Questions?
See the FAQ
or other info.

Polytope of Type {3,6,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,16}*1728b
if this polytope has a name.
Group : SmallGroup(1728,31412)
Rank : 4
Schlafli Type : {3,6,16}
Number of vertices, edges, etc : 3, 27, 144, 48
Order of s0s1s2s3 : 48
Order of s0s1s2s3s2s1 : 6
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 : {3,6,8}*864b
   4-fold quotients : {3,6,4}*432b
   9-fold quotients : {3,2,16}*192
   18-fold quotients : {3,2,8}*96
   36-fold quotients : {3,2,4}*48
   72-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)(  8,330)
(  9,329)( 10,343)( 11,345)( 12,344)( 13,349)( 14,351)( 15,350)( 16,346)
( 17,348)( 18,347)( 19,334)( 20,336)( 21,335)( 22,340)( 23,342)( 24,341)
( 25,337)( 26,339)( 27,338)( 28,352)( 29,354)( 30,353)( 31,358)( 32,360)
( 33,359)( 34,355)( 35,357)( 36,356)( 37,370)( 38,372)( 39,371)( 40,376)
( 41,378)( 42,377)( 43,373)( 44,375)( 45,374)( 46,361)( 47,363)( 48,362)
( 49,367)( 50,369)( 51,368)( 52,364)( 53,366)( 54,365)( 55,379)( 56,381)
( 57,380)( 58,385)( 59,387)( 60,386)( 61,382)( 62,384)( 63,383)( 64,397)
( 65,399)( 66,398)( 67,403)( 68,405)( 69,404)( 70,400)( 71,402)( 72,401)
( 73,388)( 74,390)( 75,389)( 76,394)( 77,396)( 78,395)( 79,391)( 80,393)
( 81,392)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)( 88,409)
( 89,411)( 90,410)( 91,424)( 92,426)( 93,425)( 94,430)( 95,432)( 96,431)
( 97,427)( 98,429)( 99,428)(100,415)(101,417)(102,416)(103,421)(104,423)
(105,422)(106,418)(107,420)(108,419)(109,298)(110,300)(111,299)(112,304)
(113,306)(114,305)(115,301)(116,303)(117,302)(118,316)(119,318)(120,317)
(121,322)(122,324)(123,323)(124,319)(125,321)(126,320)(127,307)(128,309)
(129,308)(130,313)(131,315)(132,314)(133,310)(134,312)(135,311)(136,271)
(137,273)(138,272)(139,277)(140,279)(141,278)(142,274)(143,276)(144,275)
(145,289)(146,291)(147,290)(148,295)(149,297)(150,296)(151,292)(152,294)
(153,293)(154,280)(155,282)(156,281)(157,286)(158,288)(159,287)(160,283)
(161,285)(162,284)(163,217)(164,219)(165,218)(166,223)(167,225)(168,224)
(169,220)(170,222)(171,221)(172,235)(173,237)(174,236)(175,241)(176,243)
(177,242)(178,238)(179,240)(180,239)(181,226)(182,228)(183,227)(184,232)
(185,234)(186,233)(187,229)(188,231)(189,230)(190,244)(191,246)(192,245)
(193,250)(194,252)(195,251)(196,247)(197,249)(198,248)(199,262)(200,264)
(201,263)(202,268)(203,270)(204,269)(205,265)(206,267)(207,266)(208,253)
(209,255)(210,254)(211,259)(212,261)(213,260)(214,256)(215,258)(216,257)
(433,757)(434,759)(435,758)(436,763)(437,765)(438,764)(439,760)(440,762)
(441,761)(442,775)(443,777)(444,776)(445,781)(446,783)(447,782)(448,778)
(449,780)(450,779)(451,766)(452,768)(453,767)(454,772)(455,774)(456,773)
(457,769)(458,771)(459,770)(460,784)(461,786)(462,785)(463,790)(464,792)
(465,791)(466,787)(467,789)(468,788)(469,802)(470,804)(471,803)(472,808)
(473,810)(474,809)(475,805)(476,807)(477,806)(478,793)(479,795)(480,794)
(481,799)(482,801)(483,800)(484,796)(485,798)(486,797)(487,811)(488,813)
(489,812)(490,817)(491,819)(492,818)(493,814)(494,816)(495,815)(496,829)
(497,831)(498,830)(499,835)(500,837)(501,836)(502,832)(503,834)(504,833)
(505,820)(506,822)(507,821)(508,826)(509,828)(510,827)(511,823)(512,825)
(513,824)(514,838)(515,840)(516,839)(517,844)(518,846)(519,845)(520,841)
(521,843)(522,842)(523,856)(524,858)(525,857)(526,862)(527,864)(528,863)
(529,859)(530,861)(531,860)(532,847)(533,849)(534,848)(535,853)(536,855)
(537,854)(538,850)(539,852)(540,851)(541,730)(542,732)(543,731)(544,736)
(545,738)(546,737)(547,733)(548,735)(549,734)(550,748)(551,750)(552,749)
(553,754)(554,756)(555,755)(556,751)(557,753)(558,752)(559,739)(560,741)
(561,740)(562,745)(563,747)(564,746)(565,742)(566,744)(567,743)(568,703)
(569,705)(570,704)(571,709)(572,711)(573,710)(574,706)(575,708)(576,707)
(577,721)(578,723)(579,722)(580,727)(581,729)(582,728)(583,724)(584,726)
(585,725)(586,712)(587,714)(588,713)(589,718)(590,720)(591,719)(592,715)
(593,717)(594,716)(595,649)(596,651)(597,650)(598,655)(599,657)(600,656)
(601,652)(602,654)(603,653)(604,667)(605,669)(606,668)(607,673)(608,675)
(609,674)(610,670)(611,672)(612,671)(613,658)(614,660)(615,659)(616,664)
(617,666)(618,665)(619,661)(620,663)(621,662)(622,676)(623,678)(624,677)
(625,682)(626,684)(627,683)(628,679)(629,681)(630,680)(631,694)(632,696)
(633,695)(634,700)(635,702)(636,701)(637,697)(638,699)(639,698)(640,685)
(641,687)(642,686)(643,691)(644,693)(645,692)(646,688)(647,690)(648,689);;
s1 := (  1,337)(  2,339)(  3,338)(  4,334)(  5,336)(  6,335)(  7,340)(  8,342)
(  9,341)( 10,328)( 11,330)( 12,329)( 13,325)( 14,327)( 15,326)( 16,331)
( 17,333)( 18,332)( 19,346)( 20,348)( 21,347)( 22,343)( 23,345)( 24,344)
( 25,349)( 26,351)( 27,350)( 28,364)( 29,366)( 30,365)( 31,361)( 32,363)
( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)( 40,352)
( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,373)( 47,375)( 48,374)
( 49,370)( 50,372)( 51,371)( 52,376)( 53,378)( 54,377)( 55,391)( 56,393)
( 57,392)( 58,388)( 59,390)( 60,389)( 61,394)( 62,396)( 63,395)( 64,382)
( 65,384)( 66,383)( 67,379)( 68,381)( 69,380)( 70,385)( 71,387)( 72,386)
( 73,400)( 74,402)( 75,401)( 76,397)( 77,399)( 78,398)( 79,403)( 80,405)
( 81,404)( 82,418)( 83,420)( 84,419)( 85,415)( 86,417)( 87,416)( 88,421)
( 89,423)( 90,422)( 91,409)( 92,411)( 93,410)( 94,406)( 95,408)( 96,407)
( 97,412)( 98,414)( 99,413)(100,427)(101,429)(102,428)(103,424)(104,426)
(105,425)(106,430)(107,432)(108,431)(109,310)(110,312)(111,311)(112,307)
(113,309)(114,308)(115,313)(116,315)(117,314)(118,301)(119,303)(120,302)
(121,298)(122,300)(123,299)(124,304)(125,306)(126,305)(127,319)(128,321)
(129,320)(130,316)(131,318)(132,317)(133,322)(134,324)(135,323)(136,283)
(137,285)(138,284)(139,280)(140,282)(141,281)(142,286)(143,288)(144,287)
(145,274)(146,276)(147,275)(148,271)(149,273)(150,272)(151,277)(152,279)
(153,278)(154,292)(155,294)(156,293)(157,289)(158,291)(159,290)(160,295)
(161,297)(162,296)(163,229)(164,231)(165,230)(166,226)(167,228)(168,227)
(169,232)(170,234)(171,233)(172,220)(173,222)(174,221)(175,217)(176,219)
(177,218)(178,223)(179,225)(180,224)(181,238)(182,240)(183,239)(184,235)
(185,237)(186,236)(187,241)(188,243)(189,242)(190,256)(191,258)(192,257)
(193,253)(194,255)(195,254)(196,259)(197,261)(198,260)(199,247)(200,249)
(201,248)(202,244)(203,246)(204,245)(205,250)(206,252)(207,251)(208,265)
(209,267)(210,266)(211,262)(212,264)(213,263)(214,268)(215,270)(216,269)
(433,769)(434,771)(435,770)(436,766)(437,768)(438,767)(439,772)(440,774)
(441,773)(442,760)(443,762)(444,761)(445,757)(446,759)(447,758)(448,763)
(449,765)(450,764)(451,778)(452,780)(453,779)(454,775)(455,777)(456,776)
(457,781)(458,783)(459,782)(460,796)(461,798)(462,797)(463,793)(464,795)
(465,794)(466,799)(467,801)(468,800)(469,787)(470,789)(471,788)(472,784)
(473,786)(474,785)(475,790)(476,792)(477,791)(478,805)(479,807)(480,806)
(481,802)(482,804)(483,803)(484,808)(485,810)(486,809)(487,823)(488,825)
(489,824)(490,820)(491,822)(492,821)(493,826)(494,828)(495,827)(496,814)
(497,816)(498,815)(499,811)(500,813)(501,812)(502,817)(503,819)(504,818)
(505,832)(506,834)(507,833)(508,829)(509,831)(510,830)(511,835)(512,837)
(513,836)(514,850)(515,852)(516,851)(517,847)(518,849)(519,848)(520,853)
(521,855)(522,854)(523,841)(524,843)(525,842)(526,838)(527,840)(528,839)
(529,844)(530,846)(531,845)(532,859)(533,861)(534,860)(535,856)(536,858)
(537,857)(538,862)(539,864)(540,863)(541,742)(542,744)(543,743)(544,739)
(545,741)(546,740)(547,745)(548,747)(549,746)(550,733)(551,735)(552,734)
(553,730)(554,732)(555,731)(556,736)(557,738)(558,737)(559,751)(560,753)
(561,752)(562,748)(563,750)(564,749)(565,754)(566,756)(567,755)(568,715)
(569,717)(570,716)(571,712)(572,714)(573,713)(574,718)(575,720)(576,719)
(577,706)(578,708)(579,707)(580,703)(581,705)(582,704)(583,709)(584,711)
(585,710)(586,724)(587,726)(588,725)(589,721)(590,723)(591,722)(592,727)
(593,729)(594,728)(595,661)(596,663)(597,662)(598,658)(599,660)(600,659)
(601,664)(602,666)(603,665)(604,652)(605,654)(606,653)(607,649)(608,651)
(609,650)(610,655)(611,657)(612,656)(613,670)(614,672)(615,671)(616,667)
(617,669)(618,668)(619,673)(620,675)(621,674)(622,688)(623,690)(624,689)
(625,685)(626,687)(627,686)(628,691)(629,693)(630,692)(631,679)(632,681)
(633,680)(634,676)(635,678)(636,677)(637,682)(638,684)(639,683)(640,697)
(641,699)(642,698)(643,694)(644,696)(645,695)(646,700)(647,702)(648,701);;
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)(109,595)(110,596)(111,597)(112,603)
(113,601)(114,602)(115,599)(116,600)(117,598)(118,604)(119,605)(120,606)
(121,612)(122,610)(123,611)(124,608)(125,609)(126,607)(127,613)(128,614)
(129,615)(130,621)(131,619)(132,620)(133,617)(134,618)(135,616)(136,622)
(137,623)(138,624)(139,630)(140,628)(141,629)(142,626)(143,627)(144,625)
(145,631)(146,632)(147,633)(148,639)(149,637)(150,638)(151,635)(152,636)
(153,634)(154,640)(155,641)(156,642)(157,648)(158,646)(159,647)(160,644)
(161,645)(162,643)(163,541)(164,542)(165,543)(166,549)(167,547)(168,548)
(169,545)(170,546)(171,544)(172,550)(173,551)(174,552)(175,558)(176,556)
(177,557)(178,554)(179,555)(180,553)(181,559)(182,560)(183,561)(184,567)
(185,565)(186,566)(187,563)(188,564)(189,562)(190,568)(191,569)(192,570)
(193,576)(194,574)(195,575)(196,572)(197,573)(198,571)(199,577)(200,578)
(201,579)(202,585)(203,583)(204,584)(205,581)(206,582)(207,580)(208,586)
(209,587)(210,588)(211,594)(212,592)(213,593)(214,590)(215,591)(216,589)
(217,730)(218,731)(219,732)(220,738)(221,736)(222,737)(223,734)(224,735)
(225,733)(226,739)(227,740)(228,741)(229,747)(230,745)(231,746)(232,743)
(233,744)(234,742)(235,748)(236,749)(237,750)(238,756)(239,754)(240,755)
(241,752)(242,753)(243,751)(244,703)(245,704)(246,705)(247,711)(248,709)
(249,710)(250,707)(251,708)(252,706)(253,712)(254,713)(255,714)(256,720)
(257,718)(258,719)(259,716)(260,717)(261,715)(262,721)(263,722)(264,723)
(265,729)(266,727)(267,728)(268,725)(269,726)(270,724)(271,676)(272,677)
(273,678)(274,684)(275,682)(276,683)(277,680)(278,681)(279,679)(280,685)
(281,686)(282,687)(283,693)(284,691)(285,692)(286,689)(287,690)(288,688)
(289,694)(290,695)(291,696)(292,702)(293,700)(294,701)(295,698)(296,699)
(297,697)(298,649)(299,650)(300,651)(301,657)(302,655)(303,656)(304,653)
(305,654)(306,652)(307,658)(308,659)(309,660)(310,666)(311,664)(312,665)
(313,662)(314,663)(315,661)(316,667)(317,668)(318,669)(319,675)(320,673)
(321,674)(322,671)(323,672)(324,670)(325,757)(326,758)(327,759)(328,765)
(329,763)(330,764)(331,761)(332,762)(333,760)(334,766)(335,767)(336,768)
(337,774)(338,772)(339,773)(340,770)(341,771)(342,769)(343,775)(344,776)
(345,777)(346,783)(347,781)(348,782)(349,779)(350,780)(351,778)(352,784)
(353,785)(354,786)(355,792)(356,790)(357,791)(358,788)(359,789)(360,787)
(361,793)(362,794)(363,795)(364,801)(365,799)(366,800)(367,797)(368,798)
(369,796)(370,802)(371,803)(372,804)(373,810)(374,808)(375,809)(376,806)
(377,807)(378,805)(379,838)(380,839)(381,840)(382,846)(383,844)(384,845)
(385,842)(386,843)(387,841)(388,847)(389,848)(390,849)(391,855)(392,853)
(393,854)(394,851)(395,852)(396,850)(397,856)(398,857)(399,858)(400,864)
(401,862)(402,863)(403,860)(404,861)(405,859)(406,811)(407,812)(408,813)
(409,819)(410,817)(411,818)(412,815)(413,816)(414,814)(415,820)(416,821)
(417,822)(418,828)(419,826)(420,827)(421,824)(422,825)(423,823)(424,829)
(425,830)(426,831)(427,837)(428,835)(429,836)(430,833)(431,834)(432,832);;
s3 := (  1,109)(  2,114)(  3,116)(  4,112)(  5,117)(  6,110)(  7,115)(  8,111)
(  9,113)( 10,118)( 11,123)( 12,125)( 13,121)( 14,126)( 15,119)( 16,124)
( 17,120)( 18,122)( 19,127)( 20,132)( 21,134)( 22,130)( 23,135)( 24,128)
( 25,133)( 26,129)( 27,131)( 28,136)( 29,141)( 30,143)( 31,139)( 32,144)
( 33,137)( 34,142)( 35,138)( 36,140)( 37,145)( 38,150)( 39,152)( 40,148)
( 41,153)( 42,146)( 43,151)( 44,147)( 45,149)( 46,154)( 47,159)( 48,161)
( 49,157)( 50,162)( 51,155)( 52,160)( 53,156)( 54,158)( 55,190)( 56,195)
( 57,197)( 58,193)( 59,198)( 60,191)( 61,196)( 62,192)( 63,194)( 64,199)
( 65,204)( 66,206)( 67,202)( 68,207)( 69,200)( 70,205)( 71,201)( 72,203)
( 73,208)( 74,213)( 75,215)( 76,211)( 77,216)( 78,209)( 79,214)( 80,210)
( 81,212)( 82,163)( 83,168)( 84,170)( 85,166)( 86,171)( 87,164)( 88,169)
( 89,165)( 90,167)( 91,172)( 92,177)( 93,179)( 94,175)( 95,180)( 96,173)
( 97,178)( 98,174)( 99,176)(100,181)(101,186)(102,188)(103,184)(104,189)
(105,182)(106,187)(107,183)(108,185)(217,406)(218,411)(219,413)(220,409)
(221,414)(222,407)(223,412)(224,408)(225,410)(226,415)(227,420)(228,422)
(229,418)(230,423)(231,416)(232,421)(233,417)(234,419)(235,424)(236,429)
(237,431)(238,427)(239,432)(240,425)(241,430)(242,426)(243,428)(244,379)
(245,384)(246,386)(247,382)(248,387)(249,380)(250,385)(251,381)(252,383)
(253,388)(254,393)(255,395)(256,391)(257,396)(258,389)(259,394)(260,390)
(261,392)(262,397)(263,402)(264,404)(265,400)(266,405)(267,398)(268,403)
(269,399)(270,401)(271,352)(272,357)(273,359)(274,355)(275,360)(276,353)
(277,358)(278,354)(279,356)(280,361)(281,366)(282,368)(283,364)(284,369)
(285,362)(286,367)(287,363)(288,365)(289,370)(290,375)(291,377)(292,373)
(293,378)(294,371)(295,376)(296,372)(297,374)(298,325)(299,330)(300,332)
(301,328)(302,333)(303,326)(304,331)(305,327)(306,329)(307,334)(308,339)
(309,341)(310,337)(311,342)(312,335)(313,340)(314,336)(315,338)(316,343)
(317,348)(318,350)(319,346)(320,351)(321,344)(322,349)(323,345)(324,347)
(433,514)(434,519)(435,521)(436,517)(437,522)(438,515)(439,520)(440,516)
(441,518)(442,523)(443,528)(444,530)(445,526)(446,531)(447,524)(448,529)
(449,525)(450,527)(451,532)(452,537)(453,539)(454,535)(455,540)(456,533)
(457,538)(458,534)(459,536)(460,487)(461,492)(462,494)(463,490)(464,495)
(465,488)(466,493)(467,489)(468,491)(469,496)(470,501)(471,503)(472,499)
(473,504)(474,497)(475,502)(476,498)(477,500)(478,505)(479,510)(480,512)
(481,508)(482,513)(483,506)(484,511)(485,507)(486,509)(542,546)(543,548)
(545,549)(551,555)(552,557)(554,558)(560,564)(561,566)(563,567)(569,573)
(570,575)(572,576)(578,582)(579,584)(581,585)(587,591)(588,593)(590,594)
(595,622)(596,627)(597,629)(598,625)(599,630)(600,623)(601,628)(602,624)
(603,626)(604,631)(605,636)(606,638)(607,634)(608,639)(609,632)(610,637)
(611,633)(612,635)(613,640)(614,645)(615,647)(616,643)(617,648)(618,641)
(619,646)(620,642)(621,644)(649,676)(650,681)(651,683)(652,679)(653,684)
(654,677)(655,682)(656,678)(657,680)(658,685)(659,690)(660,692)(661,688)
(662,693)(663,686)(664,691)(665,687)(666,689)(667,694)(668,699)(669,701)
(670,697)(671,702)(672,695)(673,700)(674,696)(675,698)(704,708)(705,710)
(707,711)(713,717)(714,719)(716,720)(722,726)(723,728)(725,729)(731,735)
(732,737)(734,738)(740,744)(741,746)(743,747)(749,753)(750,755)(752,756)
(757,838)(758,843)(759,845)(760,841)(761,846)(762,839)(763,844)(764,840)
(765,842)(766,847)(767,852)(768,854)(769,850)(770,855)(771,848)(772,853)
(773,849)(774,851)(775,856)(776,861)(777,863)(778,859)(779,864)(780,857)
(781,862)(782,858)(783,860)(784,811)(785,816)(786,818)(787,814)(788,819)
(789,812)(790,817)(791,813)(792,815)(793,820)(794,825)(795,827)(796,823)
(797,828)(798,821)(799,826)(800,822)(801,824)(802,829)(803,834)(804,836)
(805,832)(806,837)(807,830)(808,835)(809,831)(810,833);;
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*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)!(  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)
(  8,330)(  9,329)( 10,343)( 11,345)( 12,344)( 13,349)( 14,351)( 15,350)
( 16,346)( 17,348)( 18,347)( 19,334)( 20,336)( 21,335)( 22,340)( 23,342)
( 24,341)( 25,337)( 26,339)( 27,338)( 28,352)( 29,354)( 30,353)( 31,358)
( 32,360)( 33,359)( 34,355)( 35,357)( 36,356)( 37,370)( 38,372)( 39,371)
( 40,376)( 41,378)( 42,377)( 43,373)( 44,375)( 45,374)( 46,361)( 47,363)
( 48,362)( 49,367)( 50,369)( 51,368)( 52,364)( 53,366)( 54,365)( 55,379)
( 56,381)( 57,380)( 58,385)( 59,387)( 60,386)( 61,382)( 62,384)( 63,383)
( 64,397)( 65,399)( 66,398)( 67,403)( 68,405)( 69,404)( 70,400)( 71,402)
( 72,401)( 73,388)( 74,390)( 75,389)( 76,394)( 77,396)( 78,395)( 79,391)
( 80,393)( 81,392)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)
( 88,409)( 89,411)( 90,410)( 91,424)( 92,426)( 93,425)( 94,430)( 95,432)
( 96,431)( 97,427)( 98,429)( 99,428)(100,415)(101,417)(102,416)(103,421)
(104,423)(105,422)(106,418)(107,420)(108,419)(109,298)(110,300)(111,299)
(112,304)(113,306)(114,305)(115,301)(116,303)(117,302)(118,316)(119,318)
(120,317)(121,322)(122,324)(123,323)(124,319)(125,321)(126,320)(127,307)
(128,309)(129,308)(130,313)(131,315)(132,314)(133,310)(134,312)(135,311)
(136,271)(137,273)(138,272)(139,277)(140,279)(141,278)(142,274)(143,276)
(144,275)(145,289)(146,291)(147,290)(148,295)(149,297)(150,296)(151,292)
(152,294)(153,293)(154,280)(155,282)(156,281)(157,286)(158,288)(159,287)
(160,283)(161,285)(162,284)(163,217)(164,219)(165,218)(166,223)(167,225)
(168,224)(169,220)(170,222)(171,221)(172,235)(173,237)(174,236)(175,241)
(176,243)(177,242)(178,238)(179,240)(180,239)(181,226)(182,228)(183,227)
(184,232)(185,234)(186,233)(187,229)(188,231)(189,230)(190,244)(191,246)
(192,245)(193,250)(194,252)(195,251)(196,247)(197,249)(198,248)(199,262)
(200,264)(201,263)(202,268)(203,270)(204,269)(205,265)(206,267)(207,266)
(208,253)(209,255)(210,254)(211,259)(212,261)(213,260)(214,256)(215,258)
(216,257)(433,757)(434,759)(435,758)(436,763)(437,765)(438,764)(439,760)
(440,762)(441,761)(442,775)(443,777)(444,776)(445,781)(446,783)(447,782)
(448,778)(449,780)(450,779)(451,766)(452,768)(453,767)(454,772)(455,774)
(456,773)(457,769)(458,771)(459,770)(460,784)(461,786)(462,785)(463,790)
(464,792)(465,791)(466,787)(467,789)(468,788)(469,802)(470,804)(471,803)
(472,808)(473,810)(474,809)(475,805)(476,807)(477,806)(478,793)(479,795)
(480,794)(481,799)(482,801)(483,800)(484,796)(485,798)(486,797)(487,811)
(488,813)(489,812)(490,817)(491,819)(492,818)(493,814)(494,816)(495,815)
(496,829)(497,831)(498,830)(499,835)(500,837)(501,836)(502,832)(503,834)
(504,833)(505,820)(506,822)(507,821)(508,826)(509,828)(510,827)(511,823)
(512,825)(513,824)(514,838)(515,840)(516,839)(517,844)(518,846)(519,845)
(520,841)(521,843)(522,842)(523,856)(524,858)(525,857)(526,862)(527,864)
(528,863)(529,859)(530,861)(531,860)(532,847)(533,849)(534,848)(535,853)
(536,855)(537,854)(538,850)(539,852)(540,851)(541,730)(542,732)(543,731)
(544,736)(545,738)(546,737)(547,733)(548,735)(549,734)(550,748)(551,750)
(552,749)(553,754)(554,756)(555,755)(556,751)(557,753)(558,752)(559,739)
(560,741)(561,740)(562,745)(563,747)(564,746)(565,742)(566,744)(567,743)
(568,703)(569,705)(570,704)(571,709)(572,711)(573,710)(574,706)(575,708)
(576,707)(577,721)(578,723)(579,722)(580,727)(581,729)(582,728)(583,724)
(584,726)(585,725)(586,712)(587,714)(588,713)(589,718)(590,720)(591,719)
(592,715)(593,717)(594,716)(595,649)(596,651)(597,650)(598,655)(599,657)
(600,656)(601,652)(602,654)(603,653)(604,667)(605,669)(606,668)(607,673)
(608,675)(609,674)(610,670)(611,672)(612,671)(613,658)(614,660)(615,659)
(616,664)(617,666)(618,665)(619,661)(620,663)(621,662)(622,676)(623,678)
(624,677)(625,682)(626,684)(627,683)(628,679)(629,681)(630,680)(631,694)
(632,696)(633,695)(634,700)(635,702)(636,701)(637,697)(638,699)(639,698)
(640,685)(641,687)(642,686)(643,691)(644,693)(645,692)(646,688)(647,690)
(648,689);
s1 := Sym(864)!(  1,337)(  2,339)(  3,338)(  4,334)(  5,336)(  6,335)(  7,340)
(  8,342)(  9,341)( 10,328)( 11,330)( 12,329)( 13,325)( 14,327)( 15,326)
( 16,331)( 17,333)( 18,332)( 19,346)( 20,348)( 21,347)( 22,343)( 23,345)
( 24,344)( 25,349)( 26,351)( 27,350)( 28,364)( 29,366)( 30,365)( 31,361)
( 32,363)( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)
( 40,352)( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,373)( 47,375)
( 48,374)( 49,370)( 50,372)( 51,371)( 52,376)( 53,378)( 54,377)( 55,391)
( 56,393)( 57,392)( 58,388)( 59,390)( 60,389)( 61,394)( 62,396)( 63,395)
( 64,382)( 65,384)( 66,383)( 67,379)( 68,381)( 69,380)( 70,385)( 71,387)
( 72,386)( 73,400)( 74,402)( 75,401)( 76,397)( 77,399)( 78,398)( 79,403)
( 80,405)( 81,404)( 82,418)( 83,420)( 84,419)( 85,415)( 86,417)( 87,416)
( 88,421)( 89,423)( 90,422)( 91,409)( 92,411)( 93,410)( 94,406)( 95,408)
( 96,407)( 97,412)( 98,414)( 99,413)(100,427)(101,429)(102,428)(103,424)
(104,426)(105,425)(106,430)(107,432)(108,431)(109,310)(110,312)(111,311)
(112,307)(113,309)(114,308)(115,313)(116,315)(117,314)(118,301)(119,303)
(120,302)(121,298)(122,300)(123,299)(124,304)(125,306)(126,305)(127,319)
(128,321)(129,320)(130,316)(131,318)(132,317)(133,322)(134,324)(135,323)
(136,283)(137,285)(138,284)(139,280)(140,282)(141,281)(142,286)(143,288)
(144,287)(145,274)(146,276)(147,275)(148,271)(149,273)(150,272)(151,277)
(152,279)(153,278)(154,292)(155,294)(156,293)(157,289)(158,291)(159,290)
(160,295)(161,297)(162,296)(163,229)(164,231)(165,230)(166,226)(167,228)
(168,227)(169,232)(170,234)(171,233)(172,220)(173,222)(174,221)(175,217)
(176,219)(177,218)(178,223)(179,225)(180,224)(181,238)(182,240)(183,239)
(184,235)(185,237)(186,236)(187,241)(188,243)(189,242)(190,256)(191,258)
(192,257)(193,253)(194,255)(195,254)(196,259)(197,261)(198,260)(199,247)
(200,249)(201,248)(202,244)(203,246)(204,245)(205,250)(206,252)(207,251)
(208,265)(209,267)(210,266)(211,262)(212,264)(213,263)(214,268)(215,270)
(216,269)(433,769)(434,771)(435,770)(436,766)(437,768)(438,767)(439,772)
(440,774)(441,773)(442,760)(443,762)(444,761)(445,757)(446,759)(447,758)
(448,763)(449,765)(450,764)(451,778)(452,780)(453,779)(454,775)(455,777)
(456,776)(457,781)(458,783)(459,782)(460,796)(461,798)(462,797)(463,793)
(464,795)(465,794)(466,799)(467,801)(468,800)(469,787)(470,789)(471,788)
(472,784)(473,786)(474,785)(475,790)(476,792)(477,791)(478,805)(479,807)
(480,806)(481,802)(482,804)(483,803)(484,808)(485,810)(486,809)(487,823)
(488,825)(489,824)(490,820)(491,822)(492,821)(493,826)(494,828)(495,827)
(496,814)(497,816)(498,815)(499,811)(500,813)(501,812)(502,817)(503,819)
(504,818)(505,832)(506,834)(507,833)(508,829)(509,831)(510,830)(511,835)
(512,837)(513,836)(514,850)(515,852)(516,851)(517,847)(518,849)(519,848)
(520,853)(521,855)(522,854)(523,841)(524,843)(525,842)(526,838)(527,840)
(528,839)(529,844)(530,846)(531,845)(532,859)(533,861)(534,860)(535,856)
(536,858)(537,857)(538,862)(539,864)(540,863)(541,742)(542,744)(543,743)
(544,739)(545,741)(546,740)(547,745)(548,747)(549,746)(550,733)(551,735)
(552,734)(553,730)(554,732)(555,731)(556,736)(557,738)(558,737)(559,751)
(560,753)(561,752)(562,748)(563,750)(564,749)(565,754)(566,756)(567,755)
(568,715)(569,717)(570,716)(571,712)(572,714)(573,713)(574,718)(575,720)
(576,719)(577,706)(578,708)(579,707)(580,703)(581,705)(582,704)(583,709)
(584,711)(585,710)(586,724)(587,726)(588,725)(589,721)(590,723)(591,722)
(592,727)(593,729)(594,728)(595,661)(596,663)(597,662)(598,658)(599,660)
(600,659)(601,664)(602,666)(603,665)(604,652)(605,654)(606,653)(607,649)
(608,651)(609,650)(610,655)(611,657)(612,656)(613,670)(614,672)(615,671)
(616,667)(617,669)(618,668)(619,673)(620,675)(621,674)(622,688)(623,690)
(624,689)(625,685)(626,687)(627,686)(628,691)(629,693)(630,692)(631,679)
(632,681)(633,680)(634,676)(635,678)(636,677)(637,682)(638,684)(639,683)
(640,697)(641,699)(642,698)(643,694)(644,696)(645,695)(646,700)(647,702)
(648,701);
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)(109,595)(110,596)(111,597)
(112,603)(113,601)(114,602)(115,599)(116,600)(117,598)(118,604)(119,605)
(120,606)(121,612)(122,610)(123,611)(124,608)(125,609)(126,607)(127,613)
(128,614)(129,615)(130,621)(131,619)(132,620)(133,617)(134,618)(135,616)
(136,622)(137,623)(138,624)(139,630)(140,628)(141,629)(142,626)(143,627)
(144,625)(145,631)(146,632)(147,633)(148,639)(149,637)(150,638)(151,635)
(152,636)(153,634)(154,640)(155,641)(156,642)(157,648)(158,646)(159,647)
(160,644)(161,645)(162,643)(163,541)(164,542)(165,543)(166,549)(167,547)
(168,548)(169,545)(170,546)(171,544)(172,550)(173,551)(174,552)(175,558)
(176,556)(177,557)(178,554)(179,555)(180,553)(181,559)(182,560)(183,561)
(184,567)(185,565)(186,566)(187,563)(188,564)(189,562)(190,568)(191,569)
(192,570)(193,576)(194,574)(195,575)(196,572)(197,573)(198,571)(199,577)
(200,578)(201,579)(202,585)(203,583)(204,584)(205,581)(206,582)(207,580)
(208,586)(209,587)(210,588)(211,594)(212,592)(213,593)(214,590)(215,591)
(216,589)(217,730)(218,731)(219,732)(220,738)(221,736)(222,737)(223,734)
(224,735)(225,733)(226,739)(227,740)(228,741)(229,747)(230,745)(231,746)
(232,743)(233,744)(234,742)(235,748)(236,749)(237,750)(238,756)(239,754)
(240,755)(241,752)(242,753)(243,751)(244,703)(245,704)(246,705)(247,711)
(248,709)(249,710)(250,707)(251,708)(252,706)(253,712)(254,713)(255,714)
(256,720)(257,718)(258,719)(259,716)(260,717)(261,715)(262,721)(263,722)
(264,723)(265,729)(266,727)(267,728)(268,725)(269,726)(270,724)(271,676)
(272,677)(273,678)(274,684)(275,682)(276,683)(277,680)(278,681)(279,679)
(280,685)(281,686)(282,687)(283,693)(284,691)(285,692)(286,689)(287,690)
(288,688)(289,694)(290,695)(291,696)(292,702)(293,700)(294,701)(295,698)
(296,699)(297,697)(298,649)(299,650)(300,651)(301,657)(302,655)(303,656)
(304,653)(305,654)(306,652)(307,658)(308,659)(309,660)(310,666)(311,664)
(312,665)(313,662)(314,663)(315,661)(316,667)(317,668)(318,669)(319,675)
(320,673)(321,674)(322,671)(323,672)(324,670)(325,757)(326,758)(327,759)
(328,765)(329,763)(330,764)(331,761)(332,762)(333,760)(334,766)(335,767)
(336,768)(337,774)(338,772)(339,773)(340,770)(341,771)(342,769)(343,775)
(344,776)(345,777)(346,783)(347,781)(348,782)(349,779)(350,780)(351,778)
(352,784)(353,785)(354,786)(355,792)(356,790)(357,791)(358,788)(359,789)
(360,787)(361,793)(362,794)(363,795)(364,801)(365,799)(366,800)(367,797)
(368,798)(369,796)(370,802)(371,803)(372,804)(373,810)(374,808)(375,809)
(376,806)(377,807)(378,805)(379,838)(380,839)(381,840)(382,846)(383,844)
(384,845)(385,842)(386,843)(387,841)(388,847)(389,848)(390,849)(391,855)
(392,853)(393,854)(394,851)(395,852)(396,850)(397,856)(398,857)(399,858)
(400,864)(401,862)(402,863)(403,860)(404,861)(405,859)(406,811)(407,812)
(408,813)(409,819)(410,817)(411,818)(412,815)(413,816)(414,814)(415,820)
(416,821)(417,822)(418,828)(419,826)(420,827)(421,824)(422,825)(423,823)
(424,829)(425,830)(426,831)(427,837)(428,835)(429,836)(430,833)(431,834)
(432,832);
s3 := Sym(864)!(  1,109)(  2,114)(  3,116)(  4,112)(  5,117)(  6,110)(  7,115)
(  8,111)(  9,113)( 10,118)( 11,123)( 12,125)( 13,121)( 14,126)( 15,119)
( 16,124)( 17,120)( 18,122)( 19,127)( 20,132)( 21,134)( 22,130)( 23,135)
( 24,128)( 25,133)( 26,129)( 27,131)( 28,136)( 29,141)( 30,143)( 31,139)
( 32,144)( 33,137)( 34,142)( 35,138)( 36,140)( 37,145)( 38,150)( 39,152)
( 40,148)( 41,153)( 42,146)( 43,151)( 44,147)( 45,149)( 46,154)( 47,159)
( 48,161)( 49,157)( 50,162)( 51,155)( 52,160)( 53,156)( 54,158)( 55,190)
( 56,195)( 57,197)( 58,193)( 59,198)( 60,191)( 61,196)( 62,192)( 63,194)
( 64,199)( 65,204)( 66,206)( 67,202)( 68,207)( 69,200)( 70,205)( 71,201)
( 72,203)( 73,208)( 74,213)( 75,215)( 76,211)( 77,216)( 78,209)( 79,214)
( 80,210)( 81,212)( 82,163)( 83,168)( 84,170)( 85,166)( 86,171)( 87,164)
( 88,169)( 89,165)( 90,167)( 91,172)( 92,177)( 93,179)( 94,175)( 95,180)
( 96,173)( 97,178)( 98,174)( 99,176)(100,181)(101,186)(102,188)(103,184)
(104,189)(105,182)(106,187)(107,183)(108,185)(217,406)(218,411)(219,413)
(220,409)(221,414)(222,407)(223,412)(224,408)(225,410)(226,415)(227,420)
(228,422)(229,418)(230,423)(231,416)(232,421)(233,417)(234,419)(235,424)
(236,429)(237,431)(238,427)(239,432)(240,425)(241,430)(242,426)(243,428)
(244,379)(245,384)(246,386)(247,382)(248,387)(249,380)(250,385)(251,381)
(252,383)(253,388)(254,393)(255,395)(256,391)(257,396)(258,389)(259,394)
(260,390)(261,392)(262,397)(263,402)(264,404)(265,400)(266,405)(267,398)
(268,403)(269,399)(270,401)(271,352)(272,357)(273,359)(274,355)(275,360)
(276,353)(277,358)(278,354)(279,356)(280,361)(281,366)(282,368)(283,364)
(284,369)(285,362)(286,367)(287,363)(288,365)(289,370)(290,375)(291,377)
(292,373)(293,378)(294,371)(295,376)(296,372)(297,374)(298,325)(299,330)
(300,332)(301,328)(302,333)(303,326)(304,331)(305,327)(306,329)(307,334)
(308,339)(309,341)(310,337)(311,342)(312,335)(313,340)(314,336)(315,338)
(316,343)(317,348)(318,350)(319,346)(320,351)(321,344)(322,349)(323,345)
(324,347)(433,514)(434,519)(435,521)(436,517)(437,522)(438,515)(439,520)
(440,516)(441,518)(442,523)(443,528)(444,530)(445,526)(446,531)(447,524)
(448,529)(449,525)(450,527)(451,532)(452,537)(453,539)(454,535)(455,540)
(456,533)(457,538)(458,534)(459,536)(460,487)(461,492)(462,494)(463,490)
(464,495)(465,488)(466,493)(467,489)(468,491)(469,496)(470,501)(471,503)
(472,499)(473,504)(474,497)(475,502)(476,498)(477,500)(478,505)(479,510)
(480,512)(481,508)(482,513)(483,506)(484,511)(485,507)(486,509)(542,546)
(543,548)(545,549)(551,555)(552,557)(554,558)(560,564)(561,566)(563,567)
(569,573)(570,575)(572,576)(578,582)(579,584)(581,585)(587,591)(588,593)
(590,594)(595,622)(596,627)(597,629)(598,625)(599,630)(600,623)(601,628)
(602,624)(603,626)(604,631)(605,636)(606,638)(607,634)(608,639)(609,632)
(610,637)(611,633)(612,635)(613,640)(614,645)(615,647)(616,643)(617,648)
(618,641)(619,646)(620,642)(621,644)(649,676)(650,681)(651,683)(652,679)
(653,684)(654,677)(655,682)(656,678)(657,680)(658,685)(659,690)(660,692)
(661,688)(662,693)(663,686)(664,691)(665,687)(666,689)(667,694)(668,699)
(669,701)(670,697)(671,702)(672,695)(673,700)(674,696)(675,698)(704,708)
(705,710)(707,711)(713,717)(714,719)(716,720)(722,726)(723,728)(725,729)
(731,735)(732,737)(734,738)(740,744)(741,746)(743,747)(749,753)(750,755)
(752,756)(757,838)(758,843)(759,845)(760,841)(761,846)(762,839)(763,844)
(764,840)(765,842)(766,847)(767,852)(768,854)(769,850)(770,855)(771,848)
(772,853)(773,849)(774,851)(775,856)(776,861)(777,863)(778,859)(779,864)
(780,857)(781,862)(782,858)(783,860)(784,811)(785,816)(786,818)(787,814)
(788,819)(789,812)(790,817)(791,813)(792,815)(793,820)(794,825)(795,827)
(796,823)(797,828)(798,821)(799,826)(800,822)(801,824)(802,829)(803,834)
(804,836)(805,832)(806,837)(807,830)(808,835)(809,831)(810,833);
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope