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