(181,550)(182,552)(183,551)
(184,556)(185,558)(186,557)(187,553)(188,555)(189,554)(190,568)(191,570)
(192,569)(193,574)(194,576)(195,575)(196,571)(197,573)(198,572)(199,586)
(200,588)(201,587)(202,592)(203,594)(204,593)(205,589)(206,591)(207,590)
(208,577)(209,579)(210,578)(211,583)(212,585)(213,584)(214,580)(215,582)
(216,581)(217,730)(218,732)(219,731)(220,736)(221,738)(222,737)(223,733)
(224,735)(225,734)(226,748)(227,750)(228,749)(229,754)(230,756)(231,755)
(232,751)(233,753)(234,752)(235,739)(236,741)(237,740)(238,745)(239,747)
(240,746)(241,742)(242,744)(243,743)(244,703)(245,705)(246,704)(247,709)
(248,711)(249,710)(250,706)(251,708)(252,707)(253,721)(254,723)(255,722)
(256,727)(257,729)(258,728)(259,724)(260,726)(261,725)(262,712)(263,714)
(264,713)(265,718)(266,720)(267,719)(268,715)(269,717)(270,716)(271,649)
(272,651)(273,650)(274,655)(275,657)(276,656)(277,652)(278,654)(279,653)
(280,667)(281,669)(282,668)(283,673)(284,675)(285,674)(286,670)(287,672)
(288,671)(289,658)(290,660)(291,659)(292,664)(293,666)(294,665)(295,661)
(296,663)(297,662)(298,676)(299,678)(300,677)(301,682)(302,684)(303,683)
(304,679)(305,681)(306,680)(307,694)(308,696)(309,695)(310,700)(311,702)
(312,701)(313,697)(314,699)(315,698)(316,685)(317,687)(318,686)(319,691)
(320,693)(321,692)(322,688)(323,690)(324,689)(325,811)(326,813)(327,812)
(328,817)(329,819)(330,818)(331,814)(332,816)(333,815)(334,829)(335,831)
(336,830)(337,835)(338,837)(339,836)(340,832)(341,834)(342,833)(343,820)
(344,822)(345,821)(346,826)(347,828)(348,827)(349,823)(350,825)(351,824)
(352,838)(353,840)(354,839)(355,844)(356,846)(357,845)(358,841)(359,843)
(360,842)(361,856)(362,858)(363,857)(364,862)(365,864)(366,863)(367,859)
(368,861)(369,860)(370,847)(371,849)(372,848)(373,853)(374,855)(375,854)
(376,850)(377,852)(378,851)(379,784)(380,786)(381,785)(382,790)(383,792)
(384,791)(385,787)(386,789)(387,788)(388,802)(389,804)(390,803)(391,808)
(392,810)(393,809)(394,805)(395,807)(396,806)(397,793)(398,795)(399,794)
(400,799)(401,801)(402,800)(403,796)(404,798)(405,797)(406,757)(407,759)
(408,758)(409,763)(410,765)(411,764)(412,760)(413,762)(414,761)(415,775)
(416,777)(417,776)(418,781)(419,783)(420,782)(421,778)(422,780)(423,779)
(424,766)(425,768)(426,767)(427,772)(428,774)(429,773)(430,769)(431,771)
(432,770);
s1 := Sym(864)!( 1,497)( 2,496)( 3,498)( 4,503)( 5,502)( 6,504)( 7,500)
( 8,499)( 9,501)( 10,488)( 11,487)( 12,489)( 13,494)( 14,493)( 15,495)
( 16,491)( 17,490)( 18,492)( 19,506)( 20,505)( 21,507)( 22,512)( 23,511)
( 24,513)( 25,509)( 26,508)( 27,510)( 28,524)( 29,523)( 30,525)( 31,530)
( 32,529)( 33,531)( 34,527)( 35,526)( 36,528)( 37,515)( 38,514)( 39,516)
( 40,521)( 41,520)( 42,522)( 43,518)( 44,517)( 45,519)( 46,533)( 47,532)
( 48,534)( 49,539)( 50,538)( 51,540)( 52,536)( 53,535)( 54,537)( 55,470)
( 56,469)( 57,471)( 58,476)( 59,475)( 60,477)( 61,473)( 62,472)( 63,474)
( 64,461)( 65,460)( 66,462)( 67,467)( 68,466)( 69,468)( 70,464)( 71,463)
( 72,465)( 73,479)( 74,478)( 75,480)( 76,485)( 77,484)( 78,486)( 79,482)
( 80,481)( 81,483)( 82,443)( 83,442)( 84,444)( 85,449)( 86,448)( 87,450)
( 88,446)( 89,445)( 90,447)( 91,434)( 92,433)( 93,435)( 94,440)( 95,439)
( 96,441)( 97,437)( 98,436)( 99,438)(100,452)(101,451)(102,453)(103,458)
(104,457)(105,459)(106,455)(107,454)(108,456)(109,632)(110,631)(111,633)
(112,638)(113,637)(114,639)(115,635)(116,634)(117,636)(118,623)(119,622)
(120,624)(121,629)(122,628)(123,630)(124,626)(125,625)(126,627)(127,641)
(128,640)(129,642)(130,647)(131,646)(132,648)(133,644)(134,643)(135,645)
(136,605)(137,604)(138,606)(139,611)(140,610)(141,612)(142,608)(143,607)
(144,609)(145,596)(146,595)(147,597)(148,602)(149,601)(150,603)(151,599)
(152,598)(153,600)(154,614)(155,613)(156,615)(157,620)(158,619)(159,621)
(160,617)(161,616)(162,618)(163,551)(164,550)(165,552)(166,557)(167,556)
(168,558)(169,554)(170,553)(171,555)(172,542)(173,541)(174,543)(175,548)
(176,547)(177,549)(178,545)(179,544)(180,546)(181,560)(182,559)(183,561)
(184,566)(185,565)(186,567)(187,563)(188,562)(189,564)(190,578)(191,577)
(192,579)(193,584)(194,583)(195,585)(196,581)(197,580)(198,582)(199,569)
(200,568)(201,570)(202,575)(203,574)(204,576)(205,572)(206,571)(207,573)
(208,587)(209,586)(210,588)(211,593)(212,592)(213,594)(214,590)(215,589)
(216,591)(217,740)(218,739)(219,741)(220,746)(221,745)(222,747)(223,743)
(224,742)(225,744)(226,731)(227,730)(228,732)(229,737)(230,736)(231,738)
(232,734)(233,733)(234,735)(235,749)(236,748)(237,750)(238,755)(239,754)
(240,756)(241,752)(242,751)(243,753)(244,713)(245,712)(246,714)(247,719)
(248,718)(249,720)(250,716)(251,715)(252,717)(253,704)(254,703)(255,705)
(256,710)(257,709)(258,711)(259,707)(260,706)(261,708)(262,722)(263,721)
(264,723)(265,728)(266,727)(267,729)(268,725)(269,724)(270,726)(271,659)
(272,658)(273,660)(274,665)(275,664)(276,666)(277,662)(278,661)(279,663)
(280,650)(281,649)(282,651)(283,656)(284,655)(285,657)(286,653)(287,652)
(288,654)(289,668)(290,667)(291,669)(292,674)(293,673)(294,675)(295,671)
(296,670)(297,672)(298,686)(299,685)(300,687)(301,692)(302,691)(303,693)
(304,689)(305,688)(306,690)(307,677)(308,676)(309,678)(310,683)(311,682)
(312,684)(313,680)(314,679)(315,681)(316,695)(317,694)(318,696)(319,701)
(320,700)(321,702)(322,698)(323,697)(324,699)(325,821)(326,820)(327,822)
(328,827)(329,826)(330,828)(331,824)(332,823)(333,825)(334,812)(335,811)
(336,813)(337,818)(338,817)(339,819)(340,815)(341,814)(342,816)(343,830)
(344,829)(345,831)(346,836)(347,835)(348,837)(349,833)(350,832)(351,834)
(352,848)(353,847)(354,849)(355,854)(356,853)(357,855)(358,851)(359,850)
(360,852)(361,839)(362,838)(363,840)(364,845)(365,844)(366,846)(367,842)
(368,841)(369,843)(370,857)(371,856)(372,858)(373,863)(374,862)(375,864)
(376,860)(377,859)(378,861)(379,794)(380,793)(381,795)(382,800)(383,799)
(384,801)(385,797)(386,796)(387,798)(388,785)(389,784)(390,786)(391,791)
(392,790)(393,792)(394,788)(395,787)(396,789)(397,803)(398,802)(399,804)
(400,809)(401,808)(402,810)(403,806)(404,805)(405,807)(406,767)(407,766)
(408,768)(409,773)(410,772)(411,774)(412,770)(413,769)(414,771)(415,758)
(416,757)(417,759)(418,764)(419,763)(420,765)(421,761)(422,760)(423,762)
(424,776)(425,775)(426,777)(427,782)(428,781)(429,783)(430,779)(431,778)
(432,780);
s2 := Sym(864)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,133)( 11,134)( 12,135)( 13,127)( 14,128)( 15,129)
( 16,130)( 17,131)( 18,132)( 19,121)( 20,122)( 21,123)( 22,124)( 23,125)
( 24,126)( 25,118)( 26,119)( 27,120)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,160)( 38,161)( 39,162)
( 40,154)( 41,155)( 42,156)( 43,157)( 44,158)( 45,159)( 46,148)( 47,149)
( 48,150)( 49,151)( 50,152)( 51,153)( 52,145)( 53,146)( 54,147)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,214)( 65,215)( 66,216)( 67,208)( 68,209)( 69,210)( 70,211)( 71,212)
( 72,213)( 73,202)( 74,203)( 75,204)( 76,205)( 77,206)( 78,207)( 79,199)
( 80,200)( 81,201)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,187)( 92,188)( 93,189)( 94,181)( 95,182)
( 96,183)( 97,184)( 98,185)( 99,186)(100,175)(101,176)(102,177)(103,178)
(104,179)(105,180)(106,172)(107,173)(108,174)(217,325)(218,326)(219,327)
(220,328)(221,329)(222,330)(223,331)(224,332)(225,333)(226,349)(227,350)
(228,351)(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,337)
(236,338)(237,339)(238,340)(239,341)(240,342)(241,334)(242,335)(243,336)
(244,352)(245,353)(246,354)(247,355)(248,356)(249,357)(250,358)(251,359)
(252,360)(253,376)(254,377)(255,378)(256,370)(257,371)(258,372)(259,373)
(260,374)(261,375)(262,364)(263,365)(264,366)(265,367)(266,368)(267,369)
(268,361)(269,362)(270,363)(271,406)(272,407)(273,408)(274,409)(275,410)
(276,411)(277,412)(278,413)(279,414)(280,430)(281,431)(282,432)(283,424)
(284,425)(285,426)(286,427)(287,428)(288,429)(289,418)(290,419)(291,420)
(292,421)(293,422)(294,423)(295,415)(296,416)(297,417)(298,379)(299,380)
(300,381)(301,382)(302,383)(303,384)(304,385)(305,386)(306,387)(307,403)
(308,404)(309,405)(310,397)(311,398)(312,399)(313,400)(314,401)(315,402)
(316,391)(317,392)(318,393)(319,394)(320,395)(321,396)(322,388)(323,389)
(324,390)(433,541)(434,542)(435,543)(436,544)(437,545)(438,546)(439,547)
(440,548)(441,549)(442,565)(443,566)(444,567)(445,559)(446,560)(447,561)
(448,562)(449,563)(450,564)(451,553)(452,554)(453,555)(454,556)(455,557)
(456,558)(457,550)(458,551)(459,552)(460,568)(461,569)(462,570)(463,571)
(464,572)(465,573)(466,574)(467,575)(468,576)(469,592)(470,593)(471,594)
(472,586)(473,587)(474,588)(475,589)(476,590)(477,591)(478,580)(479,581)
(480,582)(481,583)(482,584)(483,585)(484,577)(485,578)(486,579)(487,622)
(488,623)(489,624)(490,625)(491,626)(492,627)(493,628)(494,629)(495,630)
(496,646)(497,647)(498,648)(499,640)(500,641)(501,642)(502,643)(503,644)
(504,645)(505,634)(506,635)(507,636)(508,637)(509,638)(510,639)(511,631)
(512,632)(513,633)(514,595)(515,596)(516,597)(517,598)(518,599)(519,600)
(520,601)(521,602)(522,603)(523,619)(524,620)(525,621)(526,613)(527,614)
(528,615)(529,616)(530,617)(531,618)(532,607)(533,608)(534,609)(535,610)
(536,611)(537,612)(538,604)(539,605)(540,606)(649,757)(650,758)(651,759)
(652,760)(653,761)(654,762)(655,763)(656,764)(657,765)(658,781)(659,782)
(660,783)(661,775)(662,776)(663,777)(664,778)(665,779)(666,780)(667,769)
(668,770)(669,771)(670,772)(671,773)(672,774)(673,766)(674,767)(675,768)
(676,784)(677,785)(678,786)(679,787)(680,788)(681,789)(682,790)(683,791)
(684,792)(685,808)(686,809)(687,810)(688,802)(689,803)(690,804)(691,805)
(692,806)(693,807)(694,796)(695,797)(696,798)(697,799)(698,800)(699,801)
(700,793)(701,794)(702,795)(703,838)(704,839)(705,840)(706,841)(707,842)
(708,843)(709,844)(710,845)(711,846)(712,862)(713,863)(714,864)(715,856)
(716,857)(717,858)(718,859)(719,860)(720,861)(721,850)(722,851)(723,852)
(724,853)(725,854)(726,855)(727,847)(728,848)(729,849)(730,811)(731,812)
(732,813)(733,814)(734,815)(735,816)(736,817)(737,818)(738,819)(739,835)
(740,836)(741,837)(742,829)(743,830)(744,831)(745,832)(746,833)(747,834)
(748,823)(749,824)(750,825)(751,826)(752,827)(753,828)(754,820)(755,821)
(756,822);
s3 := Sym(864)!( 1,271)( 2,272)( 3,273)( 4,286)( 5,287)( 6,288)( 7,292)
( 8,293)( 9,294)( 10,280)( 11,281)( 12,282)( 13,295)( 14,296)( 15,297)
( 16,274)( 17,275)( 18,276)( 19,289)( 20,290)( 21,291)( 22,277)( 23,278)
( 24,279)( 25,283)( 26,284)( 27,285)( 28,298)( 29,299)( 30,300)( 31,313)
( 32,314)( 33,315)( 34,319)( 35,320)( 36,321)( 37,307)( 38,308)( 39,309)
( 40,322)( 41,323)( 42,324)( 43,301)( 44,302)( 45,303)( 46,316)( 47,317)
( 48,318)( 49,304)( 50,305)( 51,306)( 52,310)( 53,311)( 54,312)( 55,217)
( 56,218)( 57,219)( 58,232)( 59,233)( 60,234)( 61,238)( 62,239)( 63,240)
( 64,226)( 65,227)( 66,228)( 67,241)( 68,242)( 69,243)( 70,220)( 71,221)
( 72,222)( 73,235)( 74,236)( 75,237)( 76,223)( 77,224)( 78,225)( 79,229)
( 80,230)( 81,231)( 82,244)( 83,245)( 84,246)( 85,259)( 86,260)( 87,261)
( 88,265)( 89,266)( 90,267)( 91,253)( 92,254)( 93,255)( 94,268)( 95,269)
( 96,270)( 97,247)( 98,248)( 99,249)(100,262)(101,263)(102,264)(103,250)
(104,251)(105,252)(106,256)(107,257)(108,258)(109,352)(110,353)(111,354)
(112,367)(113,368)(114,369)(115,373)(116,374)(117,375)(118,361)(119,362)
(120,363)(121,376)(122,377)(123,378)(124,355)(125,356)(126,357)(127,370)
(128,371)(129,372)(130,358)(131,359)(132,360)(133,364)(134,365)(135,366)
(136,325)(137,326)(138,327)(139,340)(140,341)(141,342)(142,346)(143,347)
(144,348)(145,334)(146,335)(147,336)(148,349)(149,350)(150,351)(151,328)
(152,329)(153,330)(154,343)(155,344)(156,345)(157,331)(158,332)(159,333)
(160,337)(161,338)(162,339)(163,379)(164,380)(165,381)(166,394)(167,395)
(168,396)(169,400)(170,401)(171,402)(172,388)(173,389)(174,390)(175,403)
(176,404)(177,405)(178,382)(179,383)(180,384)(181,397)(182,398)(183,399)
(184,385)(185,386)(186,387)(187,391)(188,392)(189,393)(190,406)(191,407)
(192,408)(193,421)(194,422)(195,423)(196,427)(197,428)(198,429)(199,415)
(200,416)(201,417)(202,430)(203,431)(204,432)(205,409)(206,410)(207,411)
(208,424)(209,425)(210,426)(211,412)(212,413)(213,414)(214,418)(215,419)
(216,420)(433,703)(434,704)(435,705)(436,718)(437,719)(438,720)(439,724)
(440,725)(441,726)(442,712)(443,713)(444,714)(445,727)(446,728)(447,729)
(448,706)(449,707)(450,708)(451,721)(452,722)(453,723)(454,709)(455,710)
(456,711)(457,715)(458,716)(459,717)(460,730)(461,731)(462,732)(463,745)
(464,746)(465,747)(466,751)(467,752)(468,753)(469,739)(470,740)(471,741)
(472,754)(473,755)(474,756)(475,733)(476,734)(477,735)(478,748)(479,749)
(480,750)(481,736)(482,737)(483,738)(484,742)(485,743)(486,744)(487,649)
(488,650)(489,651)(490,664)(491,665)(492,666)(493,670)(494,671)(495,672)
(496,658)(497,659)(498,660)(499,673)(500,674)(501,675)(502,652)(503,653)
(504,654)(505,667)(506,668)(507,669)(508,655)(509,656)(510,657)(511,661)
(512,662)(513,663)(514,676)(515,677)(516,678)(517,691)(518,692)(519,693)
(520,697)(521,698)(522,699)(523,685)(524,686)(525,687)(526,700)(527,701)
(528,702)(529,679)(530,680)(531,681)(532,694)(533,695)(534,696)(535,682)
(536,683)(537,684)(538,688)(539,689)(540,690)(541,784)(542,785)(543,786)
(544,799)(545,800)(546,801)(547,805)(548,806)(549,807)(550,793)(551,794)
(552,795)(553,808)(554,809)(555,810)(556,787)(557,788)(558,789)(559,802)
(560,803)(561,804)(562,790)(563,791)(564,792)(565,796)(566,797)(567,798)
(568,757)(569,758)(570,759)(571,772)(572,773)(573,774)(574,778)(575,779)
(576,780)(577,766)(578,767)(579,768)(580,781)(581,782)(582,783)(583,760)
(584,761)(585,762)(586,775)(587,776)(588,777)(589,763)(590,764)(591,765)
(592,769)(593,770)(594,771)(595,811)(596,812)(597,813)(598,826)(599,827)
(600,828)(601,832)(602,833)(603,834)(604,820)(605,821)(606,822)(607,835)
(608,836)(609,837)(610,814)(611,815)(612,816)(613,829)(614,830)(615,831)
(616,817)(617,818)(618,819)(619,823)(620,824)(621,825)(622,838)(623,839)
(624,840)(625,853)(626,854)(627,855)(628,859)(629,860)(630,861)(631,847)
(632,848)(633,849)(634,862)(635,863)(636,864)(637,841)(638,842)(639,843)
(640,856)(641,857)(642,858)(643,844)(644,845)(645,846)(646,850)(647,851)
(648,852);
poly := sub<Sym(864)|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,
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope