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Polytope of Type {3,6,40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,40}*1440
if this polytope has a name.
Group : SmallGroup(1440,1992)
Rank : 4
Schlafli Type : {3,6,40}
Number of vertices, edges, etc : 3, 9, 120, 40
Order of s0s1s2s3 : 120
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 : {3,6,20}*720
   3-fold quotients : {3,2,40}*480
   4-fold quotients : {3,6,10}*360
   5-fold quotients : {3,6,8}*288
   6-fold quotients : {3,2,20}*240
   10-fold quotients : {3,6,4}*144
   12-fold quotients : {3,2,10}*120
   15-fold quotients : {3,2,8}*96
   20-fold quotients : {3,6,2}*72
   24-fold quotients : {3,2,5}*60
   30-fold quotients : {3,2,4}*48
   60-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,541)(  2,542)(  3,543)(  4,544)(  5,545)(  6,551)(  7,552)(  8,553)
(  9,554)( 10,555)( 11,546)( 12,547)( 13,548)( 14,549)( 15,550)( 16,571)
( 17,572)( 18,573)( 19,574)( 20,575)( 21,581)( 22,582)( 23,583)( 24,584)
( 25,585)( 26,576)( 27,577)( 28,578)( 29,579)( 30,580)( 31,556)( 32,557)
( 33,558)( 34,559)( 35,560)( 36,566)( 37,567)( 38,568)( 39,569)( 40,570)
( 41,561)( 42,562)( 43,563)( 44,564)( 45,565)( 46,586)( 47,587)( 48,588)
( 49,589)( 50,590)( 51,596)( 52,597)( 53,598)( 54,599)( 55,600)( 56,591)
( 57,592)( 58,593)( 59,594)( 60,595)( 61,616)( 62,617)( 63,618)( 64,619)
( 65,620)( 66,626)( 67,627)( 68,628)( 69,629)( 70,630)( 71,621)( 72,622)
( 73,623)( 74,624)( 75,625)( 76,601)( 77,602)( 78,603)( 79,604)( 80,605)
( 81,611)( 82,612)( 83,613)( 84,614)( 85,615)( 86,606)( 87,607)( 88,608)
( 89,609)( 90,610)( 91,631)( 92,632)( 93,633)( 94,634)( 95,635)( 96,641)
( 97,642)( 98,643)( 99,644)(100,645)(101,636)(102,637)(103,638)(104,639)
(105,640)(106,661)(107,662)(108,663)(109,664)(110,665)(111,671)(112,672)
(113,673)(114,674)(115,675)(116,666)(117,667)(118,668)(119,669)(120,670)
(121,646)(122,647)(123,648)(124,649)(125,650)(126,656)(127,657)(128,658)
(129,659)(130,660)(131,651)(132,652)(133,653)(134,654)(135,655)(136,676)
(137,677)(138,678)(139,679)(140,680)(141,686)(142,687)(143,688)(144,689)
(145,690)(146,681)(147,682)(148,683)(149,684)(150,685)(151,706)(152,707)
(153,708)(154,709)(155,710)(156,716)(157,717)(158,718)(159,719)(160,720)
(161,711)(162,712)(163,713)(164,714)(165,715)(166,691)(167,692)(168,693)
(169,694)(170,695)(171,701)(172,702)(173,703)(174,704)(175,705)(176,696)
(177,697)(178,698)(179,699)(180,700)(181,451)(182,452)(183,453)(184,454)
(185,455)(186,461)(187,462)(188,463)(189,464)(190,465)(191,456)(192,457)
(193,458)(194,459)(195,460)(196,481)(197,482)(198,483)(199,484)(200,485)
(201,491)(202,492)(203,493)(204,494)(205,495)(206,486)(207,487)(208,488)
(209,489)(210,490)(211,466)(212,467)(213,468)(214,469)(215,470)(216,476)
(217,477)(218,478)(219,479)(220,480)(221,471)(222,472)(223,473)(224,474)
(225,475)(226,496)(227,497)(228,498)(229,499)(230,500)(231,506)(232,507)
(233,508)(234,509)(235,510)(236,501)(237,502)(238,503)(239,504)(240,505)
(241,526)(242,527)(243,528)(244,529)(245,530)(246,536)(247,537)(248,538)
(249,539)(250,540)(251,531)(252,532)(253,533)(254,534)(255,535)(256,511)
(257,512)(258,513)(259,514)(260,515)(261,521)(262,522)(263,523)(264,524)
(265,525)(266,516)(267,517)(268,518)(269,519)(270,520)(271,406)(272,407)
(273,408)(274,409)(275,410)(276,416)(277,417)(278,418)(279,419)(280,420)
(281,411)(282,412)(283,413)(284,414)(285,415)(286,436)(287,437)(288,438)
(289,439)(290,440)(291,446)(292,447)(293,448)(294,449)(295,450)(296,441)
(297,442)(298,443)(299,444)(300,445)(301,421)(302,422)(303,423)(304,424)
(305,425)(306,431)(307,432)(308,433)(309,434)(310,435)(311,426)(312,427)
(313,428)(314,429)(315,430)(316,361)(317,362)(318,363)(319,364)(320,365)
(321,371)(322,372)(323,373)(324,374)(325,375)(326,366)(327,367)(328,368)
(329,369)(330,370)(331,391)(332,392)(333,393)(334,394)(335,395)(336,401)
(337,402)(338,403)(339,404)(340,405)(341,396)(342,397)(343,398)(344,399)
(345,400)(346,376)(347,377)(348,378)(349,379)(350,380)(351,386)(352,387)
(353,388)(354,389)(355,390)(356,381)(357,382)(358,383)(359,384)(360,385);;
s1 := (  1,561)(  2,562)(  3,563)(  4,564)(  5,565)(  6,556)(  7,557)(  8,558)
(  9,559)( 10,560)( 11,566)( 12,567)( 13,568)( 14,569)( 15,570)( 16,546)
( 17,547)( 18,548)( 19,549)( 20,550)( 21,541)( 22,542)( 23,543)( 24,544)
( 25,545)( 26,551)( 27,552)( 28,553)( 29,554)( 30,555)( 31,576)( 32,577)
( 33,578)( 34,579)( 35,580)( 36,571)( 37,572)( 38,573)( 39,574)( 40,575)
( 41,581)( 42,582)( 43,583)( 44,584)( 45,585)( 46,606)( 47,607)( 48,608)
( 49,609)( 50,610)( 51,601)( 52,602)( 53,603)( 54,604)( 55,605)( 56,611)
( 57,612)( 58,613)( 59,614)( 60,615)( 61,591)( 62,592)( 63,593)( 64,594)
( 65,595)( 66,586)( 67,587)( 68,588)( 69,589)( 70,590)( 71,596)( 72,597)
( 73,598)( 74,599)( 75,600)( 76,621)( 77,622)( 78,623)( 79,624)( 80,625)
( 81,616)( 82,617)( 83,618)( 84,619)( 85,620)( 86,626)( 87,627)( 88,628)
( 89,629)( 90,630)( 91,651)( 92,652)( 93,653)( 94,654)( 95,655)( 96,646)
( 97,647)( 98,648)( 99,649)(100,650)(101,656)(102,657)(103,658)(104,659)
(105,660)(106,636)(107,637)(108,638)(109,639)(110,640)(111,631)(112,632)
(113,633)(114,634)(115,635)(116,641)(117,642)(118,643)(119,644)(120,645)
(121,666)(122,667)(123,668)(124,669)(125,670)(126,661)(127,662)(128,663)
(129,664)(130,665)(131,671)(132,672)(133,673)(134,674)(135,675)(136,696)
(137,697)(138,698)(139,699)(140,700)(141,691)(142,692)(143,693)(144,694)
(145,695)(146,701)(147,702)(148,703)(149,704)(150,705)(151,681)(152,682)
(153,683)(154,684)(155,685)(156,676)(157,677)(158,678)(159,679)(160,680)
(161,686)(162,687)(163,688)(164,689)(165,690)(166,711)(167,712)(168,713)
(169,714)(170,715)(171,706)(172,707)(173,708)(174,709)(175,710)(176,716)
(177,717)(178,718)(179,719)(180,720)(181,471)(182,472)(183,473)(184,474)
(185,475)(186,466)(187,467)(188,468)(189,469)(190,470)(191,476)(192,477)
(193,478)(194,479)(195,480)(196,456)(197,457)(198,458)(199,459)(200,460)
(201,451)(202,452)(203,453)(204,454)(205,455)(206,461)(207,462)(208,463)
(209,464)(210,465)(211,486)(212,487)(213,488)(214,489)(215,490)(216,481)
(217,482)(218,483)(219,484)(220,485)(221,491)(222,492)(223,493)(224,494)
(225,495)(226,516)(227,517)(228,518)(229,519)(230,520)(231,511)(232,512)
(233,513)(234,514)(235,515)(236,521)(237,522)(238,523)(239,524)(240,525)
(241,501)(242,502)(243,503)(244,504)(245,505)(246,496)(247,497)(248,498)
(249,499)(250,500)(251,506)(252,507)(253,508)(254,509)(255,510)(256,531)
(257,532)(258,533)(259,534)(260,535)(261,526)(262,527)(263,528)(264,529)
(265,530)(266,536)(267,537)(268,538)(269,539)(270,540)(271,426)(272,427)
(273,428)(274,429)(275,430)(276,421)(277,422)(278,423)(279,424)(280,425)
(281,431)(282,432)(283,433)(284,434)(285,435)(286,411)(287,412)(288,413)
(289,414)(290,415)(291,406)(292,407)(293,408)(294,409)(295,410)(296,416)
(297,417)(298,418)(299,419)(300,420)(301,441)(302,442)(303,443)(304,444)
(305,445)(306,436)(307,437)(308,438)(309,439)(310,440)(311,446)(312,447)
(313,448)(314,449)(315,450)(316,381)(317,382)(318,383)(319,384)(320,385)
(321,376)(322,377)(323,378)(324,379)(325,380)(326,386)(327,387)(328,388)
(329,389)(330,390)(331,366)(332,367)(333,368)(334,369)(335,370)(336,361)
(337,362)(338,363)(339,364)(340,365)(341,371)(342,372)(343,373)(344,374)
(345,375)(346,396)(347,397)(348,398)(349,399)(350,400)(351,391)(352,392)
(353,393)(354,394)(355,395)(356,401)(357,402)(358,403)(359,404)(360,405);;
s2 := (  1,361)(  2,365)(  3,364)(  4,363)(  5,362)(  6,366)(  7,370)(  8,369)
(  9,368)( 10,367)( 11,371)( 12,375)( 13,374)( 14,373)( 15,372)( 16,391)
( 17,395)( 18,394)( 19,393)( 20,392)( 21,396)( 22,400)( 23,399)( 24,398)
( 25,397)( 26,401)( 27,405)( 28,404)( 29,403)( 30,402)( 31,376)( 32,380)
( 33,379)( 34,378)( 35,377)( 36,381)( 37,385)( 38,384)( 39,383)( 40,382)
( 41,386)( 42,390)( 43,389)( 44,388)( 45,387)( 46,406)( 47,410)( 48,409)
( 49,408)( 50,407)( 51,411)( 52,415)( 53,414)( 54,413)( 55,412)( 56,416)
( 57,420)( 58,419)( 59,418)( 60,417)( 61,436)( 62,440)( 63,439)( 64,438)
( 65,437)( 66,441)( 67,445)( 68,444)( 69,443)( 70,442)( 71,446)( 72,450)
( 73,449)( 74,448)( 75,447)( 76,421)( 77,425)( 78,424)( 79,423)( 80,422)
( 81,426)( 82,430)( 83,429)( 84,428)( 85,427)( 86,431)( 87,435)( 88,434)
( 89,433)( 90,432)( 91,496)( 92,500)( 93,499)( 94,498)( 95,497)( 96,501)
( 97,505)( 98,504)( 99,503)(100,502)(101,506)(102,510)(103,509)(104,508)
(105,507)(106,526)(107,530)(108,529)(109,528)(110,527)(111,531)(112,535)
(113,534)(114,533)(115,532)(116,536)(117,540)(118,539)(119,538)(120,537)
(121,511)(122,515)(123,514)(124,513)(125,512)(126,516)(127,520)(128,519)
(129,518)(130,517)(131,521)(132,525)(133,524)(134,523)(135,522)(136,451)
(137,455)(138,454)(139,453)(140,452)(141,456)(142,460)(143,459)(144,458)
(145,457)(146,461)(147,465)(148,464)(149,463)(150,462)(151,481)(152,485)
(153,484)(154,483)(155,482)(156,486)(157,490)(158,489)(159,488)(160,487)
(161,491)(162,495)(163,494)(164,493)(165,492)(166,466)(167,470)(168,469)
(169,468)(170,467)(171,471)(172,475)(173,474)(174,473)(175,472)(176,476)
(177,480)(178,479)(179,478)(180,477)(181,676)(182,680)(183,679)(184,678)
(185,677)(186,681)(187,685)(188,684)(189,683)(190,682)(191,686)(192,690)
(193,689)(194,688)(195,687)(196,706)(197,710)(198,709)(199,708)(200,707)
(201,711)(202,715)(203,714)(204,713)(205,712)(206,716)(207,720)(208,719)
(209,718)(210,717)(211,691)(212,695)(213,694)(214,693)(215,692)(216,696)
(217,700)(218,699)(219,698)(220,697)(221,701)(222,705)(223,704)(224,703)
(225,702)(226,631)(227,635)(228,634)(229,633)(230,632)(231,636)(232,640)
(233,639)(234,638)(235,637)(236,641)(237,645)(238,644)(239,643)(240,642)
(241,661)(242,665)(243,664)(244,663)(245,662)(246,666)(247,670)(248,669)
(249,668)(250,667)(251,671)(252,675)(253,674)(254,673)(255,672)(256,646)
(257,650)(258,649)(259,648)(260,647)(261,651)(262,655)(263,654)(264,653)
(265,652)(266,656)(267,660)(268,659)(269,658)(270,657)(271,586)(272,590)
(273,589)(274,588)(275,587)(276,591)(277,595)(278,594)(279,593)(280,592)
(281,596)(282,600)(283,599)(284,598)(285,597)(286,616)(287,620)(288,619)
(289,618)(290,617)(291,621)(292,625)(293,624)(294,623)(295,622)(296,626)
(297,630)(298,629)(299,628)(300,627)(301,601)(302,605)(303,604)(304,603)
(305,602)(306,606)(307,610)(308,609)(309,608)(310,607)(311,611)(312,615)
(313,614)(314,613)(315,612)(316,541)(317,545)(318,544)(319,543)(320,542)
(321,546)(322,550)(323,549)(324,548)(325,547)(326,551)(327,555)(328,554)
(329,553)(330,552)(331,571)(332,575)(333,574)(334,573)(335,572)(336,576)
(337,580)(338,579)(339,578)(340,577)(341,581)(342,585)(343,584)(344,583)
(345,582)(346,556)(347,560)(348,559)(349,558)(350,557)(351,561)(352,565)
(353,564)(354,563)(355,562)(356,566)(357,570)(358,569)(359,568)(360,567);;
s3 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91,137)( 92,136)( 93,140)( 94,139)
( 95,138)( 96,142)( 97,141)( 98,145)( 99,144)(100,143)(101,147)(102,146)
(103,150)(104,149)(105,148)(106,152)(107,151)(108,155)(109,154)(110,153)
(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)(118,165)
(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)(126,172)
(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)(134,179)
(135,178)(181,317)(182,316)(183,320)(184,319)(185,318)(186,322)(187,321)
(188,325)(189,324)(190,323)(191,327)(192,326)(193,330)(194,329)(195,328)
(196,332)(197,331)(198,335)(199,334)(200,333)(201,337)(202,336)(203,340)
(204,339)(205,338)(206,342)(207,341)(208,345)(209,344)(210,343)(211,347)
(212,346)(213,350)(214,349)(215,348)(216,352)(217,351)(218,355)(219,354)
(220,353)(221,357)(222,356)(223,360)(224,359)(225,358)(226,272)(227,271)
(228,275)(229,274)(230,273)(231,277)(232,276)(233,280)(234,279)(235,278)
(236,282)(237,281)(238,285)(239,284)(240,283)(241,287)(242,286)(243,290)
(244,289)(245,288)(246,292)(247,291)(248,295)(249,294)(250,293)(251,297)
(252,296)(253,300)(254,299)(255,298)(256,302)(257,301)(258,305)(259,304)
(260,303)(261,307)(262,306)(263,310)(264,309)(265,308)(266,312)(267,311)
(268,315)(269,314)(270,313)(361,452)(362,451)(363,455)(364,454)(365,453)
(366,457)(367,456)(368,460)(369,459)(370,458)(371,462)(372,461)(373,465)
(374,464)(375,463)(376,467)(377,466)(378,470)(379,469)(380,468)(381,472)
(382,471)(383,475)(384,474)(385,473)(386,477)(387,476)(388,480)(389,479)
(390,478)(391,482)(392,481)(393,485)(394,484)(395,483)(396,487)(397,486)
(398,490)(399,489)(400,488)(401,492)(402,491)(403,495)(404,494)(405,493)
(406,497)(407,496)(408,500)(409,499)(410,498)(411,502)(412,501)(413,505)
(414,504)(415,503)(416,507)(417,506)(418,510)(419,509)(420,508)(421,512)
(422,511)(423,515)(424,514)(425,513)(426,517)(427,516)(428,520)(429,519)
(430,518)(431,522)(432,521)(433,525)(434,524)(435,523)(436,527)(437,526)
(438,530)(439,529)(440,528)(441,532)(442,531)(443,535)(444,534)(445,533)
(446,537)(447,536)(448,540)(449,539)(450,538)(541,542)(543,545)(546,547)
(548,550)(551,552)(553,555)(556,557)(558,560)(561,562)(563,565)(566,567)
(568,570)(571,572)(573,575)(576,577)(578,580)(581,582)(583,585)(586,587)
(588,590)(591,592)(593,595)(596,597)(598,600)(601,602)(603,605)(606,607)
(608,610)(611,612)(613,615)(616,617)(618,620)(621,622)(623,625)(626,627)
(628,630)(631,677)(632,676)(633,680)(634,679)(635,678)(636,682)(637,681)
(638,685)(639,684)(640,683)(641,687)(642,686)(643,690)(644,689)(645,688)
(646,692)(647,691)(648,695)(649,694)(650,693)(651,697)(652,696)(653,700)
(654,699)(655,698)(656,702)(657,701)(658,705)(659,704)(660,703)(661,707)
(662,706)(663,710)(664,709)(665,708)(666,712)(667,711)(668,715)(669,714)
(670,713)(671,717)(672,716)(673,720)(674,719)(675,718);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(720)!(  1,541)(  2,542)(  3,543)(  4,544)(  5,545)(  6,551)(  7,552)
(  8,553)(  9,554)( 10,555)( 11,546)( 12,547)( 13,548)( 14,549)( 15,550)
( 16,571)( 17,572)( 18,573)( 19,574)( 20,575)( 21,581)( 22,582)( 23,583)
( 24,584)( 25,585)( 26,576)( 27,577)( 28,578)( 29,579)( 30,580)( 31,556)
( 32,557)( 33,558)( 34,559)( 35,560)( 36,566)( 37,567)( 38,568)( 39,569)
( 40,570)( 41,561)( 42,562)( 43,563)( 44,564)( 45,565)( 46,586)( 47,587)
( 48,588)( 49,589)( 50,590)( 51,596)( 52,597)( 53,598)( 54,599)( 55,600)
( 56,591)( 57,592)( 58,593)( 59,594)( 60,595)( 61,616)( 62,617)( 63,618)
( 64,619)( 65,620)( 66,626)( 67,627)( 68,628)( 69,629)( 70,630)( 71,621)
( 72,622)( 73,623)( 74,624)( 75,625)( 76,601)( 77,602)( 78,603)( 79,604)
( 80,605)( 81,611)( 82,612)( 83,613)( 84,614)( 85,615)( 86,606)( 87,607)
( 88,608)( 89,609)( 90,610)( 91,631)( 92,632)( 93,633)( 94,634)( 95,635)
( 96,641)( 97,642)( 98,643)( 99,644)(100,645)(101,636)(102,637)(103,638)
(104,639)(105,640)(106,661)(107,662)(108,663)(109,664)(110,665)(111,671)
(112,672)(113,673)(114,674)(115,675)(116,666)(117,667)(118,668)(119,669)
(120,670)(121,646)(122,647)(123,648)(124,649)(125,650)(126,656)(127,657)
(128,658)(129,659)(130,660)(131,651)(132,652)(133,653)(134,654)(135,655)
(136,676)(137,677)(138,678)(139,679)(140,680)(141,686)(142,687)(143,688)
(144,689)(145,690)(146,681)(147,682)(148,683)(149,684)(150,685)(151,706)
(152,707)(153,708)(154,709)(155,710)(156,716)(157,717)(158,718)(159,719)
(160,720)(161,711)(162,712)(163,713)(164,714)(165,715)(166,691)(167,692)
(168,693)(169,694)(170,695)(171,701)(172,702)(173,703)(174,704)(175,705)
(176,696)(177,697)(178,698)(179,699)(180,700)(181,451)(182,452)(183,453)
(184,454)(185,455)(186,461)(187,462)(188,463)(189,464)(190,465)(191,456)
(192,457)(193,458)(194,459)(195,460)(196,481)(197,482)(198,483)(199,484)
(200,485)(201,491)(202,492)(203,493)(204,494)(205,495)(206,486)(207,487)
(208,488)(209,489)(210,490)(211,466)(212,467)(213,468)(214,469)(215,470)
(216,476)(217,477)(218,478)(219,479)(220,480)(221,471)(222,472)(223,473)
(224,474)(225,475)(226,496)(227,497)(228,498)(229,499)(230,500)(231,506)
(232,507)(233,508)(234,509)(235,510)(236,501)(237,502)(238,503)(239,504)
(240,505)(241,526)(242,527)(243,528)(244,529)(245,530)(246,536)(247,537)
(248,538)(249,539)(250,540)(251,531)(252,532)(253,533)(254,534)(255,535)
(256,511)(257,512)(258,513)(259,514)(260,515)(261,521)(262,522)(263,523)
(264,524)(265,525)(266,516)(267,517)(268,518)(269,519)(270,520)(271,406)
(272,407)(273,408)(274,409)(275,410)(276,416)(277,417)(278,418)(279,419)
(280,420)(281,411)(282,412)(283,413)(284,414)(285,415)(286,436)(287,437)
(288,438)(289,439)(290,440)(291,446)(292,447)(293,448)(294,449)(295,450)
(296,441)(297,442)(298,443)(299,444)(300,445)(301,421)(302,422)(303,423)
(304,424)(305,425)(306,431)(307,432)(308,433)(309,434)(310,435)(311,426)
(312,427)(313,428)(314,429)(315,430)(316,361)(317,362)(318,363)(319,364)
(320,365)(321,371)(322,372)(323,373)(324,374)(325,375)(326,366)(327,367)
(328,368)(329,369)(330,370)(331,391)(332,392)(333,393)(334,394)(335,395)
(336,401)(337,402)(338,403)(339,404)(340,405)(341,396)(342,397)(343,398)
(344,399)(345,400)(346,376)(347,377)(348,378)(349,379)(350,380)(351,386)
(352,387)(353,388)(354,389)(355,390)(356,381)(357,382)(358,383)(359,384)
(360,385);
s1 := Sym(720)!(  1,561)(  2,562)(  3,563)(  4,564)(  5,565)(  6,556)(  7,557)
(  8,558)(  9,559)( 10,560)( 11,566)( 12,567)( 13,568)( 14,569)( 15,570)
( 16,546)( 17,547)( 18,548)( 19,549)( 20,550)( 21,541)( 22,542)( 23,543)
( 24,544)( 25,545)( 26,551)( 27,552)( 28,553)( 29,554)( 30,555)( 31,576)
( 32,577)( 33,578)( 34,579)( 35,580)( 36,571)( 37,572)( 38,573)( 39,574)
( 40,575)( 41,581)( 42,582)( 43,583)( 44,584)( 45,585)( 46,606)( 47,607)
( 48,608)( 49,609)( 50,610)( 51,601)( 52,602)( 53,603)( 54,604)( 55,605)
( 56,611)( 57,612)( 58,613)( 59,614)( 60,615)( 61,591)( 62,592)( 63,593)
( 64,594)( 65,595)( 66,586)( 67,587)( 68,588)( 69,589)( 70,590)( 71,596)
( 72,597)( 73,598)( 74,599)( 75,600)( 76,621)( 77,622)( 78,623)( 79,624)
( 80,625)( 81,616)( 82,617)( 83,618)( 84,619)( 85,620)( 86,626)( 87,627)
( 88,628)( 89,629)( 90,630)( 91,651)( 92,652)( 93,653)( 94,654)( 95,655)
( 96,646)( 97,647)( 98,648)( 99,649)(100,650)(101,656)(102,657)(103,658)
(104,659)(105,660)(106,636)(107,637)(108,638)(109,639)(110,640)(111,631)
(112,632)(113,633)(114,634)(115,635)(116,641)(117,642)(118,643)(119,644)
(120,645)(121,666)(122,667)(123,668)(124,669)(125,670)(126,661)(127,662)
(128,663)(129,664)(130,665)(131,671)(132,672)(133,673)(134,674)(135,675)
(136,696)(137,697)(138,698)(139,699)(140,700)(141,691)(142,692)(143,693)
(144,694)(145,695)(146,701)(147,702)(148,703)(149,704)(150,705)(151,681)
(152,682)(153,683)(154,684)(155,685)(156,676)(157,677)(158,678)(159,679)
(160,680)(161,686)(162,687)(163,688)(164,689)(165,690)(166,711)(167,712)
(168,713)(169,714)(170,715)(171,706)(172,707)(173,708)(174,709)(175,710)
(176,716)(177,717)(178,718)(179,719)(180,720)(181,471)(182,472)(183,473)
(184,474)(185,475)(186,466)(187,467)(188,468)(189,469)(190,470)(191,476)
(192,477)(193,478)(194,479)(195,480)(196,456)(197,457)(198,458)(199,459)
(200,460)(201,451)(202,452)(203,453)(204,454)(205,455)(206,461)(207,462)
(208,463)(209,464)(210,465)(211,486)(212,487)(213,488)(214,489)(215,490)
(216,481)(217,482)(218,483)(219,484)(220,485)(221,491)(222,492)(223,493)
(224,494)(225,495)(226,516)(227,517)(228,518)(229,519)(230,520)(231,511)
(232,512)(233,513)(234,514)(235,515)(236,521)(237,522)(238,523)(239,524)
(240,525)(241,501)(242,502)(243,503)(244,504)(245,505)(246,496)(247,497)
(248,498)(249,499)(250,500)(251,506)(252,507)(253,508)(254,509)(255,510)
(256,531)(257,532)(258,533)(259,534)(260,535)(261,526)(262,527)(263,528)
(264,529)(265,530)(266,536)(267,537)(268,538)(269,539)(270,540)(271,426)
(272,427)(273,428)(274,429)(275,430)(276,421)(277,422)(278,423)(279,424)
(280,425)(281,431)(282,432)(283,433)(284,434)(285,435)(286,411)(287,412)
(288,413)(289,414)(290,415)(291,406)(292,407)(293,408)(294,409)(295,410)
(296,416)(297,417)(298,418)(299,419)(300,420)(301,441)(302,442)(303,443)
(304,444)(305,445)(306,436)(307,437)(308,438)(309,439)(310,440)(311,446)
(312,447)(313,448)(314,449)(315,450)(316,381)(317,382)(318,383)(319,384)
(320,385)(321,376)(322,377)(323,378)(324,379)(325,380)(326,386)(327,387)
(328,388)(329,389)(330,390)(331,366)(332,367)(333,368)(334,369)(335,370)
(336,361)(337,362)(338,363)(339,364)(340,365)(341,371)(342,372)(343,373)
(344,374)(345,375)(346,396)(347,397)(348,398)(349,399)(350,400)(351,391)
(352,392)(353,393)(354,394)(355,395)(356,401)(357,402)(358,403)(359,404)
(360,405);
s2 := Sym(720)!(  1,361)(  2,365)(  3,364)(  4,363)(  5,362)(  6,366)(  7,370)
(  8,369)(  9,368)( 10,367)( 11,371)( 12,375)( 13,374)( 14,373)( 15,372)
( 16,391)( 17,395)( 18,394)( 19,393)( 20,392)( 21,396)( 22,400)( 23,399)
( 24,398)( 25,397)( 26,401)( 27,405)( 28,404)( 29,403)( 30,402)( 31,376)
( 32,380)( 33,379)( 34,378)( 35,377)( 36,381)( 37,385)( 38,384)( 39,383)
( 40,382)( 41,386)( 42,390)( 43,389)( 44,388)( 45,387)( 46,406)( 47,410)
( 48,409)( 49,408)( 50,407)( 51,411)( 52,415)( 53,414)( 54,413)( 55,412)
( 56,416)( 57,420)( 58,419)( 59,418)( 60,417)( 61,436)( 62,440)( 63,439)
( 64,438)( 65,437)( 66,441)( 67,445)( 68,444)( 69,443)( 70,442)( 71,446)
( 72,450)( 73,449)( 74,448)( 75,447)( 76,421)( 77,425)( 78,424)( 79,423)
( 80,422)( 81,426)( 82,430)( 83,429)( 84,428)( 85,427)( 86,431)( 87,435)
( 88,434)( 89,433)( 90,432)( 91,496)( 92,500)( 93,499)( 94,498)( 95,497)
( 96,501)( 97,505)( 98,504)( 99,503)(100,502)(101,506)(102,510)(103,509)
(104,508)(105,507)(106,526)(107,530)(108,529)(109,528)(110,527)(111,531)
(112,535)(113,534)(114,533)(115,532)(116,536)(117,540)(118,539)(119,538)
(120,537)(121,511)(122,515)(123,514)(124,513)(125,512)(126,516)(127,520)
(128,519)(129,518)(130,517)(131,521)(132,525)(133,524)(134,523)(135,522)
(136,451)(137,455)(138,454)(139,453)(140,452)(141,456)(142,460)(143,459)
(144,458)(145,457)(146,461)(147,465)(148,464)(149,463)(150,462)(151,481)
(152,485)(153,484)(154,483)(155,482)(156,486)(157,490)(158,489)(159,488)
(160,487)(161,491)(162,495)(163,494)(164,493)(165,492)(166,466)(167,470)
(168,469)(169,468)(170,467)(171,471)(172,475)(173,474)(174,473)(175,472)
(176,476)(177,480)(178,479)(179,478)(180,477)(181,676)(182,680)(183,679)
(184,678)(185,677)(186,681)(187,685)(188,684)(189,683)(190,682)(191,686)
(192,690)(193,689)(194,688)(195,687)(196,706)(197,710)(198,709)(199,708)
(200,707)(201,711)(202,715)(203,714)(204,713)(205,712)(206,716)(207,720)
(208,719)(209,718)(210,717)(211,691)(212,695)(213,694)(214,693)(215,692)
(216,696)(217,700)(218,699)(219,698)(220,697)(221,701)(222,705)(223,704)
(224,703)(225,702)(226,631)(227,635)(228,634)(229,633)(230,632)(231,636)
(232,640)(233,639)(234,638)(235,637)(236,641)(237,645)(238,644)(239,643)
(240,642)(241,661)(242,665)(243,664)(244,663)(245,662)(246,666)(247,670)
(248,669)(249,668)(250,667)(251,671)(252,675)(253,674)(254,673)(255,672)
(256,646)(257,650)(258,649)(259,648)(260,647)(261,651)(262,655)(263,654)
(264,653)(265,652)(266,656)(267,660)(268,659)(269,658)(270,657)(271,586)
(272,590)(273,589)(274,588)(275,587)(276,591)(277,595)(278,594)(279,593)
(280,592)(281,596)(282,600)(283,599)(284,598)(285,597)(286,616)(287,620)
(288,619)(289,618)(290,617)(291,621)(292,625)(293,624)(294,623)(295,622)
(296,626)(297,630)(298,629)(299,628)(300,627)(301,601)(302,605)(303,604)
(304,603)(305,602)(306,606)(307,610)(308,609)(309,608)(310,607)(311,611)
(312,615)(313,614)(314,613)(315,612)(316,541)(317,545)(318,544)(319,543)
(320,542)(321,546)(322,550)(323,549)(324,548)(325,547)(326,551)(327,555)
(328,554)(329,553)(330,552)(331,571)(332,575)(333,574)(334,573)(335,572)
(336,576)(337,580)(338,579)(339,578)(340,577)(341,581)(342,585)(343,584)
(344,583)(345,582)(346,556)(347,560)(348,559)(349,558)(350,557)(351,561)
(352,565)(353,564)(354,563)(355,562)(356,566)(357,570)(358,569)(359,568)
(360,567);
s3 := Sym(720)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91,137)( 92,136)( 93,140)
( 94,139)( 95,138)( 96,142)( 97,141)( 98,145)( 99,144)(100,143)(101,147)
(102,146)(103,150)(104,149)(105,148)(106,152)(107,151)(108,155)(109,154)
(110,153)(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)
(118,165)(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)
(126,172)(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)
(134,179)(135,178)(181,317)(182,316)(183,320)(184,319)(185,318)(186,322)
(187,321)(188,325)(189,324)(190,323)(191,327)(192,326)(193,330)(194,329)
(195,328)(196,332)(197,331)(198,335)(199,334)(200,333)(201,337)(202,336)
(203,340)(204,339)(205,338)(206,342)(207,341)(208,345)(209,344)(210,343)
(211,347)(212,346)(213,350)(214,349)(215,348)(216,352)(217,351)(218,355)
(219,354)(220,353)(221,357)(222,356)(223,360)(224,359)(225,358)(226,272)
(227,271)(228,275)(229,274)(230,273)(231,277)(232,276)(233,280)(234,279)
(235,278)(236,282)(237,281)(238,285)(239,284)(240,283)(241,287)(242,286)
(243,290)(244,289)(245,288)(246,292)(247,291)(248,295)(249,294)(250,293)
(251,297)(252,296)(253,300)(254,299)(255,298)(256,302)(257,301)(258,305)
(259,304)(260,303)(261,307)(262,306)(263,310)(264,309)(265,308)(266,312)
(267,311)(268,315)(269,314)(270,313)(361,452)(362,451)(363,455)(364,454)
(365,453)(366,457)(367,456)(368,460)(369,459)(370,458)(371,462)(372,461)
(373,465)(374,464)(375,463)(376,467)(377,466)(378,470)(379,469)(380,468)
(381,472)(382,471)(383,475)(384,474)(385,473)(386,477)(387,476)(388,480)
(389,479)(390,478)(391,482)(392,481)(393,485)(394,484)(395,483)(396,487)
(397,486)(398,490)(399,489)(400,488)(401,492)(402,491)(403,495)(404,494)
(405,493)(406,497)(407,496)(408,500)(409,499)(410,498)(411,502)(412,501)
(413,505)(414,504)(415,503)(416,507)(417,506)(418,510)(419,509)(420,508)
(421,512)(422,511)(423,515)(424,514)(425,513)(426,517)(427,516)(428,520)
(429,519)(430,518)(431,522)(432,521)(433,525)(434,524)(435,523)(436,527)
(437,526)(438,530)(439,529)(440,528)(441,532)(442,531)(443,535)(444,534)
(445,533)(446,537)(447,536)(448,540)(449,539)(450,538)(541,542)(543,545)
(546,547)(548,550)(551,552)(553,555)(556,557)(558,560)(561,562)(563,565)
(566,567)(568,570)(571,572)(573,575)(576,577)(578,580)(581,582)(583,585)
(586,587)(588,590)(591,592)(593,595)(596,597)(598,600)(601,602)(603,605)
(606,607)(608,610)(611,612)(613,615)(616,617)(618,620)(621,622)(623,625)
(626,627)(628,630)(631,677)(632,676)(633,680)(634,679)(635,678)(636,682)
(637,681)(638,685)(639,684)(640,683)(641,687)(642,686)(643,690)(644,689)
(645,688)(646,692)(647,691)(648,695)(649,694)(650,693)(651,697)(652,696)
(653,700)(654,699)(655,698)(656,702)(657,701)(658,705)(659,704)(660,703)
(661,707)(662,706)(663,710)(664,709)(665,708)(666,712)(667,711)(668,715)
(669,714)(670,713)(671,717)(672,716)(673,720)(674,719)(675,718);
poly := sub<Sym(720)|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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope