Polytope of Type {4,54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54,4}*1728g
if this polytope has a name.
Group : SmallGroup(1728,20783)
Rank : 4
Schlafli Type : {4,54,4}
Number of vertices, edges, etc : 4, 108, 108, 4
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,27,4}*864
3-fold quotients : {4,18,4}*576g
6-fold quotients : {4,9,4}*288
9-fold quotients : {4,6,4}*192g
18-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,441)( 2,442)( 3,443)( 4,444)( 5,445)( 6,446)( 7,447)( 8,448)
( 9,433)( 10,434)( 11,435)( 12,436)( 13,437)( 14,438)( 15,439)( 16,440)
( 17,457)( 18,458)( 19,459)( 20,460)( 21,461)( 22,462)( 23,463)( 24,464)
( 25,449)( 26,450)( 27,451)( 28,452)( 29,453)( 30,454)( 31,455)( 32,456)
( 33,473)( 34,474)( 35,475)( 36,476)( 37,477)( 38,478)( 39,479)( 40,480)
( 41,465)( 42,466)( 43,467)( 44,468)( 45,469)( 46,470)( 47,471)( 48,472)
( 49,489)( 50,490)( 51,491)( 52,492)( 53,493)( 54,494)( 55,495)( 56,496)
( 57,481)( 58,482)( 59,483)( 60,484)( 61,485)( 62,486)( 63,487)( 64,488)
( 65,505)( 66,506)( 67,507)( 68,508)( 69,509)( 70,510)( 71,511)( 72,512)
( 73,497)( 74,498)( 75,499)( 76,500)( 77,501)( 78,502)( 79,503)( 80,504)
( 81,521)( 82,522)( 83,523)( 84,524)( 85,525)( 86,526)( 87,527)( 88,528)
( 89,513)( 90,514)( 91,515)( 92,516)( 93,517)( 94,518)( 95,519)( 96,520)
( 97,537)( 98,538)( 99,539)(100,540)(101,541)(102,542)(103,543)(104,544)
(105,529)(106,530)(107,531)(108,532)(109,533)(110,534)(111,535)(112,536)
(113,553)(114,554)(115,555)(116,556)(117,557)(118,558)(119,559)(120,560)
(121,545)(122,546)(123,547)(124,548)(125,549)(126,550)(127,551)(128,552)
(129,569)(130,570)(131,571)(132,572)(133,573)(134,574)(135,575)(136,576)
(137,561)(138,562)(139,563)(140,564)(141,565)(142,566)(143,567)(144,568)
(145,585)(146,586)(147,587)(148,588)(149,589)(150,590)(151,591)(152,592)
(153,577)(154,578)(155,579)(156,580)(157,581)(158,582)(159,583)(160,584)
(161,601)(162,602)(163,603)(164,604)(165,605)(166,606)(167,607)(168,608)
(169,593)(170,594)(171,595)(172,596)(173,597)(174,598)(175,599)(176,600)
(177,617)(178,618)(179,619)(180,620)(181,621)(182,622)(183,623)(184,624)
(185,609)(186,610)(187,611)(188,612)(189,613)(190,614)(191,615)(192,616)
(193,633)(194,634)(195,635)(196,636)(197,637)(198,638)(199,639)(200,640)
(201,625)(202,626)(203,627)(204,628)(205,629)(206,630)(207,631)(208,632)
(209,649)(210,650)(211,651)(212,652)(213,653)(214,654)(215,655)(216,656)
(217,641)(218,642)(219,643)(220,644)(221,645)(222,646)(223,647)(224,648)
(225,665)(226,666)(227,667)(228,668)(229,669)(230,670)(231,671)(232,672)
(233,657)(234,658)(235,659)(236,660)(237,661)(238,662)(239,663)(240,664)
(241,681)(242,682)(243,683)(244,684)(245,685)(246,686)(247,687)(248,688)
(249,673)(250,674)(251,675)(252,676)(253,677)(254,678)(255,679)(256,680)
(257,697)(258,698)(259,699)(260,700)(261,701)(262,702)(263,703)(264,704)
(265,689)(266,690)(267,691)(268,692)(269,693)(270,694)(271,695)(272,696)
(273,713)(274,714)(275,715)(276,716)(277,717)(278,718)(279,719)(280,720)
(281,705)(282,706)(283,707)(284,708)(285,709)(286,710)(287,711)(288,712)
(289,729)(290,730)(291,731)(292,732)(293,733)(294,734)(295,735)(296,736)
(297,721)(298,722)(299,723)(300,724)(301,725)(302,726)(303,727)(304,728)
(305,745)(306,746)(307,747)(308,748)(309,749)(310,750)(311,751)(312,752)
(313,737)(314,738)(315,739)(316,740)(317,741)(318,742)(319,743)(320,744)
(321,761)(322,762)(323,763)(324,764)(325,765)(326,766)(327,767)(328,768)
(329,753)(330,754)(331,755)(332,756)(333,757)(334,758)(335,759)(336,760)
(337,777)(338,778)(339,779)(340,780)(341,781)(342,782)(343,783)(344,784)
(345,769)(346,770)(347,771)(348,772)(349,773)(350,774)(351,775)(352,776)
(353,793)(354,794)(355,795)(356,796)(357,797)(358,798)(359,799)(360,800)
(361,785)(362,786)(363,787)(364,788)(365,789)(366,790)(367,791)(368,792)
(369,809)(370,810)(371,811)(372,812)(373,813)(374,814)(375,815)(376,816)
(377,801)(378,802)(379,803)(380,804)(381,805)(382,806)(383,807)(384,808)
(385,825)(386,826)(387,827)(388,828)(389,829)(390,830)(391,831)(392,832)
(393,817)(394,818)(395,819)(396,820)(397,821)(398,822)(399,823)(400,824)
(401,841)(402,842)(403,843)(404,844)(405,845)(406,846)(407,847)(408,848)
(409,833)(410,834)(411,835)(412,836)(413,837)(414,838)(415,839)(416,840)
(417,857)(418,858)(419,859)(420,860)(421,861)(422,862)(423,863)(424,864)
(425,849)(426,850)(427,851)(428,852)(429,853)(430,854)(431,855)(432,856);;
s1 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)( 50,130)
( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)( 58,142)
( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)( 66,114)
( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)( 74,126)
( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)( 82, 98)
( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)(146,418)
(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)(154,430)
(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)(162,402)
(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)(170,414)
(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)(178,386)
(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)(186,398)
(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)(194,370)
(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)(202,382)
(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)(210,354)
(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)(218,366)
(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)(226,338)
(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)(234,350)
(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)(242,322)
(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)(250,334)
(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)(258,306)
(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)(266,318)
(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)(274,290)
(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)(282,302)
(283,304)(284,303)(285,297)(286,298)(287,300)(288,299)(435,436)(439,440)
(441,445)(442,446)(443,448)(444,447)(449,465)(450,466)(451,468)(452,467)
(453,469)(454,470)(455,472)(456,471)(457,477)(458,478)(459,480)(460,479)
(461,473)(462,474)(463,476)(464,475)(481,561)(482,562)(483,564)(484,563)
(485,565)(486,566)(487,568)(488,567)(489,573)(490,574)(491,576)(492,575)
(493,569)(494,570)(495,572)(496,571)(497,545)(498,546)(499,548)(500,547)
(501,549)(502,550)(503,552)(504,551)(505,557)(506,558)(507,560)(508,559)
(509,553)(510,554)(511,556)(512,555)(513,529)(514,530)(515,532)(516,531)
(517,533)(518,534)(519,536)(520,535)(521,541)(522,542)(523,544)(524,543)
(525,537)(526,538)(527,540)(528,539)(577,849)(578,850)(579,852)(580,851)
(581,853)(582,854)(583,856)(584,855)(585,861)(586,862)(587,864)(588,863)
(589,857)(590,858)(591,860)(592,859)(593,833)(594,834)(595,836)(596,835)
(597,837)(598,838)(599,840)(600,839)(601,845)(602,846)(603,848)(604,847)
(605,841)(606,842)(607,844)(608,843)(609,817)(610,818)(611,820)(612,819)
(613,821)(614,822)(615,824)(616,823)(617,829)(618,830)(619,832)(620,831)
(621,825)(622,826)(623,828)(624,827)(625,801)(626,802)(627,804)(628,803)
(629,805)(630,806)(631,808)(632,807)(633,813)(634,814)(635,816)(636,815)
(637,809)(638,810)(639,812)(640,811)(641,785)(642,786)(643,788)(644,787)
(645,789)(646,790)(647,792)(648,791)(649,797)(650,798)(651,800)(652,799)
(653,793)(654,794)(655,796)(656,795)(657,769)(658,770)(659,772)(660,771)
(661,773)(662,774)(663,776)(664,775)(665,781)(666,782)(667,784)(668,783)
(669,777)(670,778)(671,780)(672,779)(673,753)(674,754)(675,756)(676,755)
(677,757)(678,758)(679,760)(680,759)(681,765)(682,766)(683,768)(684,767)
(685,761)(686,762)(687,764)(688,763)(689,737)(690,738)(691,740)(692,739)
(693,741)(694,742)(695,744)(696,743)(697,749)(698,750)(699,752)(700,751)
(701,745)(702,746)(703,748)(704,747)(705,721)(706,722)(707,724)(708,723)
(709,725)(710,726)(711,728)(712,727)(713,733)(714,734)(715,736)(716,735)
(717,729)(718,730)(719,732)(720,731);;
s2 := ( 1,577)( 2,580)( 3,579)( 4,578)( 5,589)( 6,592)( 7,591)( 8,590)
( 9,585)( 10,588)( 11,587)( 12,586)( 13,581)( 14,584)( 15,583)( 16,582)
( 17,609)( 18,612)( 19,611)( 20,610)( 21,621)( 22,624)( 23,623)( 24,622)
( 25,617)( 26,620)( 27,619)( 28,618)( 29,613)( 30,616)( 31,615)( 32,614)
( 33,593)( 34,596)( 35,595)( 36,594)( 37,605)( 38,608)( 39,607)( 40,606)
( 41,601)( 42,604)( 43,603)( 44,602)( 45,597)( 46,600)( 47,599)( 48,598)
( 49,705)( 50,708)( 51,707)( 52,706)( 53,717)( 54,720)( 55,719)( 56,718)
( 57,713)( 58,716)( 59,715)( 60,714)( 61,709)( 62,712)( 63,711)( 64,710)
( 65,689)( 66,692)( 67,691)( 68,690)( 69,701)( 70,704)( 71,703)( 72,702)
( 73,697)( 74,700)( 75,699)( 76,698)( 77,693)( 78,696)( 79,695)( 80,694)
( 81,673)( 82,676)( 83,675)( 84,674)( 85,685)( 86,688)( 87,687)( 88,686)
( 89,681)( 90,684)( 91,683)( 92,682)( 93,677)( 94,680)( 95,679)( 96,678)
( 97,657)( 98,660)( 99,659)(100,658)(101,669)(102,672)(103,671)(104,670)
(105,665)(106,668)(107,667)(108,666)(109,661)(110,664)(111,663)(112,662)
(113,641)(114,644)(115,643)(116,642)(117,653)(118,656)(119,655)(120,654)
(121,649)(122,652)(123,651)(124,650)(125,645)(126,648)(127,647)(128,646)
(129,625)(130,628)(131,627)(132,626)(133,637)(134,640)(135,639)(136,638)
(137,633)(138,636)(139,635)(140,634)(141,629)(142,632)(143,631)(144,630)
(145,433)(146,436)(147,435)(148,434)(149,445)(150,448)(151,447)(152,446)
(153,441)(154,444)(155,443)(156,442)(157,437)(158,440)(159,439)(160,438)
(161,465)(162,468)(163,467)(164,466)(165,477)(166,480)(167,479)(168,478)
(169,473)(170,476)(171,475)(172,474)(173,469)(174,472)(175,471)(176,470)
(177,449)(178,452)(179,451)(180,450)(181,461)(182,464)(183,463)(184,462)
(185,457)(186,460)(187,459)(188,458)(189,453)(190,456)(191,455)(192,454)
(193,561)(194,564)(195,563)(196,562)(197,573)(198,576)(199,575)(200,574)
(201,569)(202,572)(203,571)(204,570)(205,565)(206,568)(207,567)(208,566)
(209,545)(210,548)(211,547)(212,546)(213,557)(214,560)(215,559)(216,558)
(217,553)(218,556)(219,555)(220,554)(221,549)(222,552)(223,551)(224,550)
(225,529)(226,532)(227,531)(228,530)(229,541)(230,544)(231,543)(232,542)
(233,537)(234,540)(235,539)(236,538)(237,533)(238,536)(239,535)(240,534)
(241,513)(242,516)(243,515)(244,514)(245,525)(246,528)(247,527)(248,526)
(249,521)(250,524)(251,523)(252,522)(253,517)(254,520)(255,519)(256,518)
(257,497)(258,500)(259,499)(260,498)(261,509)(262,512)(263,511)(264,510)
(265,505)(266,508)(267,507)(268,506)(269,501)(270,504)(271,503)(272,502)
(273,481)(274,484)(275,483)(276,482)(277,493)(278,496)(279,495)(280,494)
(281,489)(282,492)(283,491)(284,490)(285,485)(286,488)(287,487)(288,486)
(289,849)(290,852)(291,851)(292,850)(293,861)(294,864)(295,863)(296,862)
(297,857)(298,860)(299,859)(300,858)(301,853)(302,856)(303,855)(304,854)
(305,833)(306,836)(307,835)(308,834)(309,845)(310,848)(311,847)(312,846)
(313,841)(314,844)(315,843)(316,842)(317,837)(318,840)(319,839)(320,838)
(321,817)(322,820)(323,819)(324,818)(325,829)(326,832)(327,831)(328,830)
(329,825)(330,828)(331,827)(332,826)(333,821)(334,824)(335,823)(336,822)
(337,801)(338,804)(339,803)(340,802)(341,813)(342,816)(343,815)(344,814)
(345,809)(346,812)(347,811)(348,810)(349,805)(350,808)(351,807)(352,806)
(353,785)(354,788)(355,787)(356,786)(357,797)(358,800)(359,799)(360,798)
(361,793)(362,796)(363,795)(364,794)(365,789)(366,792)(367,791)(368,790)
(369,769)(370,772)(371,771)(372,770)(373,781)(374,784)(375,783)(376,782)
(377,777)(378,780)(379,779)(380,778)(381,773)(382,776)(383,775)(384,774)
(385,753)(386,756)(387,755)(388,754)(389,765)(390,768)(391,767)(392,766)
(393,761)(394,764)(395,763)(396,762)(397,757)(398,760)(399,759)(400,758)
(401,737)(402,740)(403,739)(404,738)(405,749)(406,752)(407,751)(408,750)
(409,745)(410,748)(411,747)(412,746)(413,741)(414,744)(415,743)(416,742)
(417,721)(418,724)(419,723)(420,722)(421,733)(422,736)(423,735)(424,734)
(425,729)(426,732)(427,731)(428,730)(429,725)(430,728)(431,727)(432,726);;
s3 := ( 1,434)( 2,433)( 3,436)( 4,435)( 5,438)( 6,437)( 7,440)( 8,439)
( 9,442)( 10,441)( 11,444)( 12,443)( 13,446)( 14,445)( 15,448)( 16,447)
( 17,450)( 18,449)( 19,452)( 20,451)( 21,454)( 22,453)( 23,456)( 24,455)
( 25,458)( 26,457)( 27,460)( 28,459)( 29,462)( 30,461)( 31,464)( 32,463)
( 33,466)( 34,465)( 35,468)( 36,467)( 37,470)( 38,469)( 39,472)( 40,471)
( 41,474)( 42,473)( 43,476)( 44,475)( 45,478)( 46,477)( 47,480)( 48,479)
( 49,482)( 50,481)( 51,484)( 52,483)( 53,486)( 54,485)( 55,488)( 56,487)
( 57,490)( 58,489)( 59,492)( 60,491)( 61,494)( 62,493)( 63,496)( 64,495)
( 65,498)( 66,497)( 67,500)( 68,499)( 69,502)( 70,501)( 71,504)( 72,503)
( 73,506)( 74,505)( 75,508)( 76,507)( 77,510)( 78,509)( 79,512)( 80,511)
( 81,514)( 82,513)( 83,516)( 84,515)( 85,518)( 86,517)( 87,520)( 88,519)
( 89,522)( 90,521)( 91,524)( 92,523)( 93,526)( 94,525)( 95,528)( 96,527)
( 97,530)( 98,529)( 99,532)(100,531)(101,534)(102,533)(103,536)(104,535)
(105,538)(106,537)(107,540)(108,539)(109,542)(110,541)(111,544)(112,543)
(113,546)(114,545)(115,548)(116,547)(117,550)(118,549)(119,552)(120,551)
(121,554)(122,553)(123,556)(124,555)(125,558)(126,557)(127,560)(128,559)
(129,562)(130,561)(131,564)(132,563)(133,566)(134,565)(135,568)(136,567)
(137,570)(138,569)(139,572)(140,571)(141,574)(142,573)(143,576)(144,575)
(145,578)(146,577)(147,580)(148,579)(149,582)(150,581)(151,584)(152,583)
(153,586)(154,585)(155,588)(156,587)(157,590)(158,589)(159,592)(160,591)
(161,594)(162,593)(163,596)(164,595)(165,598)(166,597)(167,600)(168,599)
(169,602)(170,601)(171,604)(172,603)(173,606)(174,605)(175,608)(176,607)
(177,610)(178,609)(179,612)(180,611)(181,614)(182,613)(183,616)(184,615)
(185,618)(186,617)(187,620)(188,619)(189,622)(190,621)(191,624)(192,623)
(193,626)(194,625)(195,628)(196,627)(197,630)(198,629)(199,632)(200,631)
(201,634)(202,633)(203,636)(204,635)(205,638)(206,637)(207,640)(208,639)
(209,642)(210,641)(211,644)(212,643)(213,646)(214,645)(215,648)(216,647)
(217,650)(218,649)(219,652)(220,651)(221,654)(222,653)(223,656)(224,655)
(225,658)(226,657)(227,660)(228,659)(229,662)(230,661)(231,664)(232,663)
(233,666)(234,665)(235,668)(236,667)(237,670)(238,669)(239,672)(240,671)
(241,674)(242,673)(243,676)(244,675)(245,678)(246,677)(247,680)(248,679)
(249,682)(250,681)(251,684)(252,683)(253,686)(254,685)(255,688)(256,687)
(257,690)(258,689)(259,692)(260,691)(261,694)(262,693)(263,696)(264,695)
(265,698)(266,697)(267,700)(268,699)(269,702)(270,701)(271,704)(272,703)
(273,706)(274,705)(275,708)(276,707)(277,710)(278,709)(279,712)(280,711)
(281,714)(282,713)(283,716)(284,715)(285,718)(286,717)(287,720)(288,719)
(289,722)(290,721)(291,724)(292,723)(293,726)(294,725)(295,728)(296,727)
(297,730)(298,729)(299,732)(300,731)(301,734)(302,733)(303,736)(304,735)
(305,738)(306,737)(307,740)(308,739)(309,742)(310,741)(311,744)(312,743)
(313,746)(314,745)(315,748)(316,747)(317,750)(318,749)(319,752)(320,751)
(321,754)(322,753)(323,756)(324,755)(325,758)(326,757)(327,760)(328,759)
(329,762)(330,761)(331,764)(332,763)(333,766)(334,765)(335,768)(336,767)
(337,770)(338,769)(339,772)(340,771)(341,774)(342,773)(343,776)(344,775)
(345,778)(346,777)(347,780)(348,779)(349,782)(350,781)(351,784)(352,783)
(353,786)(354,785)(355,788)(356,787)(357,790)(358,789)(359,792)(360,791)
(361,794)(362,793)(363,796)(364,795)(365,798)(366,797)(367,800)(368,799)
(369,802)(370,801)(371,804)(372,803)(373,806)(374,805)(375,808)(376,807)
(377,810)(378,809)(379,812)(380,811)(381,814)(382,813)(383,816)(384,815)
(385,818)(386,817)(387,820)(388,819)(389,822)(390,821)(391,824)(392,823)
(393,826)(394,825)(395,828)(396,827)(397,830)(398,829)(399,832)(400,831)
(401,834)(402,833)(403,836)(404,835)(405,838)(406,837)(407,840)(408,839)
(409,842)(410,841)(411,844)(412,843)(413,846)(414,845)(415,848)(416,847)
(417,850)(418,849)(419,852)(420,851)(421,854)(422,853)(423,856)(424,855)
(425,858)(426,857)(427,860)(428,859)(429,862)(430,861)(431,864)(432,863);;
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*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s1*s3*s2*s1*s3*s0*s1*s2*s0*s1*s0,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*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)!( 1,441)( 2,442)( 3,443)( 4,444)( 5,445)( 6,446)( 7,447)
( 8,448)( 9,433)( 10,434)( 11,435)( 12,436)( 13,437)( 14,438)( 15,439)
( 16,440)( 17,457)( 18,458)( 19,459)( 20,460)( 21,461)( 22,462)( 23,463)
( 24,464)( 25,449)( 26,450)( 27,451)( 28,452)( 29,453)( 30,454)( 31,455)
( 32,456)( 33,473)( 34,474)( 35,475)( 36,476)( 37,477)( 38,478)( 39,479)
( 40,480)( 41,465)( 42,466)( 43,467)( 44,468)( 45,469)( 46,470)( 47,471)
( 48,472)( 49,489)( 50,490)( 51,491)( 52,492)( 53,493)( 54,494)( 55,495)
( 56,496)( 57,481)( 58,482)( 59,483)( 60,484)( 61,485)( 62,486)( 63,487)
( 64,488)( 65,505)( 66,506)( 67,507)( 68,508)( 69,509)( 70,510)( 71,511)
( 72,512)( 73,497)( 74,498)( 75,499)( 76,500)( 77,501)( 78,502)( 79,503)
( 80,504)( 81,521)( 82,522)( 83,523)( 84,524)( 85,525)( 86,526)( 87,527)
( 88,528)( 89,513)( 90,514)( 91,515)( 92,516)( 93,517)( 94,518)( 95,519)
( 96,520)( 97,537)( 98,538)( 99,539)(100,540)(101,541)(102,542)(103,543)
(104,544)(105,529)(106,530)(107,531)(108,532)(109,533)(110,534)(111,535)
(112,536)(113,553)(114,554)(115,555)(116,556)(117,557)(118,558)(119,559)
(120,560)(121,545)(122,546)(123,547)(124,548)(125,549)(126,550)(127,551)
(128,552)(129,569)(130,570)(131,571)(132,572)(133,573)(134,574)(135,575)
(136,576)(137,561)(138,562)(139,563)(140,564)(141,565)(142,566)(143,567)
(144,568)(145,585)(146,586)(147,587)(148,588)(149,589)(150,590)(151,591)
(152,592)(153,577)(154,578)(155,579)(156,580)(157,581)(158,582)(159,583)
(160,584)(161,601)(162,602)(163,603)(164,604)(165,605)(166,606)(167,607)
(168,608)(169,593)(170,594)(171,595)(172,596)(173,597)(174,598)(175,599)
(176,600)(177,617)(178,618)(179,619)(180,620)(181,621)(182,622)(183,623)
(184,624)(185,609)(186,610)(187,611)(188,612)(189,613)(190,614)(191,615)
(192,616)(193,633)(194,634)(195,635)(196,636)(197,637)(198,638)(199,639)
(200,640)(201,625)(202,626)(203,627)(204,628)(205,629)(206,630)(207,631)
(208,632)(209,649)(210,650)(211,651)(212,652)(213,653)(214,654)(215,655)
(216,656)(217,641)(218,642)(219,643)(220,644)(221,645)(222,646)(223,647)
(224,648)(225,665)(226,666)(227,667)(228,668)(229,669)(230,670)(231,671)
(232,672)(233,657)(234,658)(235,659)(236,660)(237,661)(238,662)(239,663)
(240,664)(241,681)(242,682)(243,683)(244,684)(245,685)(246,686)(247,687)
(248,688)(249,673)(250,674)(251,675)(252,676)(253,677)(254,678)(255,679)
(256,680)(257,697)(258,698)(259,699)(260,700)(261,701)(262,702)(263,703)
(264,704)(265,689)(266,690)(267,691)(268,692)(269,693)(270,694)(271,695)
(272,696)(273,713)(274,714)(275,715)(276,716)(277,717)(278,718)(279,719)
(280,720)(281,705)(282,706)(283,707)(284,708)(285,709)(286,710)(287,711)
(288,712)(289,729)(290,730)(291,731)(292,732)(293,733)(294,734)(295,735)
(296,736)(297,721)(298,722)(299,723)(300,724)(301,725)(302,726)(303,727)
(304,728)(305,745)(306,746)(307,747)(308,748)(309,749)(310,750)(311,751)
(312,752)(313,737)(314,738)(315,739)(316,740)(317,741)(318,742)(319,743)
(320,744)(321,761)(322,762)(323,763)(324,764)(325,765)(326,766)(327,767)
(328,768)(329,753)(330,754)(331,755)(332,756)(333,757)(334,758)(335,759)
(336,760)(337,777)(338,778)(339,779)(340,780)(341,781)(342,782)(343,783)
(344,784)(345,769)(346,770)(347,771)(348,772)(349,773)(350,774)(351,775)
(352,776)(353,793)(354,794)(355,795)(356,796)(357,797)(358,798)(359,799)
(360,800)(361,785)(362,786)(363,787)(364,788)(365,789)(366,790)(367,791)
(368,792)(369,809)(370,810)(371,811)(372,812)(373,813)(374,814)(375,815)
(376,816)(377,801)(378,802)(379,803)(380,804)(381,805)(382,806)(383,807)
(384,808)(385,825)(386,826)(387,827)(388,828)(389,829)(390,830)(391,831)
(392,832)(393,817)(394,818)(395,819)(396,820)(397,821)(398,822)(399,823)
(400,824)(401,841)(402,842)(403,843)(404,844)(405,845)(406,846)(407,847)
(408,848)(409,833)(410,834)(411,835)(412,836)(413,837)(414,838)(415,839)
(416,840)(417,857)(418,858)(419,859)(420,860)(421,861)(422,862)(423,863)
(424,864)(425,849)(426,850)(427,851)(428,852)(429,853)(430,854)(431,855)
(432,856);
s1 := Sym(864)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)
( 50,130)( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)
( 58,142)( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)
( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)
( 74,126)( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)
( 82, 98)( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)
(146,418)(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)
(154,430)(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)
(162,402)(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)
(170,414)(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)
(178,386)(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)
(186,398)(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)
(194,370)(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)
(202,382)(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)
(210,354)(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)
(218,366)(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)
(226,338)(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)
(234,350)(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)
(242,322)(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)
(250,334)(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)
(258,306)(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)
(266,318)(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)
(274,290)(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)
(282,302)(283,304)(284,303)(285,297)(286,298)(287,300)(288,299)(435,436)
(439,440)(441,445)(442,446)(443,448)(444,447)(449,465)(450,466)(451,468)
(452,467)(453,469)(454,470)(455,472)(456,471)(457,477)(458,478)(459,480)
(460,479)(461,473)(462,474)(463,476)(464,475)(481,561)(482,562)(483,564)
(484,563)(485,565)(486,566)(487,568)(488,567)(489,573)(490,574)(491,576)
(492,575)(493,569)(494,570)(495,572)(496,571)(497,545)(498,546)(499,548)
(500,547)(501,549)(502,550)(503,552)(504,551)(505,557)(506,558)(507,560)
(508,559)(509,553)(510,554)(511,556)(512,555)(513,529)(514,530)(515,532)
(516,531)(517,533)(518,534)(519,536)(520,535)(521,541)(522,542)(523,544)
(524,543)(525,537)(526,538)(527,540)(528,539)(577,849)(578,850)(579,852)
(580,851)(581,853)(582,854)(583,856)(584,855)(585,861)(586,862)(587,864)
(588,863)(589,857)(590,858)(591,860)(592,859)(593,833)(594,834)(595,836)
(596,835)(597,837)(598,838)(599,840)(600,839)(601,845)(602,846)(603,848)
(604,847)(605,841)(606,842)(607,844)(608,843)(609,817)(610,818)(611,820)
(612,819)(613,821)(614,822)(615,824)(616,823)(617,829)(618,830)(619,832)
(620,831)(621,825)(622,826)(623,828)(624,827)(625,801)(626,802)(627,804)
(628,803)(629,805)(630,806)(631,808)(632,807)(633,813)(634,814)(635,816)
(636,815)(637,809)(638,810)(639,812)(640,811)(641,785)(642,786)(643,788)
(644,787)(645,789)(646,790)(647,792)(648,791)(649,797)(650,798)(651,800)
(652,799)(653,793)(654,794)(655,796)(656,795)(657,769)(658,770)(659,772)
(660,771)(661,773)(662,774)(663,776)(664,775)(665,781)(666,782)(667,784)
(668,783)(669,777)(670,778)(671,780)(672,779)(673,753)(674,754)(675,756)
(676,755)(677,757)(678,758)(679,760)(680,759)(681,765)(682,766)(683,768)
(684,767)(685,761)(686,762)(687,764)(688,763)(689,737)(690,738)(691,740)
(692,739)(693,741)(694,742)(695,744)(696,743)(697,749)(698,750)(699,752)
(700,751)(701,745)(702,746)(703,748)(704,747)(705,721)(706,722)(707,724)
(708,723)(709,725)(710,726)(711,728)(712,727)(713,733)(714,734)(715,736)
(716,735)(717,729)(718,730)(719,732)(720,731);
s2 := Sym(864)!( 1,577)( 2,580)( 3,579)( 4,578)( 5,589)( 6,592)( 7,591)
( 8,590)( 9,585)( 10,588)( 11,587)( 12,586)( 13,581)( 14,584)( 15,583)
( 16,582)( 17,609)( 18,612)( 19,611)( 20,610)( 21,621)( 22,624)( 23,623)
( 24,622)( 25,617)( 26,620)( 27,619)( 28,618)( 29,613)( 30,616)( 31,615)
( 32,614)( 33,593)( 34,596)( 35,595)( 36,594)( 37,605)( 38,608)( 39,607)
( 40,606)( 41,601)( 42,604)( 43,603)( 44,602)( 45,597)( 46,600)( 47,599)
( 48,598)( 49,705)( 50,708)( 51,707)( 52,706)( 53,717)( 54,720)( 55,719)
( 56,718)( 57,713)( 58,716)( 59,715)( 60,714)( 61,709)( 62,712)( 63,711)
( 64,710)( 65,689)( 66,692)( 67,691)( 68,690)( 69,701)( 70,704)( 71,703)
( 72,702)( 73,697)( 74,700)( 75,699)( 76,698)( 77,693)( 78,696)( 79,695)
( 80,694)( 81,673)( 82,676)( 83,675)( 84,674)( 85,685)( 86,688)( 87,687)
( 88,686)( 89,681)( 90,684)( 91,683)( 92,682)( 93,677)( 94,680)( 95,679)
( 96,678)( 97,657)( 98,660)( 99,659)(100,658)(101,669)(102,672)(103,671)
(104,670)(105,665)(106,668)(107,667)(108,666)(109,661)(110,664)(111,663)
(112,662)(113,641)(114,644)(115,643)(116,642)(117,653)(118,656)(119,655)
(120,654)(121,649)(122,652)(123,651)(124,650)(125,645)(126,648)(127,647)
(128,646)(129,625)(130,628)(131,627)(132,626)(133,637)(134,640)(135,639)
(136,638)(137,633)(138,636)(139,635)(140,634)(141,629)(142,632)(143,631)
(144,630)(145,433)(146,436)(147,435)(148,434)(149,445)(150,448)(151,447)
(152,446)(153,441)(154,444)(155,443)(156,442)(157,437)(158,440)(159,439)
(160,438)(161,465)(162,468)(163,467)(164,466)(165,477)(166,480)(167,479)
(168,478)(169,473)(170,476)(171,475)(172,474)(173,469)(174,472)(175,471)
(176,470)(177,449)(178,452)(179,451)(180,450)(181,461)(182,464)(183,463)
(184,462)(185,457)(186,460)(187,459)(188,458)(189,453)(190,456)(191,455)
(192,454)(193,561)(194,564)(195,563)(196,562)(197,573)(198,576)(199,575)
(200,574)(201,569)(202,572)(203,571)(204,570)(205,565)(206,568)(207,567)
(208,566)(209,545)(210,548)(211,547)(212,546)(213,557)(214,560)(215,559)
(216,558)(217,553)(218,556)(219,555)(220,554)(221,549)(222,552)(223,551)
(224,550)(225,529)(226,532)(227,531)(228,530)(229,541)(230,544)(231,543)
(232,542)(233,537)(234,540)(235,539)(236,538)(237,533)(238,536)(239,535)
(240,534)(241,513)(242,516)(243,515)(244,514)(245,525)(246,528)(247,527)
(248,526)(249,521)(250,524)(251,523)(252,522)(253,517)(254,520)(255,519)
(256,518)(257,497)(258,500)(259,499)(260,498)(261,509)(262,512)(263,511)
(264,510)(265,505)(266,508)(267,507)(268,506)(269,501)(270,504)(271,503)
(272,502)(273,481)(274,484)(275,483)(276,482)(277,493)(278,496)(279,495)
(280,494)(281,489)(282,492)(283,491)(284,490)(285,485)(286,488)(287,487)
(288,486)(289,849)(290,852)(291,851)(292,850)(293,861)(294,864)(295,863)
(296,862)(297,857)(298,860)(299,859)(300,858)(301,853)(302,856)(303,855)
(304,854)(305,833)(306,836)(307,835)(308,834)(309,845)(310,848)(311,847)
(312,846)(313,841)(314,844)(315,843)(316,842)(317,837)(318,840)(319,839)
(320,838)(321,817)(322,820)(323,819)(324,818)(325,829)(326,832)(327,831)
(328,830)(329,825)(330,828)(331,827)(332,826)(333,821)(334,824)(335,823)
(336,822)(337,801)(338,804)(339,803)(340,802)(341,813)(342,816)(343,815)
(344,814)(345,809)(346,812)(347,811)(348,810)(349,805)(350,808)(351,807)
(352,806)(353,785)(354,788)(355,787)(356,786)(357,797)(358,800)(359,799)
(360,798)(361,793)(362,796)(363,795)(364,794)(365,789)(366,792)(367,791)
(368,790)(369,769)(370,772)(371,771)(372,770)(373,781)(374,784)(375,783)
(376,782)(377,777)(378,780)(379,779)(380,778)(381,773)(382,776)(383,775)
(384,774)(385,753)(386,756)(387,755)(388,754)(389,765)(390,768)(391,767)
(392,766)(393,761)(394,764)(395,763)(396,762)(397,757)(398,760)(399,759)
(400,758)(401,737)(402,740)(403,739)(404,738)(405,749)(406,752)(407,751)
(408,750)(409,745)(410,748)(411,747)(412,746)(413,741)(414,744)(415,743)
(416,742)(417,721)(418,724)(419,723)(420,722)(421,733)(422,736)(423,735)
(424,734)(425,729)(426,732)(427,731)(428,730)(429,725)(430,728)(431,727)
(432,726);
s3 := Sym(864)!( 1,434)( 2,433)( 3,436)( 4,435)( 5,438)( 6,437)( 7,440)
( 8,439)( 9,442)( 10,441)( 11,444)( 12,443)( 13,446)( 14,445)( 15,448)
( 16,447)( 17,450)( 18,449)( 19,452)( 20,451)( 21,454)( 22,453)( 23,456)
( 24,455)( 25,458)( 26,457)( 27,460)( 28,459)( 29,462)( 30,461)( 31,464)
( 32,463)( 33,466)( 34,465)( 35,468)( 36,467)( 37,470)( 38,469)( 39,472)
( 40,471)( 41,474)( 42,473)( 43,476)( 44,475)( 45,478)( 46,477)( 47,480)
( 48,479)( 49,482)( 50,481)( 51,484)( 52,483)( 53,486)( 54,485)( 55,488)
( 56,487)( 57,490)( 58,489)( 59,492)( 60,491)( 61,494)( 62,493)( 63,496)
( 64,495)( 65,498)( 66,497)( 67,500)( 68,499)( 69,502)( 70,501)( 71,504)
( 72,503)( 73,506)( 74,505)( 75,508)( 76,507)( 77,510)( 78,509)( 79,512)
( 80,511)( 81,514)( 82,513)( 83,516)( 84,515)( 85,518)( 86,517)( 87,520)
( 88,519)( 89,522)( 90,521)( 91,524)( 92,523)( 93,526)( 94,525)( 95,528)
( 96,527)( 97,530)( 98,529)( 99,532)(100,531)(101,534)(102,533)(103,536)
(104,535)(105,538)(106,537)(107,540)(108,539)(109,542)(110,541)(111,544)
(112,543)(113,546)(114,545)(115,548)(116,547)(117,550)(118,549)(119,552)
(120,551)(121,554)(122,553)(123,556)(124,555)(125,558)(126,557)(127,560)
(128,559)(129,562)(130,561)(131,564)(132,563)(133,566)(134,565)(135,568)
(136,567)(137,570)(138,569)(139,572)(140,571)(141,574)(142,573)(143,576)
(144,575)(145,578)(146,577)(147,580)(148,579)(149,582)(150,581)(151,584)
(152,583)(153,586)(154,585)(155,588)(156,587)(157,590)(158,589)(159,592)
(160,591)(161,594)(162,593)(163,596)(164,595)(165,598)(166,597)(167,600)
(168,599)(169,602)(170,601)(171,604)(172,603)(173,606)(174,605)(175,608)
(176,607)(177,610)(178,609)(179,612)(180,611)(181,614)(182,613)(183,616)
(184,615)(185,618)(186,617)(187,620)(188,619)(189,622)(190,621)(191,624)
(192,623)(193,626)(194,625)(195,628)(196,627)(197,630)(198,629)(199,632)
(200,631)(201,634)(202,633)(203,636)(204,635)(205,638)(206,637)(207,640)
(208,639)(209,642)(210,641)(211,644)(212,643)(213,646)(214,645)(215,648)
(216,647)(217,650)(218,649)(219,652)(220,651)(221,654)(222,653)(223,656)
(224,655)(225,658)(226,657)(227,660)(228,659)(229,662)(230,661)(231,664)
(232,663)(233,666)(234,665)(235,668)(236,667)(237,670)(238,669)(239,672)
(240,671)(241,674)(242,673)(243,676)(244,675)(245,678)(246,677)(247,680)
(248,679)(249,682)(250,681)(251,684)(252,683)(253,686)(254,685)(255,688)
(256,687)(257,690)(258,689)(259,692)(260,691)(261,694)(262,693)(263,696)
(264,695)(265,698)(266,697)(267,700)(268,699)(269,702)(270,701)(271,704)
(272,703)(273,706)(274,705)(275,708)(276,707)(277,710)(278,709)(279,712)
(280,711)(281,714)(282,713)(283,716)(284,715)(285,718)(286,717)(287,720)
(288,719)(289,722)(290,721)(291,724)(292,723)(293,726)(294,725)(295,728)
(296,727)(297,730)(298,729)(299,732)(300,731)(301,734)(302,733)(303,736)
(304,735)(305,738)(306,737)(307,740)(308,739)(309,742)(310,741)(311,744)
(312,743)(313,746)(314,745)(315,748)(316,747)(317,750)(318,749)(319,752)
(320,751)(321,754)(322,753)(323,756)(324,755)(325,758)(326,757)(327,760)
(328,759)(329,762)(330,761)(331,764)(332,763)(333,766)(334,765)(335,768)
(336,767)(337,770)(338,769)(339,772)(340,771)(341,774)(342,773)(343,776)
(344,775)(345,778)(346,777)(347,780)(348,779)(349,782)(350,781)(351,784)
(352,783)(353,786)(354,785)(355,788)(356,787)(357,790)(358,789)(359,792)
(360,791)(361,794)(362,793)(363,796)(364,795)(365,798)(366,797)(367,800)
(368,799)(369,802)(370,801)(371,804)(372,803)(373,806)(374,805)(375,808)
(376,807)(377,810)(378,809)(379,812)(380,811)(381,814)(382,813)(383,816)
(384,815)(385,818)(386,817)(387,820)(388,819)(389,822)(390,821)(391,824)
(392,823)(393,826)(394,825)(395,828)(396,827)(397,830)(398,829)(399,832)
(400,831)(401,834)(402,833)(403,836)(404,835)(405,838)(406,837)(407,840)
(408,839)(409,842)(410,841)(411,844)(412,843)(413,846)(414,845)(415,848)
(416,847)(417,850)(418,849)(419,852)(420,851)(421,854)(422,853)(423,856)
(424,855)(425,858)(426,857)(427,860)(428,859)(429,862)(430,861)(431,864)
(432,863);
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*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s1*s3*s2*s1*s3*s0*s1*s2*s0*s1*s0,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope