Polytope of Type {4,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14710)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 36, 108, 36
Order of s0s1s2s3 : 12
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 : {2,6,12}*864b, {4,6,6}*864b
3-fold quotients : {4,6,12}*576a
4-fold quotients : {2,6,6}*432b
6-fold quotients : {2,6,12}*288a, {4,6,6}*288a
8-fold quotients : {2,6,6}*216
9-fold quotients : {4,2,12}*192, {4,6,4}*192a
12-fold quotients : {2,6,6}*144a
18-fold quotients : {2,2,12}*96, {2,6,4}*96a, {4,2,6}*96, {4,6,2}*96a
27-fold quotients : {4,2,4}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,2,4}*32, {4,2,2}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)( 8,224)
( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)( 96,312)
( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)(448,664)
(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)(472,688)
(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,712)
(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)(504,720)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)(528,744)
(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)(544,814)
(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)(552,822)
(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)(560,830)
(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)(576,846)
(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)(584,854)
(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)(600,762)
(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)(608,770)
(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)(616,778)
(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)(632,794)
(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)(640,802)
(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)(648,810);;
s1 := ( 1,541)( 2,543)( 3,542)( 4,547)( 5,549)( 6,548)( 7,544)( 8,546)
( 9,545)( 10,550)( 11,552)( 12,551)( 13,556)( 14,558)( 15,557)( 16,553)
( 17,555)( 18,554)( 19,559)( 20,561)( 21,560)( 22,565)( 23,567)( 24,566)
( 25,562)( 26,564)( 27,563)( 28,568)( 29,570)( 30,569)( 31,574)( 32,576)
( 33,575)( 34,571)( 35,573)( 36,572)( 37,577)( 38,579)( 39,578)( 40,583)
( 41,585)( 42,584)( 43,580)( 44,582)( 45,581)( 46,586)( 47,588)( 48,587)
( 49,592)( 50,594)( 51,593)( 52,589)( 53,591)( 54,590)( 55,595)( 56,597)
( 57,596)( 58,601)( 59,603)( 60,602)( 61,598)( 62,600)( 63,599)( 64,604)
( 65,606)( 66,605)( 67,610)( 68,612)( 69,611)( 70,607)( 71,609)( 72,608)
( 73,613)( 74,615)( 75,614)( 76,619)( 77,621)( 78,620)( 79,616)( 80,618)
( 81,617)( 82,622)( 83,624)( 84,623)( 85,628)( 86,630)( 87,629)( 88,625)
( 89,627)( 90,626)( 91,631)( 92,633)( 93,632)( 94,637)( 95,639)( 96,638)
( 97,634)( 98,636)( 99,635)(100,640)(101,642)(102,641)(103,646)(104,648)
(105,647)(106,643)(107,645)(108,644)(109,460)(110,462)(111,461)(112,466)
(113,468)(114,467)(115,463)(116,465)(117,464)(118,469)(119,471)(120,470)
(121,475)(122,477)(123,476)(124,472)(125,474)(126,473)(127,478)(128,480)
(129,479)(130,484)(131,486)(132,485)(133,481)(134,483)(135,482)(136,433)
(137,435)(138,434)(139,439)(140,441)(141,440)(142,436)(143,438)(144,437)
(145,442)(146,444)(147,443)(148,448)(149,450)(150,449)(151,445)(152,447)
(153,446)(154,451)(155,453)(156,452)(157,457)(158,459)(159,458)(160,454)
(161,456)(162,455)(163,514)(164,516)(165,515)(166,520)(167,522)(168,521)
(169,517)(170,519)(171,518)(172,523)(173,525)(174,524)(175,529)(176,531)
(177,530)(178,526)(179,528)(180,527)(181,532)(182,534)(183,533)(184,538)
(185,540)(186,539)(187,535)(188,537)(189,536)(190,487)(191,489)(192,488)
(193,493)(194,495)(195,494)(196,490)(197,492)(198,491)(199,496)(200,498)
(201,497)(202,502)(203,504)(204,503)(205,499)(206,501)(207,500)(208,505)
(209,507)(210,506)(211,511)(212,513)(213,512)(214,508)(215,510)(216,509)
(217,757)(218,759)(219,758)(220,763)(221,765)(222,764)(223,760)(224,762)
(225,761)(226,766)(227,768)(228,767)(229,772)(230,774)(231,773)(232,769)
(233,771)(234,770)(235,775)(236,777)(237,776)(238,781)(239,783)(240,782)
(241,778)(242,780)(243,779)(244,784)(245,786)(246,785)(247,790)(248,792)
(249,791)(250,787)(251,789)(252,788)(253,793)(254,795)(255,794)(256,799)
(257,801)(258,800)(259,796)(260,798)(261,797)(262,802)(263,804)(264,803)
(265,808)(266,810)(267,809)(268,805)(269,807)(270,806)(271,811)(272,813)
(273,812)(274,817)(275,819)(276,818)(277,814)(278,816)(279,815)(280,820)
(281,822)(282,821)(283,826)(284,828)(285,827)(286,823)(287,825)(288,824)
(289,829)(290,831)(291,830)(292,835)(293,837)(294,836)(295,832)(296,834)
(297,833)(298,838)(299,840)(300,839)(301,844)(302,846)(303,845)(304,841)
(305,843)(306,842)(307,847)(308,849)(309,848)(310,853)(311,855)(312,854)
(313,850)(314,852)(315,851)(316,856)(317,858)(318,857)(319,862)(320,864)
(321,863)(322,859)(323,861)(324,860)(325,676)(326,678)(327,677)(328,682)
(329,684)(330,683)(331,679)(332,681)(333,680)(334,685)(335,687)(336,686)
(337,691)(338,693)(339,692)(340,688)(341,690)(342,689)(343,694)(344,696)
(345,695)(346,700)(347,702)(348,701)(349,697)(350,699)(351,698)(352,649)
(353,651)(354,650)(355,655)(356,657)(357,656)(358,652)(359,654)(360,653)
(361,658)(362,660)(363,659)(364,664)(365,666)(366,665)(367,661)(368,663)
(369,662)(370,667)(371,669)(372,668)(373,673)(374,675)(375,674)(376,670)
(377,672)(378,671)(379,730)(380,732)(381,731)(382,736)(383,738)(384,737)
(385,733)(386,735)(387,734)(388,739)(389,741)(390,740)(391,745)(392,747)
(393,746)(394,742)(395,744)(396,743)(397,748)(398,750)(399,749)(400,754)
(401,756)(402,755)(403,751)(404,753)(405,752)(406,703)(407,705)(408,704)
(409,709)(410,711)(411,710)(412,706)(413,708)(414,707)(415,712)(416,714)
(417,713)(418,718)(419,720)(420,719)(421,715)(422,717)(423,716)(424,721)
(425,723)(426,722)(427,727)(428,729)(429,728)(430,724)(431,726)(432,725);;
s2 := ( 1,436)( 2,437)( 3,438)( 4,433)( 5,434)( 6,435)( 7,439)( 8,440)
( 9,441)( 10,454)( 11,455)( 12,456)( 13,451)( 14,452)( 15,453)( 16,457)
( 17,458)( 18,459)( 19,445)( 20,446)( 21,447)( 22,442)( 23,443)( 24,444)
( 25,448)( 26,449)( 27,450)( 28,463)( 29,464)( 30,465)( 31,460)( 32,461)
( 33,462)( 34,466)( 35,467)( 36,468)( 37,481)( 38,482)( 39,483)( 40,478)
( 41,479)( 42,480)( 43,484)( 44,485)( 45,486)( 46,472)( 47,473)( 48,474)
( 49,469)( 50,470)( 51,471)( 52,475)( 53,476)( 54,477)( 55,490)( 56,491)
( 57,492)( 58,487)( 59,488)( 60,489)( 61,493)( 62,494)( 63,495)( 64,508)
( 65,509)( 66,510)( 67,505)( 68,506)( 69,507)( 70,511)( 71,512)( 72,513)
( 73,499)( 74,500)( 75,501)( 76,496)( 77,497)( 78,498)( 79,502)( 80,503)
( 81,504)( 82,517)( 83,518)( 84,519)( 85,514)( 86,515)( 87,516)( 88,520)
( 89,521)( 90,522)( 91,535)( 92,536)( 93,537)( 94,532)( 95,533)( 96,534)
( 97,538)( 98,539)( 99,540)(100,526)(101,527)(102,528)(103,523)(104,524)
(105,525)(106,529)(107,530)(108,531)(109,571)(110,572)(111,573)(112,568)
(113,569)(114,570)(115,574)(116,575)(117,576)(118,589)(119,590)(120,591)
(121,586)(122,587)(123,588)(124,592)(125,593)(126,594)(127,580)(128,581)
(129,582)(130,577)(131,578)(132,579)(133,583)(134,584)(135,585)(136,544)
(137,545)(138,546)(139,541)(140,542)(141,543)(142,547)(143,548)(144,549)
(145,562)(146,563)(147,564)(148,559)(149,560)(150,561)(151,565)(152,566)
(153,567)(154,553)(155,554)(156,555)(157,550)(158,551)(159,552)(160,556)
(161,557)(162,558)(163,625)(164,626)(165,627)(166,622)(167,623)(168,624)
(169,628)(170,629)(171,630)(172,643)(173,644)(174,645)(175,640)(176,641)
(177,642)(178,646)(179,647)(180,648)(181,634)(182,635)(183,636)(184,631)
(185,632)(186,633)(187,637)(188,638)(189,639)(190,598)(191,599)(192,600)
(193,595)(194,596)(195,597)(196,601)(197,602)(198,603)(199,616)(200,617)
(201,618)(202,613)(203,614)(204,615)(205,619)(206,620)(207,621)(208,607)
(209,608)(210,609)(211,604)(212,605)(213,606)(214,610)(215,611)(216,612)
(217,652)(218,653)(219,654)(220,649)(221,650)(222,651)(223,655)(224,656)
(225,657)(226,670)(227,671)(228,672)(229,667)(230,668)(231,669)(232,673)
(233,674)(234,675)(235,661)(236,662)(237,663)(238,658)(239,659)(240,660)
(241,664)(242,665)(243,666)(244,679)(245,680)(246,681)(247,676)(248,677)
(249,678)(250,682)(251,683)(252,684)(253,697)(254,698)(255,699)(256,694)
(257,695)(258,696)(259,700)(260,701)(261,702)(262,688)(263,689)(264,690)
(265,685)(266,686)(267,687)(268,691)(269,692)(270,693)(271,706)(272,707)
(273,708)(274,703)(275,704)(276,705)(277,709)(278,710)(279,711)(280,724)
(281,725)(282,726)(283,721)(284,722)(285,723)(286,727)(287,728)(288,729)
(289,715)(290,716)(291,717)(292,712)(293,713)(294,714)(295,718)(296,719)
(297,720)(298,733)(299,734)(300,735)(301,730)(302,731)(303,732)(304,736)
(305,737)(306,738)(307,751)(308,752)(309,753)(310,748)(311,749)(312,750)
(313,754)(314,755)(315,756)(316,742)(317,743)(318,744)(319,739)(320,740)
(321,741)(322,745)(323,746)(324,747)(325,787)(326,788)(327,789)(328,784)
(329,785)(330,786)(331,790)(332,791)(333,792)(334,805)(335,806)(336,807)
(337,802)(338,803)(339,804)(340,808)(341,809)(342,810)(343,796)(344,797)
(345,798)(346,793)(347,794)(348,795)(349,799)(350,800)(351,801)(352,760)
(353,761)(354,762)(355,757)(356,758)(357,759)(358,763)(359,764)(360,765)
(361,778)(362,779)(363,780)(364,775)(365,776)(366,777)(367,781)(368,782)
(369,783)(370,769)(371,770)(372,771)(373,766)(374,767)(375,768)(376,772)
(377,773)(378,774)(379,841)(380,842)(381,843)(382,838)(383,839)(384,840)
(385,844)(386,845)(387,846)(388,859)(389,860)(390,861)(391,856)(392,857)
(393,858)(394,862)(395,863)(396,864)(397,850)(398,851)(399,852)(400,847)
(401,848)(402,849)(403,853)(404,854)(405,855)(406,814)(407,815)(408,816)
(409,811)(410,812)(411,813)(412,817)(413,818)(414,819)(415,832)(416,833)
(417,834)(418,829)(419,830)(420,831)(421,835)(422,836)(423,837)(424,823)
(425,824)(426,825)(427,820)(428,821)(429,822)(430,826)(431,827)(432,828);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)( 8, 17)
( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)( 62, 71)
( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 95)
( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)(106,108)
(109,145)(110,147)(111,146)(112,149)(113,148)(114,150)(115,153)(116,152)
(117,151)(118,136)(119,138)(120,137)(121,140)(122,139)(123,141)(124,144)
(125,143)(126,142)(127,154)(128,156)(129,155)(130,158)(131,157)(132,159)
(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,203)(167,202)
(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)(175,194)
(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)(183,209)
(184,212)(185,211)(186,213)(187,216)(188,215)(189,214)(217,226)(218,228)
(219,227)(220,230)(221,229)(222,231)(223,234)(224,233)(225,232)(236,237)
(238,239)(241,243)(244,253)(245,255)(246,254)(247,257)(248,256)(249,258)
(250,261)(251,260)(252,259)(263,264)(265,266)(268,270)(271,280)(272,282)
(273,281)(274,284)(275,283)(276,285)(277,288)(278,287)(279,286)(290,291)
(292,293)(295,297)(298,307)(299,309)(300,308)(301,311)(302,310)(303,312)
(304,315)(305,314)(306,313)(317,318)(319,320)(322,324)(325,361)(326,363)
(327,362)(328,365)(329,364)(330,366)(331,369)(332,368)(333,367)(334,352)
(335,354)(336,353)(337,356)(338,355)(339,357)(340,360)(341,359)(342,358)
(343,370)(344,372)(345,371)(346,374)(347,373)(348,375)(349,378)(350,377)
(351,376)(379,415)(380,417)(381,416)(382,419)(383,418)(384,420)(385,423)
(386,422)(387,421)(388,406)(389,408)(390,407)(391,410)(392,409)(393,411)
(394,414)(395,413)(396,412)(397,424)(398,426)(399,425)(400,428)(401,427)
(402,429)(403,432)(404,431)(405,430)(433,469)(434,471)(435,470)(436,473)
(437,472)(438,474)(439,477)(440,476)(441,475)(442,460)(443,462)(444,461)
(445,464)(446,463)(447,465)(448,468)(449,467)(450,466)(451,478)(452,480)
(453,479)(454,482)(455,481)(456,483)(457,486)(458,485)(459,484)(487,523)
(488,525)(489,524)(490,527)(491,526)(492,528)(493,531)(494,530)(495,529)
(496,514)(497,516)(498,515)(499,518)(500,517)(501,519)(502,522)(503,521)
(504,520)(505,532)(506,534)(507,533)(508,536)(509,535)(510,537)(511,540)
(512,539)(513,538)(541,550)(542,552)(543,551)(544,554)(545,553)(546,555)
(547,558)(548,557)(549,556)(560,561)(562,563)(565,567)(568,577)(569,579)
(570,578)(571,581)(572,580)(573,582)(574,585)(575,584)(576,583)(587,588)
(589,590)(592,594)(595,604)(596,606)(597,605)(598,608)(599,607)(600,609)
(601,612)(602,611)(603,610)(614,615)(616,617)(619,621)(622,631)(623,633)
(624,632)(625,635)(626,634)(627,636)(628,639)(629,638)(630,637)(641,642)
(643,644)(646,648)(649,685)(650,687)(651,686)(652,689)(653,688)(654,690)
(655,693)(656,692)(657,691)(658,676)(659,678)(660,677)(661,680)(662,679)
(663,681)(664,684)(665,683)(666,682)(667,694)(668,696)(669,695)(670,698)
(671,697)(672,699)(673,702)(674,701)(675,700)(703,739)(704,741)(705,740)
(706,743)(707,742)(708,744)(709,747)(710,746)(711,745)(712,730)(713,732)
(714,731)(715,734)(716,733)(717,735)(718,738)(719,737)(720,736)(721,748)
(722,750)(723,749)(724,752)(725,751)(726,753)(727,756)(728,755)(729,754)
(757,766)(758,768)(759,767)(760,770)(761,769)(762,771)(763,774)(764,773)
(765,772)(776,777)(778,779)(781,783)(784,793)(785,795)(786,794)(787,797)
(788,796)(789,798)(790,801)(791,800)(792,799)(803,804)(805,806)(808,810)
(811,820)(812,822)(813,821)(814,824)(815,823)(816,825)(817,828)(818,827)
(819,826)(830,831)(832,833)(835,837)(838,847)(839,849)(840,848)(841,851)
(842,850)(843,852)(844,855)(845,854)(846,853)(857,858)(859,860)(862,864);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(864)!( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)
( 8,224)( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)
( 96,312)( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)
(448,664)(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)
(472,688)(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,712)(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)
(504,720)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)
(528,744)(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)
(544,814)(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)
(552,822)(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)
(560,830)(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)
(576,846)(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)
(584,854)(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)
(600,762)(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)
(608,770)(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)
(616,778)(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)
(632,794)(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)
(640,802)(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)
(648,810);
s1 := Sym(864)!( 1,541)( 2,543)( 3,542)( 4,547)( 5,549)( 6,548)( 7,544)
( 8,546)( 9,545)( 10,550)( 11,552)( 12,551)( 13,556)( 14,558)( 15,557)
( 16,553)( 17,555)( 18,554)( 19,559)( 20,561)( 21,560)( 22,565)( 23,567)
( 24,566)( 25,562)( 26,564)( 27,563)( 28,568)( 29,570)( 30,569)( 31,574)
( 32,576)( 33,575)( 34,571)( 35,573)( 36,572)( 37,577)( 38,579)( 39,578)
( 40,583)( 41,585)( 42,584)( 43,580)( 44,582)( 45,581)( 46,586)( 47,588)
( 48,587)( 49,592)( 50,594)( 51,593)( 52,589)( 53,591)( 54,590)( 55,595)
( 56,597)( 57,596)( 58,601)( 59,603)( 60,602)( 61,598)( 62,600)( 63,599)
( 64,604)( 65,606)( 66,605)( 67,610)( 68,612)( 69,611)( 70,607)( 71,609)
( 72,608)( 73,613)( 74,615)( 75,614)( 76,619)( 77,621)( 78,620)( 79,616)
( 80,618)( 81,617)( 82,622)( 83,624)( 84,623)( 85,628)( 86,630)( 87,629)
( 88,625)( 89,627)( 90,626)( 91,631)( 92,633)( 93,632)( 94,637)( 95,639)
( 96,638)( 97,634)( 98,636)( 99,635)(100,640)(101,642)(102,641)(103,646)
(104,648)(105,647)(106,643)(107,645)(108,644)(109,460)(110,462)(111,461)
(112,466)(113,468)(114,467)(115,463)(116,465)(117,464)(118,469)(119,471)
(120,470)(121,475)(122,477)(123,476)(124,472)(125,474)(126,473)(127,478)
(128,480)(129,479)(130,484)(131,486)(132,485)(133,481)(134,483)(135,482)
(136,433)(137,435)(138,434)(139,439)(140,441)(141,440)(142,436)(143,438)
(144,437)(145,442)(146,444)(147,443)(148,448)(149,450)(150,449)(151,445)
(152,447)(153,446)(154,451)(155,453)(156,452)(157,457)(158,459)(159,458)
(160,454)(161,456)(162,455)(163,514)(164,516)(165,515)(166,520)(167,522)
(168,521)(169,517)(170,519)(171,518)(172,523)(173,525)(174,524)(175,529)
(176,531)(177,530)(178,526)(179,528)(180,527)(181,532)(182,534)(183,533)
(184,538)(185,540)(186,539)(187,535)(188,537)(189,536)(190,487)(191,489)
(192,488)(193,493)(194,495)(195,494)(196,490)(197,492)(198,491)(199,496)
(200,498)(201,497)(202,502)(203,504)(204,503)(205,499)(206,501)(207,500)
(208,505)(209,507)(210,506)(211,511)(212,513)(213,512)(214,508)(215,510)
(216,509)(217,757)(218,759)(219,758)(220,763)(221,765)(222,764)(223,760)
(224,762)(225,761)(226,766)(227,768)(228,767)(229,772)(230,774)(231,773)
(232,769)(233,771)(234,770)(235,775)(236,777)(237,776)(238,781)(239,783)
(240,782)(241,778)(242,780)(243,779)(244,784)(245,786)(246,785)(247,790)
(248,792)(249,791)(250,787)(251,789)(252,788)(253,793)(254,795)(255,794)
(256,799)(257,801)(258,800)(259,796)(260,798)(261,797)(262,802)(263,804)
(264,803)(265,808)(266,810)(267,809)(268,805)(269,807)(270,806)(271,811)
(272,813)(273,812)(274,817)(275,819)(276,818)(277,814)(278,816)(279,815)
(280,820)(281,822)(282,821)(283,826)(284,828)(285,827)(286,823)(287,825)
(288,824)(289,829)(290,831)(291,830)(292,835)(293,837)(294,836)(295,832)
(296,834)(297,833)(298,838)(299,840)(300,839)(301,844)(302,846)(303,845)
(304,841)(305,843)(306,842)(307,847)(308,849)(309,848)(310,853)(311,855)
(312,854)(313,850)(314,852)(315,851)(316,856)(317,858)(318,857)(319,862)
(320,864)(321,863)(322,859)(323,861)(324,860)(325,676)(326,678)(327,677)
(328,682)(329,684)(330,683)(331,679)(332,681)(333,680)(334,685)(335,687)
(336,686)(337,691)(338,693)(339,692)(340,688)(341,690)(342,689)(343,694)
(344,696)(345,695)(346,700)(347,702)(348,701)(349,697)(350,699)(351,698)
(352,649)(353,651)(354,650)(355,655)(356,657)(357,656)(358,652)(359,654)
(360,653)(361,658)(362,660)(363,659)(364,664)(365,666)(366,665)(367,661)
(368,663)(369,662)(370,667)(371,669)(372,668)(373,673)(374,675)(375,674)
(376,670)(377,672)(378,671)(379,730)(380,732)(381,731)(382,736)(383,738)
(384,737)(385,733)(386,735)(387,734)(388,739)(389,741)(390,740)(391,745)
(392,747)(393,746)(394,742)(395,744)(396,743)(397,748)(398,750)(399,749)
(400,754)(401,756)(402,755)(403,751)(404,753)(405,752)(406,703)(407,705)
(408,704)(409,709)(410,711)(411,710)(412,706)(413,708)(414,707)(415,712)
(416,714)(417,713)(418,718)(419,720)(420,719)(421,715)(422,717)(423,716)
(424,721)(425,723)(426,722)(427,727)(428,729)(429,728)(430,724)(431,726)
(432,725);
s2 := Sym(864)!( 1,436)( 2,437)( 3,438)( 4,433)( 5,434)( 6,435)( 7,439)
( 8,440)( 9,441)( 10,454)( 11,455)( 12,456)( 13,451)( 14,452)( 15,453)
( 16,457)( 17,458)( 18,459)( 19,445)( 20,446)( 21,447)( 22,442)( 23,443)
( 24,444)( 25,448)( 26,449)( 27,450)( 28,463)( 29,464)( 30,465)( 31,460)
( 32,461)( 33,462)( 34,466)( 35,467)( 36,468)( 37,481)( 38,482)( 39,483)
( 40,478)( 41,479)( 42,480)( 43,484)( 44,485)( 45,486)( 46,472)( 47,473)
( 48,474)( 49,469)( 50,470)( 51,471)( 52,475)( 53,476)( 54,477)( 55,490)
( 56,491)( 57,492)( 58,487)( 59,488)( 60,489)( 61,493)( 62,494)( 63,495)
( 64,508)( 65,509)( 66,510)( 67,505)( 68,506)( 69,507)( 70,511)( 71,512)
( 72,513)( 73,499)( 74,500)( 75,501)( 76,496)( 77,497)( 78,498)( 79,502)
( 80,503)( 81,504)( 82,517)( 83,518)( 84,519)( 85,514)( 86,515)( 87,516)
( 88,520)( 89,521)( 90,522)( 91,535)( 92,536)( 93,537)( 94,532)( 95,533)
( 96,534)( 97,538)( 98,539)( 99,540)(100,526)(101,527)(102,528)(103,523)
(104,524)(105,525)(106,529)(107,530)(108,531)(109,571)(110,572)(111,573)
(112,568)(113,569)(114,570)(115,574)(116,575)(117,576)(118,589)(119,590)
(120,591)(121,586)(122,587)(123,588)(124,592)(125,593)(126,594)(127,580)
(128,581)(129,582)(130,577)(131,578)(132,579)(133,583)(134,584)(135,585)
(136,544)(137,545)(138,546)(139,541)(140,542)(141,543)(142,547)(143,548)
(144,549)(145,562)(146,563)(147,564)(148,559)(149,560)(150,561)(151,565)
(152,566)(153,567)(154,553)(155,554)(156,555)(157,550)(158,551)(159,552)
(160,556)(161,557)(162,558)(163,625)(164,626)(165,627)(166,622)(167,623)
(168,624)(169,628)(170,629)(171,630)(172,643)(173,644)(174,645)(175,640)
(176,641)(177,642)(178,646)(179,647)(180,648)(181,634)(182,635)(183,636)
(184,631)(185,632)(186,633)(187,637)(188,638)(189,639)(190,598)(191,599)
(192,600)(193,595)(194,596)(195,597)(196,601)(197,602)(198,603)(199,616)
(200,617)(201,618)(202,613)(203,614)(204,615)(205,619)(206,620)(207,621)
(208,607)(209,608)(210,609)(211,604)(212,605)(213,606)(214,610)(215,611)
(216,612)(217,652)(218,653)(219,654)(220,649)(221,650)(222,651)(223,655)
(224,656)(225,657)(226,670)(227,671)(228,672)(229,667)(230,668)(231,669)
(232,673)(233,674)(234,675)(235,661)(236,662)(237,663)(238,658)(239,659)
(240,660)(241,664)(242,665)(243,666)(244,679)(245,680)(246,681)(247,676)
(248,677)(249,678)(250,682)(251,683)(252,684)(253,697)(254,698)(255,699)
(256,694)(257,695)(258,696)(259,700)(260,701)(261,702)(262,688)(263,689)
(264,690)(265,685)(266,686)(267,687)(268,691)(269,692)(270,693)(271,706)
(272,707)(273,708)(274,703)(275,704)(276,705)(277,709)(278,710)(279,711)
(280,724)(281,725)(282,726)(283,721)(284,722)(285,723)(286,727)(287,728)
(288,729)(289,715)(290,716)(291,717)(292,712)(293,713)(294,714)(295,718)
(296,719)(297,720)(298,733)(299,734)(300,735)(301,730)(302,731)(303,732)
(304,736)(305,737)(306,738)(307,751)(308,752)(309,753)(310,748)(311,749)
(312,750)(313,754)(314,755)(315,756)(316,742)(317,743)(318,744)(319,739)
(320,740)(321,741)(322,745)(323,746)(324,747)(325,787)(326,788)(327,789)
(328,784)(329,785)(330,786)(331,790)(332,791)(333,792)(334,805)(335,806)
(336,807)(337,802)(338,803)(339,804)(340,808)(341,809)(342,810)(343,796)
(344,797)(345,798)(346,793)(347,794)(348,795)(349,799)(350,800)(351,801)
(352,760)(353,761)(354,762)(355,757)(356,758)(357,759)(358,763)(359,764)
(360,765)(361,778)(362,779)(363,780)(364,775)(365,776)(366,777)(367,781)
(368,782)(369,783)(370,769)(371,770)(372,771)(373,766)(374,767)(375,768)
(376,772)(377,773)(378,774)(379,841)(380,842)(381,843)(382,838)(383,839)
(384,840)(385,844)(386,845)(387,846)(388,859)(389,860)(390,861)(391,856)
(392,857)(393,858)(394,862)(395,863)(396,864)(397,850)(398,851)(399,852)
(400,847)(401,848)(402,849)(403,853)(404,854)(405,855)(406,814)(407,815)
(408,816)(409,811)(410,812)(411,813)(412,817)(413,818)(414,819)(415,832)
(416,833)(417,834)(418,829)(419,830)(420,831)(421,835)(422,836)(423,837)
(424,823)(425,824)(426,825)(427,820)(428,821)(429,822)(430,826)(431,827)
(432,828);
s3 := Sym(864)!( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)
( 8, 17)( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)
( 62, 71)( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 95)( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)
(106,108)(109,145)(110,147)(111,146)(112,149)(113,148)(114,150)(115,153)
(116,152)(117,151)(118,136)(119,138)(120,137)(121,140)(122,139)(123,141)
(124,144)(125,143)(126,142)(127,154)(128,156)(129,155)(130,158)(131,157)
(132,159)(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,203)
(167,202)(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)
(175,194)(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)
(183,209)(184,212)(185,211)(186,213)(187,216)(188,215)(189,214)(217,226)
(218,228)(219,227)(220,230)(221,229)(222,231)(223,234)(224,233)(225,232)
(236,237)(238,239)(241,243)(244,253)(245,255)(246,254)(247,257)(248,256)
(249,258)(250,261)(251,260)(252,259)(263,264)(265,266)(268,270)(271,280)
(272,282)(273,281)(274,284)(275,283)(276,285)(277,288)(278,287)(279,286)
(290,291)(292,293)(295,297)(298,307)(299,309)(300,308)(301,311)(302,310)
(303,312)(304,315)(305,314)(306,313)(317,318)(319,320)(322,324)(325,361)
(326,363)(327,362)(328,365)(329,364)(330,366)(331,369)(332,368)(333,367)
(334,352)(335,354)(336,353)(337,356)(338,355)(339,357)(340,360)(341,359)
(342,358)(343,370)(344,372)(345,371)(346,374)(347,373)(348,375)(349,378)
(350,377)(351,376)(379,415)(380,417)(381,416)(382,419)(383,418)(384,420)
(385,423)(386,422)(387,421)(388,406)(389,408)(390,407)(391,410)(392,409)
(393,411)(394,414)(395,413)(396,412)(397,424)(398,426)(399,425)(400,428)
(401,427)(402,429)(403,432)(404,431)(405,430)(433,469)(434,471)(435,470)
(436,473)(437,472)(438,474)(439,477)(440,476)(441,475)(442,460)(443,462)
(444,461)(445,464)(446,463)(447,465)(448,468)(449,467)(450,466)(451,478)
(452,480)(453,479)(454,482)(455,481)(456,483)(457,486)(458,485)(459,484)
(487,523)(488,525)(489,524)(490,527)(491,526)(492,528)(493,531)(494,530)
(495,529)(496,514)(497,516)(498,515)(499,518)(500,517)(501,519)(502,522)
(503,521)(504,520)(505,532)(506,534)(507,533)(508,536)(509,535)(510,537)
(511,540)(512,539)(513,538)(541,550)(542,552)(543,551)(544,554)(545,553)
(546,555)(547,558)(548,557)(549,556)(560,561)(562,563)(565,567)(568,577)
(569,579)(570,578)(571,581)(572,580)(573,582)(574,585)(575,584)(576,583)
(587,588)(589,590)(592,594)(595,604)(596,606)(597,605)(598,608)(599,607)
(600,609)(601,612)(602,611)(603,610)(614,615)(616,617)(619,621)(622,631)
(623,633)(624,632)(625,635)(626,634)(627,636)(628,639)(629,638)(630,637)
(641,642)(643,644)(646,648)(649,685)(650,687)(651,686)(652,689)(653,688)
(654,690)(655,693)(656,692)(657,691)(658,676)(659,678)(660,677)(661,680)
(662,679)(663,681)(664,684)(665,683)(666,682)(667,694)(668,696)(669,695)
(670,698)(671,697)(672,699)(673,702)(674,701)(675,700)(703,739)(704,741)
(705,740)(706,743)(707,742)(708,744)(709,747)(710,746)(711,745)(712,730)
(713,732)(714,731)(715,734)(716,733)(717,735)(718,738)(719,737)(720,736)
(721,748)(722,750)(723,749)(724,752)(725,751)(726,753)(727,756)(728,755)
(729,754)(757,766)(758,768)(759,767)(760,770)(761,769)(762,771)(763,774)
(764,773)(765,772)(776,777)(778,779)(781,783)(784,793)(785,795)(786,794)
(787,797)(788,796)(789,798)(790,801)(791,800)(792,799)(803,804)(805,806)
(808,810)(811,820)(812,822)(813,821)(814,824)(815,823)(816,825)(817,828)
(818,827)(819,826)(830,831)(832,833)(835,837)(838,847)(839,849)(840,848)
(841,851)(842,850)(843,852)(844,855)(845,854)(846,853)(857,858)(859,860)
(862,864);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*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 >;
References : None.
to this polytope