Questions?
See the FAQ
or other info.

Polytope of Type {6,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 4
Schlafli Type : {6,8,6}
Number of vertices, edges, etc : 6, 72, 72, 18
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {6,4,6}*864a
   3-fold quotients : {2,8,6}*576
   4-fold quotients : {6,4,6}*432b
   6-fold quotients : {2,4,6}*288
   9-fold quotients : {6,8,2}*192
   12-fold quotients : {2,4,6}*144
   18-fold quotients : {6,4,2}*96a
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {6,2,2}*48
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {3,2,2}*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)(  4,220)(  5,221)(  6,222)(  7,223)(  8,224)
(  9,225)( 10,235)( 11,236)( 12,237)( 13,238)( 14,239)( 15,240)( 16,241)
( 17,242)( 18,243)( 19,226)( 20,227)( 21,228)( 22,229)( 23,230)( 24,231)
( 25,232)( 26,233)( 27,234)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,262)( 38,263)( 39,264)( 40,265)
( 41,266)( 42,267)( 43,268)( 44,269)( 45,270)( 46,253)( 47,254)( 48,255)
( 49,256)( 50,257)( 51,258)( 52,259)( 53,260)( 54,261)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,289)
( 65,290)( 66,291)( 67,292)( 68,293)( 69,294)( 70,295)( 71,296)( 72,297)
( 73,280)( 74,281)( 75,282)( 76,283)( 77,284)( 78,285)( 79,286)( 80,287)
( 81,288)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,316)( 92,317)( 93,318)( 94,319)( 95,320)( 96,321)
( 97,322)( 98,323)( 99,324)(100,307)(101,308)(102,309)(103,310)(104,311)
(105,312)(106,313)(107,314)(108,315)(109,325)(110,326)(111,327)(112,328)
(113,329)(114,330)(115,331)(116,332)(117,333)(118,343)(119,344)(120,345)
(121,346)(122,347)(123,348)(124,349)(125,350)(126,351)(127,334)(128,335)
(129,336)(130,337)(131,338)(132,339)(133,340)(134,341)(135,342)(136,352)
(137,353)(138,354)(139,355)(140,356)(141,357)(142,358)(143,359)(144,360)
(145,370)(146,371)(147,372)(148,373)(149,374)(150,375)(151,376)(152,377)
(153,378)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)
(161,368)(162,369)(163,379)(164,380)(165,381)(166,382)(167,383)(168,384)
(169,385)(170,386)(171,387)(172,397)(173,398)(174,399)(175,400)(176,401)
(177,402)(178,403)(179,404)(180,405)(181,388)(182,389)(183,390)(184,391)
(185,392)(186,393)(187,394)(188,395)(189,396)(190,406)(191,407)(192,408)
(193,409)(194,410)(195,411)(196,412)(197,413)(198,414)(199,424)(200,425)
(201,426)(202,427)(203,428)(204,429)(205,430)(206,431)(207,432)(208,415)
(209,416)(210,417)(211,418)(212,419)(213,420)(214,421)(215,422)(216,423)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,667)(443,668)(444,669)(445,670)(446,671)(447,672)(448,673)
(449,674)(450,675)(451,658)(452,659)(453,660)(454,661)(455,662)(456,663)
(457,664)(458,665)(459,666)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,694)(470,695)(471,696)(472,697)
(473,698)(474,699)(475,700)(476,701)(477,702)(478,685)(479,686)(480,687)
(481,688)(482,689)(483,690)(484,691)(485,692)(486,693)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,721)
(497,722)(498,723)(499,724)(500,725)(501,726)(502,727)(503,728)(504,729)
(505,712)(506,713)(507,714)(508,715)(509,716)(510,717)(511,718)(512,719)
(513,720)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,748)(524,749)(525,750)(526,751)(527,752)(528,753)
(529,754)(530,755)(531,756)(532,739)(533,740)(534,741)(535,742)(536,743)
(537,744)(538,745)(539,746)(540,747)(541,757)(542,758)(543,759)(544,760)
(545,761)(546,762)(547,763)(548,764)(549,765)(550,775)(551,776)(552,777)
(553,778)(554,779)(555,780)(556,781)(557,782)(558,783)(559,766)(560,767)
(561,768)(562,769)(563,770)(564,771)(565,772)(566,773)(567,774)(568,784)
(569,785)(570,786)(571,787)(572,788)(573,789)(574,790)(575,791)(576,792)
(577,802)(578,803)(579,804)(580,805)(581,806)(582,807)(583,808)(584,809)
(585,810)(586,793)(587,794)(588,795)(589,796)(590,797)(591,798)(592,799)
(593,800)(594,801)(595,811)(596,812)(597,813)(598,814)(599,815)(600,816)
(601,817)(602,818)(603,819)(604,829)(605,830)(606,831)(607,832)(608,833)
(609,834)(610,835)(611,836)(612,837)(613,820)(614,821)(615,822)(616,823)
(617,824)(618,825)(619,826)(620,827)(621,828)(622,838)(623,839)(624,840)
(625,841)(626,842)(627,843)(628,844)(629,845)(630,846)(631,856)(632,857)
(633,858)(634,859)(635,860)(636,861)(637,862)(638,863)(639,864)(640,847)
(641,848)(642,849)(643,850)(644,851)(645,852)(646,853)(647,854)(648,855);;
s1 := (  1,226)(  2,233)(  3,231)(  4,232)(  5,230)(  6,228)(  7,229)(  8,227)
(  9,234)( 10,217)( 11,224)( 12,222)( 13,223)( 14,221)( 15,219)( 16,220)
( 17,218)( 18,225)( 19,235)( 20,242)( 21,240)( 22,241)( 23,239)( 24,237)
( 25,238)( 26,236)( 27,243)( 28,253)( 29,260)( 30,258)( 31,259)( 32,257)
( 33,255)( 34,256)( 35,254)( 36,261)( 37,244)( 38,251)( 39,249)( 40,250)
( 41,248)( 42,246)( 43,247)( 44,245)( 45,252)( 46,262)( 47,269)( 48,267)
( 49,268)( 50,266)( 51,264)( 52,265)( 53,263)( 54,270)( 55,307)( 56,314)
( 57,312)( 58,313)( 59,311)( 60,309)( 61,310)( 62,308)( 63,315)( 64,298)
( 65,305)( 66,303)( 67,304)( 68,302)( 69,300)( 70,301)( 71,299)( 72,306)
( 73,316)( 74,323)( 75,321)( 76,322)( 77,320)( 78,318)( 79,319)( 80,317)
( 81,324)( 82,280)( 83,287)( 84,285)( 85,286)( 86,284)( 87,282)( 88,283)
( 89,281)( 90,288)( 91,271)( 92,278)( 93,276)( 94,277)( 95,275)( 96,273)
( 97,274)( 98,272)( 99,279)(100,289)(101,296)(102,294)(103,295)(104,293)
(105,291)(106,292)(107,290)(108,297)(109,361)(110,368)(111,366)(112,367)
(113,365)(114,363)(115,364)(116,362)(117,369)(118,352)(119,359)(120,357)
(121,358)(122,356)(123,354)(124,355)(125,353)(126,360)(127,370)(128,377)
(129,375)(130,376)(131,374)(132,372)(133,373)(134,371)(135,378)(136,334)
(137,341)(138,339)(139,340)(140,338)(141,336)(142,337)(143,335)(144,342)
(145,325)(146,332)(147,330)(148,331)(149,329)(150,327)(151,328)(152,326)
(153,333)(154,343)(155,350)(156,348)(157,349)(158,347)(159,345)(160,346)
(161,344)(162,351)(163,388)(164,395)(165,393)(166,394)(167,392)(168,390)
(169,391)(170,389)(171,396)(172,379)(173,386)(174,384)(175,385)(176,383)
(177,381)(178,382)(179,380)(180,387)(181,397)(182,404)(183,402)(184,403)
(185,401)(186,399)(187,400)(188,398)(189,405)(190,415)(191,422)(192,420)
(193,421)(194,419)(195,417)(196,418)(197,416)(198,423)(199,406)(200,413)
(201,411)(202,412)(203,410)(204,408)(205,409)(206,407)(207,414)(208,424)
(209,431)(210,429)(211,430)(212,428)(213,426)(214,427)(215,425)(216,432)
(433,712)(434,719)(435,717)(436,718)(437,716)(438,714)(439,715)(440,713)
(441,720)(442,703)(443,710)(444,708)(445,709)(446,707)(447,705)(448,706)
(449,704)(450,711)(451,721)(452,728)(453,726)(454,727)(455,725)(456,723)
(457,724)(458,722)(459,729)(460,739)(461,746)(462,744)(463,745)(464,743)
(465,741)(466,742)(467,740)(468,747)(469,730)(470,737)(471,735)(472,736)
(473,734)(474,732)(475,733)(476,731)(477,738)(478,748)(479,755)(480,753)
(481,754)(482,752)(483,750)(484,751)(485,749)(486,756)(487,658)(488,665)
(489,663)(490,664)(491,662)(492,660)(493,661)(494,659)(495,666)(496,649)
(497,656)(498,654)(499,655)(500,653)(501,651)(502,652)(503,650)(504,657)
(505,667)(506,674)(507,672)(508,673)(509,671)(510,669)(511,670)(512,668)
(513,675)(514,685)(515,692)(516,690)(517,691)(518,689)(519,687)(520,688)
(521,686)(522,693)(523,676)(524,683)(525,681)(526,682)(527,680)(528,678)
(529,679)(530,677)(531,684)(532,694)(533,701)(534,699)(535,700)(536,698)
(537,696)(538,697)(539,695)(540,702)(541,847)(542,854)(543,852)(544,853)
(545,851)(546,849)(547,850)(548,848)(549,855)(550,838)(551,845)(552,843)
(553,844)(554,842)(555,840)(556,841)(557,839)(558,846)(559,856)(560,863)
(561,861)(562,862)(563,860)(564,858)(565,859)(566,857)(567,864)(568,820)
(569,827)(570,825)(571,826)(572,824)(573,822)(574,823)(575,821)(576,828)
(577,811)(578,818)(579,816)(580,817)(581,815)(582,813)(583,814)(584,812)
(585,819)(586,829)(587,836)(588,834)(589,835)(590,833)(591,831)(592,832)
(593,830)(594,837)(595,793)(596,800)(597,798)(598,799)(599,797)(600,795)
(601,796)(602,794)(603,801)(604,784)(605,791)(606,789)(607,790)(608,788)
(609,786)(610,787)(611,785)(612,792)(613,802)(614,809)(615,807)(616,808)
(617,806)(618,804)(619,805)(620,803)(621,810)(622,766)(623,773)(624,771)
(625,772)(626,770)(627,768)(628,769)(629,767)(630,774)(631,757)(632,764)
(633,762)(634,763)(635,761)(636,759)(637,760)(638,758)(639,765)(640,775)
(641,782)(642,780)(643,781)(644,779)(645,777)(646,778)(647,776)(648,783);;
s2 := (  1,433)(  2,434)(  3,435)(  4,441)(  5,439)(  6,440)(  7,437)(  8,438)
(  9,436)( 10,442)( 11,443)( 12,444)( 13,450)( 14,448)( 15,449)( 16,446)
( 17,447)( 18,445)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)( 24,458)
( 25,455)( 26,456)( 27,454)( 28,460)( 29,461)( 30,462)( 31,468)( 32,466)
( 33,467)( 34,464)( 35,465)( 36,463)( 37,469)( 38,470)( 39,471)( 40,477)
( 41,475)( 42,476)( 43,473)( 44,474)( 45,472)( 46,478)( 47,479)( 48,480)
( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,514)( 56,515)
( 57,516)( 58,522)( 59,520)( 60,521)( 61,518)( 62,519)( 63,517)( 64,523)
( 65,524)( 66,525)( 67,531)( 68,529)( 69,530)( 70,527)( 71,528)( 72,526)
( 73,532)( 74,533)( 75,534)( 76,540)( 77,538)( 78,539)( 79,536)( 80,537)
( 81,535)( 82,487)( 83,488)( 84,489)( 85,495)( 86,493)( 87,494)( 88,491)
( 89,492)( 90,490)( 91,496)( 92,497)( 93,498)( 94,504)( 95,502)( 96,503)
( 97,500)( 98,501)( 99,499)(100,505)(101,506)(102,507)(103,513)(104,511)
(105,512)(106,509)(107,510)(108,508)(109,568)(110,569)(111,570)(112,576)
(113,574)(114,575)(115,572)(116,573)(117,571)(118,577)(119,578)(120,579)
(121,585)(122,583)(123,584)(124,581)(125,582)(126,580)(127,586)(128,587)
(129,588)(130,594)(131,592)(132,593)(133,590)(134,591)(135,589)(136,541)
(137,542)(138,543)(139,549)(140,547)(141,548)(142,545)(143,546)(144,544)
(145,550)(146,551)(147,552)(148,558)(149,556)(150,557)(151,554)(152,555)
(153,553)(154,559)(155,560)(156,561)(157,567)(158,565)(159,566)(160,563)
(161,564)(162,562)(163,595)(164,596)(165,597)(166,603)(167,601)(168,602)
(169,599)(170,600)(171,598)(172,604)(173,605)(174,606)(175,612)(176,610)
(177,611)(178,608)(179,609)(180,607)(181,613)(182,614)(183,615)(184,621)
(185,619)(186,620)(187,617)(188,618)(189,616)(190,622)(191,623)(192,624)
(193,630)(194,628)(195,629)(196,626)(197,627)(198,625)(199,631)(200,632)
(201,633)(202,639)(203,637)(204,638)(205,635)(206,636)(207,634)(208,640)
(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)(216,643)
(217,649)(218,650)(219,651)(220,657)(221,655)(222,656)(223,653)(224,654)
(225,652)(226,658)(227,659)(228,660)(229,666)(230,664)(231,665)(232,662)
(233,663)(234,661)(235,667)(236,668)(237,669)(238,675)(239,673)(240,674)
(241,671)(242,672)(243,670)(244,676)(245,677)(246,678)(247,684)(248,682)
(249,683)(250,680)(251,681)(252,679)(253,685)(254,686)(255,687)(256,693)
(257,691)(258,692)(259,689)(260,690)(261,688)(262,694)(263,695)(264,696)
(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,730)(272,731)
(273,732)(274,738)(275,736)(276,737)(277,734)(278,735)(279,733)(280,739)
(281,740)(282,741)(283,747)(284,745)(285,746)(286,743)(287,744)(288,742)
(289,748)(290,749)(291,750)(292,756)(293,754)(294,755)(295,752)(296,753)
(297,751)(298,703)(299,704)(300,705)(301,711)(302,709)(303,710)(304,707)
(305,708)(306,706)(307,712)(308,713)(309,714)(310,720)(311,718)(312,719)
(313,716)(314,717)(315,715)(316,721)(317,722)(318,723)(319,729)(320,727)
(321,728)(322,725)(323,726)(324,724)(325,784)(326,785)(327,786)(328,792)
(329,790)(330,791)(331,788)(332,789)(333,787)(334,793)(335,794)(336,795)
(337,801)(338,799)(339,800)(340,797)(341,798)(342,796)(343,802)(344,803)
(345,804)(346,810)(347,808)(348,809)(349,806)(350,807)(351,805)(352,757)
(353,758)(354,759)(355,765)(356,763)(357,764)(358,761)(359,762)(360,760)
(361,766)(362,767)(363,768)(364,774)(365,772)(366,773)(367,770)(368,771)
(369,769)(370,775)(371,776)(372,777)(373,783)(374,781)(375,782)(376,779)
(377,780)(378,778)(379,811)(380,812)(381,813)(382,819)(383,817)(384,818)
(385,815)(386,816)(387,814)(388,820)(389,821)(390,822)(391,828)(392,826)
(393,827)(394,824)(395,825)(396,823)(397,829)(398,830)(399,831)(400,837)
(401,835)(402,836)(403,833)(404,834)(405,832)(406,838)(407,839)(408,840)
(409,846)(410,844)(411,845)(412,842)(413,843)(414,841)(415,847)(416,848)
(417,849)(418,855)(419,853)(420,854)(421,851)(422,852)(423,850)(424,856)
(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)(432,859);;
s3 := (  1,167)(  2,166)(  3,168)(  4,164)(  5,163)(  6,165)(  7,170)(  8,169)
(  9,171)( 10,176)( 11,175)( 12,177)( 13,173)( 14,172)( 15,174)( 16,179)
( 17,178)( 18,180)( 19,185)( 20,184)( 21,186)( 22,182)( 23,181)( 24,183)
( 25,188)( 26,187)( 27,189)( 28,194)( 29,193)( 30,195)( 31,191)( 32,190)
( 33,192)( 34,197)( 35,196)( 36,198)( 37,203)( 38,202)( 39,204)( 40,200)
( 41,199)( 42,201)( 43,206)( 44,205)( 45,207)( 46,212)( 47,211)( 48,213)
( 49,209)( 50,208)( 51,210)( 52,215)( 53,214)( 54,216)( 55,140)( 56,139)
( 57,141)( 58,137)( 59,136)( 60,138)( 61,143)( 62,142)( 63,144)( 64,149)
( 65,148)( 66,150)( 67,146)( 68,145)( 69,147)( 70,152)( 71,151)( 72,153)
( 73,158)( 74,157)( 75,159)( 76,155)( 77,154)( 78,156)( 79,161)( 80,160)
( 81,162)( 82,113)( 83,112)( 84,114)( 85,110)( 86,109)( 87,111)( 88,116)
( 89,115)( 90,117)( 91,122)( 92,121)( 93,123)( 94,119)( 95,118)( 96,120)
( 97,125)( 98,124)( 99,126)(100,131)(101,130)(102,132)(103,128)(104,127)
(105,129)(106,134)(107,133)(108,135)(217,383)(218,382)(219,384)(220,380)
(221,379)(222,381)(223,386)(224,385)(225,387)(226,392)(227,391)(228,393)
(229,389)(230,388)(231,390)(232,395)(233,394)(234,396)(235,401)(236,400)
(237,402)(238,398)(239,397)(240,399)(241,404)(242,403)(243,405)(244,410)
(245,409)(246,411)(247,407)(248,406)(249,408)(250,413)(251,412)(252,414)
(253,419)(254,418)(255,420)(256,416)(257,415)(258,417)(259,422)(260,421)
(261,423)(262,428)(263,427)(264,429)(265,425)(266,424)(267,426)(268,431)
(269,430)(270,432)(271,356)(272,355)(273,357)(274,353)(275,352)(276,354)
(277,359)(278,358)(279,360)(280,365)(281,364)(282,366)(283,362)(284,361)
(285,363)(286,368)(287,367)(288,369)(289,374)(290,373)(291,375)(292,371)
(293,370)(294,372)(295,377)(296,376)(297,378)(298,329)(299,328)(300,330)
(301,326)(302,325)(303,327)(304,332)(305,331)(306,333)(307,338)(308,337)
(309,339)(310,335)(311,334)(312,336)(313,341)(314,340)(315,342)(316,347)
(317,346)(318,348)(319,344)(320,343)(321,345)(322,350)(323,349)(324,351)
(433,599)(434,598)(435,600)(436,596)(437,595)(438,597)(439,602)(440,601)
(441,603)(442,608)(443,607)(444,609)(445,605)(446,604)(447,606)(448,611)
(449,610)(450,612)(451,617)(452,616)(453,618)(454,614)(455,613)(456,615)
(457,620)(458,619)(459,621)(460,626)(461,625)(462,627)(463,623)(464,622)
(465,624)(466,629)(467,628)(468,630)(469,635)(470,634)(471,636)(472,632)
(473,631)(474,633)(475,638)(476,637)(477,639)(478,644)(479,643)(480,645)
(481,641)(482,640)(483,642)(484,647)(485,646)(486,648)(487,572)(488,571)
(489,573)(490,569)(491,568)(492,570)(493,575)(494,574)(495,576)(496,581)
(497,580)(498,582)(499,578)(500,577)(501,579)(502,584)(503,583)(504,585)
(505,590)(506,589)(507,591)(508,587)(509,586)(510,588)(511,593)(512,592)
(513,594)(514,545)(515,544)(516,546)(517,542)(518,541)(519,543)(520,548)
(521,547)(522,549)(523,554)(524,553)(525,555)(526,551)(527,550)(528,552)
(529,557)(530,556)(531,558)(532,563)(533,562)(534,564)(535,560)(536,559)
(537,561)(538,566)(539,565)(540,567)(649,815)(650,814)(651,816)(652,812)
(653,811)(654,813)(655,818)(656,817)(657,819)(658,824)(659,823)(660,825)
(661,821)(662,820)(663,822)(664,827)(665,826)(666,828)(667,833)(668,832)
(669,834)(670,830)(671,829)(672,831)(673,836)(674,835)(675,837)(676,842)
(677,841)(678,843)(679,839)(680,838)(681,840)(682,845)(683,844)(684,846)
(685,851)(686,850)(687,852)(688,848)(689,847)(690,849)(691,854)(692,853)
(693,855)(694,860)(695,859)(696,861)(697,857)(698,856)(699,858)(700,863)
(701,862)(702,864)(703,788)(704,787)(705,789)(706,785)(707,784)(708,786)
(709,791)(710,790)(711,792)(712,797)(713,796)(714,798)(715,794)(716,793)
(717,795)(718,800)(719,799)(720,801)(721,806)(722,805)(723,807)(724,803)
(725,802)(726,804)(727,809)(728,808)(729,810)(730,761)(731,760)(732,762)
(733,758)(734,757)(735,759)(736,764)(737,763)(738,765)(739,770)(740,769)
(741,771)(742,767)(743,766)(744,768)(745,773)(746,772)(747,774)(748,779)
(749,778)(750,780)(751,776)(752,775)(753,777)(754,782)(755,781)(756,783);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)
(  8,224)(  9,225)( 10,235)( 11,236)( 12,237)( 13,238)( 14,239)( 15,240)
( 16,241)( 17,242)( 18,243)( 19,226)( 20,227)( 21,228)( 22,229)( 23,230)
( 24,231)( 25,232)( 26,233)( 27,234)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,262)( 38,263)( 39,264)
( 40,265)( 41,266)( 42,267)( 43,268)( 44,269)( 45,270)( 46,253)( 47,254)
( 48,255)( 49,256)( 50,257)( 51,258)( 52,259)( 53,260)( 54,261)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,289)( 65,290)( 66,291)( 67,292)( 68,293)( 69,294)( 70,295)( 71,296)
( 72,297)( 73,280)( 74,281)( 75,282)( 76,283)( 77,284)( 78,285)( 79,286)
( 80,287)( 81,288)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,316)( 92,317)( 93,318)( 94,319)( 95,320)
( 96,321)( 97,322)( 98,323)( 99,324)(100,307)(101,308)(102,309)(103,310)
(104,311)(105,312)(106,313)(107,314)(108,315)(109,325)(110,326)(111,327)
(112,328)(113,329)(114,330)(115,331)(116,332)(117,333)(118,343)(119,344)
(120,345)(121,346)(122,347)(123,348)(124,349)(125,350)(126,351)(127,334)
(128,335)(129,336)(130,337)(131,338)(132,339)(133,340)(134,341)(135,342)
(136,352)(137,353)(138,354)(139,355)(140,356)(141,357)(142,358)(143,359)
(144,360)(145,370)(146,371)(147,372)(148,373)(149,374)(150,375)(151,376)
(152,377)(153,378)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)
(160,367)(161,368)(162,369)(163,379)(164,380)(165,381)(166,382)(167,383)
(168,384)(169,385)(170,386)(171,387)(172,397)(173,398)(174,399)(175,400)
(176,401)(177,402)(178,403)(179,404)(180,405)(181,388)(182,389)(183,390)
(184,391)(185,392)(186,393)(187,394)(188,395)(189,396)(190,406)(191,407)
(192,408)(193,409)(194,410)(195,411)(196,412)(197,413)(198,414)(199,424)
(200,425)(201,426)(202,427)(203,428)(204,429)(205,430)(206,431)(207,432)
(208,415)(209,416)(210,417)(211,418)(212,419)(213,420)(214,421)(215,422)
(216,423)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,667)(443,668)(444,669)(445,670)(446,671)(447,672)
(448,673)(449,674)(450,675)(451,658)(452,659)(453,660)(454,661)(455,662)
(456,663)(457,664)(458,665)(459,666)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,694)(470,695)(471,696)
(472,697)(473,698)(474,699)(475,700)(476,701)(477,702)(478,685)(479,686)
(480,687)(481,688)(482,689)(483,690)(484,691)(485,692)(486,693)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,721)(497,722)(498,723)(499,724)(500,725)(501,726)(502,727)(503,728)
(504,729)(505,712)(506,713)(507,714)(508,715)(509,716)(510,717)(511,718)
(512,719)(513,720)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,748)(524,749)(525,750)(526,751)(527,752)
(528,753)(529,754)(530,755)(531,756)(532,739)(533,740)(534,741)(535,742)
(536,743)(537,744)(538,745)(539,746)(540,747)(541,757)(542,758)(543,759)
(544,760)(545,761)(546,762)(547,763)(548,764)(549,765)(550,775)(551,776)
(552,777)(553,778)(554,779)(555,780)(556,781)(557,782)(558,783)(559,766)
(560,767)(561,768)(562,769)(563,770)(564,771)(565,772)(566,773)(567,774)
(568,784)(569,785)(570,786)(571,787)(572,788)(573,789)(574,790)(575,791)
(576,792)(577,802)(578,803)(579,804)(580,805)(581,806)(582,807)(583,808)
(584,809)(585,810)(586,793)(587,794)(588,795)(589,796)(590,797)(591,798)
(592,799)(593,800)(594,801)(595,811)(596,812)(597,813)(598,814)(599,815)
(600,816)(601,817)(602,818)(603,819)(604,829)(605,830)(606,831)(607,832)
(608,833)(609,834)(610,835)(611,836)(612,837)(613,820)(614,821)(615,822)
(616,823)(617,824)(618,825)(619,826)(620,827)(621,828)(622,838)(623,839)
(624,840)(625,841)(626,842)(627,843)(628,844)(629,845)(630,846)(631,856)
(632,857)(633,858)(634,859)(635,860)(636,861)(637,862)(638,863)(639,864)
(640,847)(641,848)(642,849)(643,850)(644,851)(645,852)(646,853)(647,854)
(648,855);
s1 := Sym(864)!(  1,226)(  2,233)(  3,231)(  4,232)(  5,230)(  6,228)(  7,229)
(  8,227)(  9,234)( 10,217)( 11,224)( 12,222)( 13,223)( 14,221)( 15,219)
( 16,220)( 17,218)( 18,225)( 19,235)( 20,242)( 21,240)( 22,241)( 23,239)
( 24,237)( 25,238)( 26,236)( 27,243)( 28,253)( 29,260)( 30,258)( 31,259)
( 32,257)( 33,255)( 34,256)( 35,254)( 36,261)( 37,244)( 38,251)( 39,249)
( 40,250)( 41,248)( 42,246)( 43,247)( 44,245)( 45,252)( 46,262)( 47,269)
( 48,267)( 49,268)( 50,266)( 51,264)( 52,265)( 53,263)( 54,270)( 55,307)
( 56,314)( 57,312)( 58,313)( 59,311)( 60,309)( 61,310)( 62,308)( 63,315)
( 64,298)( 65,305)( 66,303)( 67,304)( 68,302)( 69,300)( 70,301)( 71,299)
( 72,306)( 73,316)( 74,323)( 75,321)( 76,322)( 77,320)( 78,318)( 79,319)
( 80,317)( 81,324)( 82,280)( 83,287)( 84,285)( 85,286)( 86,284)( 87,282)
( 88,283)( 89,281)( 90,288)( 91,271)( 92,278)( 93,276)( 94,277)( 95,275)
( 96,273)( 97,274)( 98,272)( 99,279)(100,289)(101,296)(102,294)(103,295)
(104,293)(105,291)(106,292)(107,290)(108,297)(109,361)(110,368)(111,366)
(112,367)(113,365)(114,363)(115,364)(116,362)(117,369)(118,352)(119,359)
(120,357)(121,358)(122,356)(123,354)(124,355)(125,353)(126,360)(127,370)
(128,377)(129,375)(130,376)(131,374)(132,372)(133,373)(134,371)(135,378)
(136,334)(137,341)(138,339)(139,340)(140,338)(141,336)(142,337)(143,335)
(144,342)(145,325)(146,332)(147,330)(148,331)(149,329)(150,327)(151,328)
(152,326)(153,333)(154,343)(155,350)(156,348)(157,349)(158,347)(159,345)
(160,346)(161,344)(162,351)(163,388)(164,395)(165,393)(166,394)(167,392)
(168,390)(169,391)(170,389)(171,396)(172,379)(173,386)(174,384)(175,385)
(176,383)(177,381)(178,382)(179,380)(180,387)(181,397)(182,404)(183,402)
(184,403)(185,401)(186,399)(187,400)(188,398)(189,405)(190,415)(191,422)
(192,420)(193,421)(194,419)(195,417)(196,418)(197,416)(198,423)(199,406)
(200,413)(201,411)(202,412)(203,410)(204,408)(205,409)(206,407)(207,414)
(208,424)(209,431)(210,429)(211,430)(212,428)(213,426)(214,427)(215,425)
(216,432)(433,712)(434,719)(435,717)(436,718)(437,716)(438,714)(439,715)
(440,713)(441,720)(442,703)(443,710)(444,708)(445,709)(446,707)(447,705)
(448,706)(449,704)(450,711)(451,721)(452,728)(453,726)(454,727)(455,725)
(456,723)(457,724)(458,722)(459,729)(460,739)(461,746)(462,744)(463,745)
(464,743)(465,741)(466,742)(467,740)(468,747)(469,730)(470,737)(471,735)
(472,736)(473,734)(474,732)(475,733)(476,731)(477,738)(478,748)(479,755)
(480,753)(481,754)(482,752)(483,750)(484,751)(485,749)(486,756)(487,658)
(488,665)(489,663)(490,664)(491,662)(492,660)(493,661)(494,659)(495,666)
(496,649)(497,656)(498,654)(499,655)(500,653)(501,651)(502,652)(503,650)
(504,657)(505,667)(506,674)(507,672)(508,673)(509,671)(510,669)(511,670)
(512,668)(513,675)(514,685)(515,692)(516,690)(517,691)(518,689)(519,687)
(520,688)(521,686)(522,693)(523,676)(524,683)(525,681)(526,682)(527,680)
(528,678)(529,679)(530,677)(531,684)(532,694)(533,701)(534,699)(535,700)
(536,698)(537,696)(538,697)(539,695)(540,702)(541,847)(542,854)(543,852)
(544,853)(545,851)(546,849)(547,850)(548,848)(549,855)(550,838)(551,845)
(552,843)(553,844)(554,842)(555,840)(556,841)(557,839)(558,846)(559,856)
(560,863)(561,861)(562,862)(563,860)(564,858)(565,859)(566,857)(567,864)
(568,820)(569,827)(570,825)(571,826)(572,824)(573,822)(574,823)(575,821)
(576,828)(577,811)(578,818)(579,816)(580,817)(581,815)(582,813)(583,814)
(584,812)(585,819)(586,829)(587,836)(588,834)(589,835)(590,833)(591,831)
(592,832)(593,830)(594,837)(595,793)(596,800)(597,798)(598,799)(599,797)
(600,795)(601,796)(602,794)(603,801)(604,784)(605,791)(606,789)(607,790)
(608,788)(609,786)(610,787)(611,785)(612,792)(613,802)(614,809)(615,807)
(616,808)(617,806)(618,804)(619,805)(620,803)(621,810)(622,766)(623,773)
(624,771)(625,772)(626,770)(627,768)(628,769)(629,767)(630,774)(631,757)
(632,764)(633,762)(634,763)(635,761)(636,759)(637,760)(638,758)(639,765)
(640,775)(641,782)(642,780)(643,781)(644,779)(645,777)(646,778)(647,776)
(648,783);
s2 := Sym(864)!(  1,433)(  2,434)(  3,435)(  4,441)(  5,439)(  6,440)(  7,437)
(  8,438)(  9,436)( 10,442)( 11,443)( 12,444)( 13,450)( 14,448)( 15,449)
( 16,446)( 17,447)( 18,445)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)
( 24,458)( 25,455)( 26,456)( 27,454)( 28,460)( 29,461)( 30,462)( 31,468)
( 32,466)( 33,467)( 34,464)( 35,465)( 36,463)( 37,469)( 38,470)( 39,471)
( 40,477)( 41,475)( 42,476)( 43,473)( 44,474)( 45,472)( 46,478)( 47,479)
( 48,480)( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,514)
( 56,515)( 57,516)( 58,522)( 59,520)( 60,521)( 61,518)( 62,519)( 63,517)
( 64,523)( 65,524)( 66,525)( 67,531)( 68,529)( 69,530)( 70,527)( 71,528)
( 72,526)( 73,532)( 74,533)( 75,534)( 76,540)( 77,538)( 78,539)( 79,536)
( 80,537)( 81,535)( 82,487)( 83,488)( 84,489)( 85,495)( 86,493)( 87,494)
( 88,491)( 89,492)( 90,490)( 91,496)( 92,497)( 93,498)( 94,504)( 95,502)
( 96,503)( 97,500)( 98,501)( 99,499)(100,505)(101,506)(102,507)(103,513)
(104,511)(105,512)(106,509)(107,510)(108,508)(109,568)(110,569)(111,570)
(112,576)(113,574)(114,575)(115,572)(116,573)(117,571)(118,577)(119,578)
(120,579)(121,585)(122,583)(123,584)(124,581)(125,582)(126,580)(127,586)
(128,587)(129,588)(130,594)(131,592)(132,593)(133,590)(134,591)(135,589)
(136,541)(137,542)(138,543)(139,549)(140,547)(141,548)(142,545)(143,546)
(144,544)(145,550)(146,551)(147,552)(148,558)(149,556)(150,557)(151,554)
(152,555)(153,553)(154,559)(155,560)(156,561)(157,567)(158,565)(159,566)
(160,563)(161,564)(162,562)(163,595)(164,596)(165,597)(166,603)(167,601)
(168,602)(169,599)(170,600)(171,598)(172,604)(173,605)(174,606)(175,612)
(176,610)(177,611)(178,608)(179,609)(180,607)(181,613)(182,614)(183,615)
(184,621)(185,619)(186,620)(187,617)(188,618)(189,616)(190,622)(191,623)
(192,624)(193,630)(194,628)(195,629)(196,626)(197,627)(198,625)(199,631)
(200,632)(201,633)(202,639)(203,637)(204,638)(205,635)(206,636)(207,634)
(208,640)(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)
(216,643)(217,649)(218,650)(219,651)(220,657)(221,655)(222,656)(223,653)
(224,654)(225,652)(226,658)(227,659)(228,660)(229,666)(230,664)(231,665)
(232,662)(233,663)(234,661)(235,667)(236,668)(237,669)(238,675)(239,673)
(240,674)(241,671)(242,672)(243,670)(244,676)(245,677)(246,678)(247,684)
(248,682)(249,683)(250,680)(251,681)(252,679)(253,685)(254,686)(255,687)
(256,693)(257,691)(258,692)(259,689)(260,690)(261,688)(262,694)(263,695)
(264,696)(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,730)
(272,731)(273,732)(274,738)(275,736)(276,737)(277,734)(278,735)(279,733)
(280,739)(281,740)(282,741)(283,747)(284,745)(285,746)(286,743)(287,744)
(288,742)(289,748)(290,749)(291,750)(292,756)(293,754)(294,755)(295,752)
(296,753)(297,751)(298,703)(299,704)(300,705)(301,711)(302,709)(303,710)
(304,707)(305,708)(306,706)(307,712)(308,713)(309,714)(310,720)(311,718)
(312,719)(313,716)(314,717)(315,715)(316,721)(317,722)(318,723)(319,729)
(320,727)(321,728)(322,725)(323,726)(324,724)(325,784)(326,785)(327,786)
(328,792)(329,790)(330,791)(331,788)(332,789)(333,787)(334,793)(335,794)
(336,795)(337,801)(338,799)(339,800)(340,797)(341,798)(342,796)(343,802)
(344,803)(345,804)(346,810)(347,808)(348,809)(349,806)(350,807)(351,805)
(352,757)(353,758)(354,759)(355,765)(356,763)(357,764)(358,761)(359,762)
(360,760)(361,766)(362,767)(363,768)(364,774)(365,772)(366,773)(367,770)
(368,771)(369,769)(370,775)(371,776)(372,777)(373,783)(374,781)(375,782)
(376,779)(377,780)(378,778)(379,811)(380,812)(381,813)(382,819)(383,817)
(384,818)(385,815)(386,816)(387,814)(388,820)(389,821)(390,822)(391,828)
(392,826)(393,827)(394,824)(395,825)(396,823)(397,829)(398,830)(399,831)
(400,837)(401,835)(402,836)(403,833)(404,834)(405,832)(406,838)(407,839)
(408,840)(409,846)(410,844)(411,845)(412,842)(413,843)(414,841)(415,847)
(416,848)(417,849)(418,855)(419,853)(420,854)(421,851)(422,852)(423,850)
(424,856)(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)
(432,859);
s3 := Sym(864)!(  1,167)(  2,166)(  3,168)(  4,164)(  5,163)(  6,165)(  7,170)
(  8,169)(  9,171)( 10,176)( 11,175)( 12,177)( 13,173)( 14,172)( 15,174)
( 16,179)( 17,178)( 18,180)( 19,185)( 20,184)( 21,186)( 22,182)( 23,181)
( 24,183)( 25,188)( 26,187)( 27,189)( 28,194)( 29,193)( 30,195)( 31,191)
( 32,190)( 33,192)( 34,197)( 35,196)( 36,198)( 37,203)( 38,202)( 39,204)
( 40,200)( 41,199)( 42,201)( 43,206)( 44,205)( 45,207)( 46,212)( 47,211)
( 48,213)( 49,209)( 50,208)( 51,210)( 52,215)( 53,214)( 54,216)( 55,140)
( 56,139)( 57,141)( 58,137)( 59,136)( 60,138)( 61,143)( 62,142)( 63,144)
( 64,149)( 65,148)( 66,150)( 67,146)( 68,145)( 69,147)( 70,152)( 71,151)
( 72,153)( 73,158)( 74,157)( 75,159)( 76,155)( 77,154)( 78,156)( 79,161)
( 80,160)( 81,162)( 82,113)( 83,112)( 84,114)( 85,110)( 86,109)( 87,111)
( 88,116)( 89,115)( 90,117)( 91,122)( 92,121)( 93,123)( 94,119)( 95,118)
( 96,120)( 97,125)( 98,124)( 99,126)(100,131)(101,130)(102,132)(103,128)
(104,127)(105,129)(106,134)(107,133)(108,135)(217,383)(218,382)(219,384)
(220,380)(221,379)(222,381)(223,386)(224,385)(225,387)(226,392)(227,391)
(228,393)(229,389)(230,388)(231,390)(232,395)(233,394)(234,396)(235,401)
(236,400)(237,402)(238,398)(239,397)(240,399)(241,404)(242,403)(243,405)
(244,410)(245,409)(246,411)(247,407)(248,406)(249,408)(250,413)(251,412)
(252,414)(253,419)(254,418)(255,420)(256,416)(257,415)(258,417)(259,422)
(260,421)(261,423)(262,428)(263,427)(264,429)(265,425)(266,424)(267,426)
(268,431)(269,430)(270,432)(271,356)(272,355)(273,357)(274,353)(275,352)
(276,354)(277,359)(278,358)(279,360)(280,365)(281,364)(282,366)(283,362)
(284,361)(285,363)(286,368)(287,367)(288,369)(289,374)(290,373)(291,375)
(292,371)(293,370)(294,372)(295,377)(296,376)(297,378)(298,329)(299,328)
(300,330)(301,326)(302,325)(303,327)(304,332)(305,331)(306,333)(307,338)
(308,337)(309,339)(310,335)(311,334)(312,336)(313,341)(314,340)(315,342)
(316,347)(317,346)(318,348)(319,344)(320,343)(321,345)(322,350)(323,349)
(324,351)(433,599)(434,598)(435,600)(436,596)(437,595)(438,597)(439,602)
(440,601)(441,603)(442,608)(443,607)(444,609)(445,605)(446,604)(447,606)
(448,611)(449,610)(450,612)(451,617)(452,616)(453,618)(454,614)(455,613)
(456,615)(457,620)(458,619)(459,621)(460,626)(461,625)(462,627)(463,623)
(464,622)(465,624)(466,629)(467,628)(468,630)(469,635)(470,634)(471,636)
(472,632)(473,631)(474,633)(475,638)(476,637)(477,639)(478,644)(479,643)
(480,645)(481,641)(482,640)(483,642)(484,647)(485,646)(486,648)(487,572)
(488,571)(489,573)(490,569)(491,568)(492,570)(493,575)(494,574)(495,576)
(496,581)(497,580)(498,582)(499,578)(500,577)(501,579)(502,584)(503,583)
(504,585)(505,590)(506,589)(507,591)(508,587)(509,586)(510,588)(511,593)
(512,592)(513,594)(514,545)(515,544)(516,546)(517,542)(518,541)(519,543)
(520,548)(521,547)(522,549)(523,554)(524,553)(525,555)(526,551)(527,550)
(528,552)(529,557)(530,556)(531,558)(532,563)(533,562)(534,564)(535,560)
(536,559)(537,561)(538,566)(539,565)(540,567)(649,815)(650,814)(651,816)
(652,812)(653,811)(654,813)(655,818)(656,817)(657,819)(658,824)(659,823)
(660,825)(661,821)(662,820)(663,822)(664,827)(665,826)(666,828)(667,833)
(668,832)(669,834)(670,830)(671,829)(672,831)(673,836)(674,835)(675,837)
(676,842)(677,841)(678,843)(679,839)(680,838)(681,840)(682,845)(683,844)
(684,846)(685,851)(686,850)(687,852)(688,848)(689,847)(690,849)(691,854)
(692,853)(693,855)(694,860)(695,859)(696,861)(697,857)(698,856)(699,858)
(700,863)(701,862)(702,864)(703,788)(704,787)(705,789)(706,785)(707,784)
(708,786)(709,791)(710,790)(711,792)(712,797)(713,796)(714,798)(715,794)
(716,793)(717,795)(718,800)(719,799)(720,801)(721,806)(722,805)(723,807)
(724,803)(725,802)(726,804)(727,809)(728,808)(729,810)(730,761)(731,760)
(732,762)(733,758)(734,757)(735,759)(736,764)(737,763)(738,765)(739,770)
(740,769)(741,771)(742,767)(743,766)(744,768)(745,773)(746,772)(747,774)
(748,779)(749,778)(750,780)(751,776)(752,775)(753,777)(754,782)(755,781)
(756,783);
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*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope