Polytope of Type {6,12,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,46303)
Rank : 4
Schlafli Type : {6,12,3}
Number of vertices, edges, etc : 6, 144, 72, 12
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,3}*864
3-fold quotients : {2,12,3}*576, {6,12,3}*576
6-fold quotients : {2,6,3}*288, {6,6,3}*288
8-fold quotients : {6,6,3}*216b
9-fold quotients : {2,12,3}*192
18-fold quotients : {2,6,3}*96
24-fold quotients : {2,6,3}*72, {6,2,3}*72
36-fold quotients : {2,3,3}*48
48-fold quotients : {3,2,3}*36
72-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)( 32, 56)
( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)( 40, 64)
( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 69)( 46, 70)( 47, 71)( 48, 72)
( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)(104,128)
(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)
(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)
(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200)
(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)(184,208)
(185,209)(186,210)(187,211)(188,212)(189,213)(190,214)(191,215)(192,216)
(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)(248,272)
(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)(256,280)
(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)(264,288)
(313,337)(314,338)(315,339)(316,340)(317,341)(318,342)(319,343)(320,344)
(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)(328,352)
(329,353)(330,354)(331,355)(332,356)(333,357)(334,358)(335,359)(336,360)
(385,409)(386,410)(387,411)(388,412)(389,413)(390,414)(391,415)(392,416)
(393,417)(394,418)(395,419)(396,420)(397,421)(398,422)(399,423)(400,424)
(401,425)(402,426)(403,427)(404,428)(405,429)(406,430)(407,431)(408,432)
(457,481)(458,482)(459,483)(460,484)(461,485)(462,486)(463,487)(464,488)
(465,489)(466,490)(467,491)(468,492)(469,493)(470,494)(471,495)(472,496)
(473,497)(474,498)(475,499)(476,500)(477,501)(478,502)(479,503)(480,504)
(529,553)(530,554)(531,555)(532,556)(533,557)(534,558)(535,559)(536,560)
(537,561)(538,562)(539,563)(540,564)(541,565)(542,566)(543,567)(544,568)
(545,569)(546,570)(547,571)(548,572)(549,573)(550,574)(551,575)(552,576)
(601,625)(602,626)(603,627)(604,628)(605,629)(606,630)(607,631)(608,632)
(609,633)(610,634)(611,635)(612,636)(613,637)(614,638)(615,639)(616,640)
(617,641)(618,642)(619,643)(620,644)(621,645)(622,646)(623,647)(624,648)
(673,697)(674,698)(675,699)(676,700)(677,701)(678,702)(679,703)(680,704)
(681,705)(682,706)(683,707)(684,708)(685,709)(686,710)(687,711)(688,712)
(689,713)(690,714)(691,715)(692,716)(693,717)(694,718)(695,719)(696,720)
(745,769)(746,770)(747,771)(748,772)(749,773)(750,774)(751,775)(752,776)
(753,777)(754,778)(755,779)(756,780)(757,781)(758,782)(759,783)(760,784)
(761,785)(762,786)(763,787)(764,788)(765,789)(766,790)(767,791)(768,792)
(817,841)(818,842)(819,843)(820,844)(821,845)(822,846)(823,847)(824,848)
(825,849)(826,850)(827,851)(828,852)(829,853)(830,854)(831,855)(832,856)
(833,857)(834,858)(835,859)(836,860)(837,861)(838,862)(839,863)(840,864);;
s1 := ( 1,241)( 2,242)( 3,244)( 4,243)( 5,247)( 6,248)( 7,245)( 8,246)
( 9,249)( 10,250)( 11,252)( 12,251)( 13,255)( 14,256)( 15,253)( 16,254)
( 17,257)( 18,258)( 19,260)( 20,259)( 21,263)( 22,264)( 23,261)( 24,262)
( 25,217)( 26,218)( 27,220)( 28,219)( 29,223)( 30,224)( 31,221)( 32,222)
( 33,225)( 34,226)( 35,228)( 36,227)( 37,231)( 38,232)( 39,229)( 40,230)
( 41,233)( 42,234)( 43,236)( 44,235)( 45,239)( 46,240)( 47,237)( 48,238)
( 49,265)( 50,266)( 51,268)( 52,267)( 53,271)( 54,272)( 55,269)( 56,270)
( 57,273)( 58,274)( 59,276)( 60,275)( 61,279)( 62,280)( 63,277)( 64,278)
( 65,281)( 66,282)( 67,284)( 68,283)( 69,287)( 70,288)( 71,285)( 72,286)
( 73,385)( 74,386)( 75,388)( 76,387)( 77,391)( 78,392)( 79,389)( 80,390)
( 81,393)( 82,394)( 83,396)( 84,395)( 85,399)( 86,400)( 87,397)( 88,398)
( 89,401)( 90,402)( 91,404)( 92,403)( 93,407)( 94,408)( 95,405)( 96,406)
( 97,361)( 98,362)( 99,364)(100,363)(101,367)(102,368)(103,365)(104,366)
(105,369)(106,370)(107,372)(108,371)(109,375)(110,376)(111,373)(112,374)
(113,377)(114,378)(115,380)(116,379)(117,383)(118,384)(119,381)(120,382)
(121,409)(122,410)(123,412)(124,411)(125,415)(126,416)(127,413)(128,414)
(129,417)(130,418)(131,420)(132,419)(133,423)(134,424)(135,421)(136,422)
(137,425)(138,426)(139,428)(140,427)(141,431)(142,432)(143,429)(144,430)
(145,313)(146,314)(147,316)(148,315)(149,319)(150,320)(151,317)(152,318)
(153,321)(154,322)(155,324)(156,323)(157,327)(158,328)(159,325)(160,326)
(161,329)(162,330)(163,332)(164,331)(165,335)(166,336)(167,333)(168,334)
(169,289)(170,290)(171,292)(172,291)(173,295)(174,296)(175,293)(176,294)
(177,297)(178,298)(179,300)(180,299)(181,303)(182,304)(183,301)(184,302)
(185,305)(186,306)(187,308)(188,307)(189,311)(190,312)(191,309)(192,310)
(193,337)(194,338)(195,340)(196,339)(197,343)(198,344)(199,341)(200,342)
(201,345)(202,346)(203,348)(204,347)(205,351)(206,352)(207,349)(208,350)
(209,353)(210,354)(211,356)(212,355)(213,359)(214,360)(215,357)(216,358)
(433,673)(434,674)(435,676)(436,675)(437,679)(438,680)(439,677)(440,678)
(441,681)(442,682)(443,684)(444,683)(445,687)(446,688)(447,685)(448,686)
(449,689)(450,690)(451,692)(452,691)(453,695)(454,696)(455,693)(456,694)
(457,649)(458,650)(459,652)(460,651)(461,655)(462,656)(463,653)(464,654)
(465,657)(466,658)(467,660)(468,659)(469,663)(470,664)(471,661)(472,662)
(473,665)(474,666)(475,668)(476,667)(477,671)(478,672)(479,669)(480,670)
(481,697)(482,698)(483,700)(484,699)(485,703)(486,704)(487,701)(488,702)
(489,705)(490,706)(491,708)(492,707)(493,711)(494,712)(495,709)(496,710)
(497,713)(498,714)(499,716)(500,715)(501,719)(502,720)(503,717)(504,718)
(505,817)(506,818)(507,820)(508,819)(509,823)(510,824)(511,821)(512,822)
(513,825)(514,826)(515,828)(516,827)(517,831)(518,832)(519,829)(520,830)
(521,833)(522,834)(523,836)(524,835)(525,839)(526,840)(527,837)(528,838)
(529,793)(530,794)(531,796)(532,795)(533,799)(534,800)(535,797)(536,798)
(537,801)(538,802)(539,804)(540,803)(541,807)(542,808)(543,805)(544,806)
(545,809)(546,810)(547,812)(548,811)(549,815)(550,816)(551,813)(552,814)
(553,841)(554,842)(555,844)(556,843)(557,847)(558,848)(559,845)(560,846)
(561,849)(562,850)(563,852)(564,851)(565,855)(566,856)(567,853)(568,854)
(569,857)(570,858)(571,860)(572,859)(573,863)(574,864)(575,861)(576,862)
(577,745)(578,746)(579,748)(580,747)(581,751)(582,752)(583,749)(584,750)
(585,753)(586,754)(587,756)(588,755)(589,759)(590,760)(591,757)(592,758)
(593,761)(594,762)(595,764)(596,763)(597,767)(598,768)(599,765)(600,766)
(601,721)(602,722)(603,724)(604,723)(605,727)(606,728)(607,725)(608,726)
(609,729)(610,730)(611,732)(612,731)(613,735)(614,736)(615,733)(616,734)
(617,737)(618,738)(619,740)(620,739)(621,743)(622,744)(623,741)(624,742)
(625,769)(626,770)(627,772)(628,771)(629,775)(630,776)(631,773)(632,774)
(633,777)(634,778)(635,780)(636,779)(637,783)(638,784)(639,781)(640,782)
(641,785)(642,786)(643,788)(644,787)(645,791)(646,792)(647,789)(648,790);;
s2 := ( 1,721)( 2,722)( 3,727)( 4,728)( 5,726)( 6,725)( 7,723)( 8,724)
( 9,737)( 10,738)( 11,743)( 12,744)( 13,742)( 14,741)( 15,739)( 16,740)
( 17,729)( 18,730)( 19,735)( 20,736)( 21,734)( 22,733)( 23,731)( 24,732)
( 25,745)( 26,746)( 27,751)( 28,752)( 29,750)( 30,749)( 31,747)( 32,748)
( 33,761)( 34,762)( 35,767)( 36,768)( 37,766)( 38,765)( 39,763)( 40,764)
( 41,753)( 42,754)( 43,759)( 44,760)( 45,758)( 46,757)( 47,755)( 48,756)
( 49,769)( 50,770)( 51,775)( 52,776)( 53,774)( 54,773)( 55,771)( 56,772)
( 57,785)( 58,786)( 59,791)( 60,792)( 61,790)( 62,789)( 63,787)( 64,788)
( 65,777)( 66,778)( 67,783)( 68,784)( 69,782)( 70,781)( 71,779)( 72,780)
( 73,649)( 74,650)( 75,655)( 76,656)( 77,654)( 78,653)( 79,651)( 80,652)
( 81,665)( 82,666)( 83,671)( 84,672)( 85,670)( 86,669)( 87,667)( 88,668)
( 89,657)( 90,658)( 91,663)( 92,664)( 93,662)( 94,661)( 95,659)( 96,660)
( 97,673)( 98,674)( 99,679)(100,680)(101,678)(102,677)(103,675)(104,676)
(105,689)(106,690)(107,695)(108,696)(109,694)(110,693)(111,691)(112,692)
(113,681)(114,682)(115,687)(116,688)(117,686)(118,685)(119,683)(120,684)
(121,697)(122,698)(123,703)(124,704)(125,702)(126,701)(127,699)(128,700)
(129,713)(130,714)(131,719)(132,720)(133,718)(134,717)(135,715)(136,716)
(137,705)(138,706)(139,711)(140,712)(141,710)(142,709)(143,707)(144,708)
(145,793)(146,794)(147,799)(148,800)(149,798)(150,797)(151,795)(152,796)
(153,809)(154,810)(155,815)(156,816)(157,814)(158,813)(159,811)(160,812)
(161,801)(162,802)(163,807)(164,808)(165,806)(166,805)(167,803)(168,804)
(169,817)(170,818)(171,823)(172,824)(173,822)(174,821)(175,819)(176,820)
(177,833)(178,834)(179,839)(180,840)(181,838)(182,837)(183,835)(184,836)
(185,825)(186,826)(187,831)(188,832)(189,830)(190,829)(191,827)(192,828)
(193,841)(194,842)(195,847)(196,848)(197,846)(198,845)(199,843)(200,844)
(201,857)(202,858)(203,863)(204,864)(205,862)(206,861)(207,859)(208,860)
(209,849)(210,850)(211,855)(212,856)(213,854)(214,853)(215,851)(216,852)
(217,506)(218,505)(219,512)(220,511)(221,509)(222,510)(223,508)(224,507)
(225,522)(226,521)(227,528)(228,527)(229,525)(230,526)(231,524)(232,523)
(233,514)(234,513)(235,520)(236,519)(237,517)(238,518)(239,516)(240,515)
(241,530)(242,529)(243,536)(244,535)(245,533)(246,534)(247,532)(248,531)
(249,546)(250,545)(251,552)(252,551)(253,549)(254,550)(255,548)(256,547)
(257,538)(258,537)(259,544)(260,543)(261,541)(262,542)(263,540)(264,539)
(265,554)(266,553)(267,560)(268,559)(269,557)(270,558)(271,556)(272,555)
(273,570)(274,569)(275,576)(276,575)(277,573)(278,574)(279,572)(280,571)
(281,562)(282,561)(283,568)(284,567)(285,565)(286,566)(287,564)(288,563)
(289,434)(290,433)(291,440)(292,439)(293,437)(294,438)(295,436)(296,435)
(297,450)(298,449)(299,456)(300,455)(301,453)(302,454)(303,452)(304,451)
(305,442)(306,441)(307,448)(308,447)(309,445)(310,446)(311,444)(312,443)
(313,458)(314,457)(315,464)(316,463)(317,461)(318,462)(319,460)(320,459)
(321,474)(322,473)(323,480)(324,479)(325,477)(326,478)(327,476)(328,475)
(329,466)(330,465)(331,472)(332,471)(333,469)(334,470)(335,468)(336,467)
(337,482)(338,481)(339,488)(340,487)(341,485)(342,486)(343,484)(344,483)
(345,498)(346,497)(347,504)(348,503)(349,501)(350,502)(351,500)(352,499)
(353,490)(354,489)(355,496)(356,495)(357,493)(358,494)(359,492)(360,491)
(361,578)(362,577)(363,584)(364,583)(365,581)(366,582)(367,580)(368,579)
(369,594)(370,593)(371,600)(372,599)(373,597)(374,598)(375,596)(376,595)
(377,586)(378,585)(379,592)(380,591)(381,589)(382,590)(383,588)(384,587)
(385,602)(386,601)(387,608)(388,607)(389,605)(390,606)(391,604)(392,603)
(393,618)(394,617)(395,624)(396,623)(397,621)(398,622)(399,620)(400,619)
(401,610)(402,609)(403,616)(404,615)(405,613)(406,614)(407,612)(408,611)
(409,626)(410,625)(411,632)(412,631)(413,629)(414,630)(415,628)(416,627)
(417,642)(418,641)(419,648)(420,647)(421,645)(422,646)(423,644)(424,643)
(425,634)(426,633)(427,640)(428,639)(429,637)(430,638)(431,636)(432,635);;
s3 := ( 1,659)( 2,660)( 3,657)( 4,658)( 5,662)( 6,661)( 7,663)( 8,664)
( 9,651)( 10,652)( 11,649)( 12,650)( 13,654)( 14,653)( 15,655)( 16,656)
( 17,667)( 18,668)( 19,665)( 20,666)( 21,670)( 22,669)( 23,671)( 24,672)
( 25,683)( 26,684)( 27,681)( 28,682)( 29,686)( 30,685)( 31,687)( 32,688)
( 33,675)( 34,676)( 35,673)( 36,674)( 37,678)( 38,677)( 39,679)( 40,680)
( 41,691)( 42,692)( 43,689)( 44,690)( 45,694)( 46,693)( 47,695)( 48,696)
( 49,707)( 50,708)( 51,705)( 52,706)( 53,710)( 54,709)( 55,711)( 56,712)
( 57,699)( 58,700)( 59,697)( 60,698)( 61,702)( 62,701)( 63,703)( 64,704)
( 65,715)( 66,716)( 67,713)( 68,714)( 69,718)( 70,717)( 71,719)( 72,720)
( 73,803)( 74,804)( 75,801)( 76,802)( 77,806)( 78,805)( 79,807)( 80,808)
( 81,795)( 82,796)( 83,793)( 84,794)( 85,798)( 86,797)( 87,799)( 88,800)
( 89,811)( 90,812)( 91,809)( 92,810)( 93,814)( 94,813)( 95,815)( 96,816)
( 97,827)( 98,828)( 99,825)(100,826)(101,830)(102,829)(103,831)(104,832)
(105,819)(106,820)(107,817)(108,818)(109,822)(110,821)(111,823)(112,824)
(113,835)(114,836)(115,833)(116,834)(117,838)(118,837)(119,839)(120,840)
(121,851)(122,852)(123,849)(124,850)(125,854)(126,853)(127,855)(128,856)
(129,843)(130,844)(131,841)(132,842)(133,846)(134,845)(135,847)(136,848)
(137,859)(138,860)(139,857)(140,858)(141,862)(142,861)(143,863)(144,864)
(145,731)(146,732)(147,729)(148,730)(149,734)(150,733)(151,735)(152,736)
(153,723)(154,724)(155,721)(156,722)(157,726)(158,725)(159,727)(160,728)
(161,739)(162,740)(163,737)(164,738)(165,742)(166,741)(167,743)(168,744)
(169,755)(170,756)(171,753)(172,754)(173,758)(174,757)(175,759)(176,760)
(177,747)(178,748)(179,745)(180,746)(181,750)(182,749)(183,751)(184,752)
(185,763)(186,764)(187,761)(188,762)(189,766)(190,765)(191,767)(192,768)
(193,779)(194,780)(195,777)(196,778)(197,782)(198,781)(199,783)(200,784)
(201,771)(202,772)(203,769)(204,770)(205,774)(206,773)(207,775)(208,776)
(209,787)(210,788)(211,785)(212,786)(213,790)(214,789)(215,791)(216,792)
(217,444)(218,443)(219,442)(220,441)(221,445)(222,446)(223,448)(224,447)
(225,436)(226,435)(227,434)(228,433)(229,437)(230,438)(231,440)(232,439)
(233,452)(234,451)(235,450)(236,449)(237,453)(238,454)(239,456)(240,455)
(241,468)(242,467)(243,466)(244,465)(245,469)(246,470)(247,472)(248,471)
(249,460)(250,459)(251,458)(252,457)(253,461)(254,462)(255,464)(256,463)
(257,476)(258,475)(259,474)(260,473)(261,477)(262,478)(263,480)(264,479)
(265,492)(266,491)(267,490)(268,489)(269,493)(270,494)(271,496)(272,495)
(273,484)(274,483)(275,482)(276,481)(277,485)(278,486)(279,488)(280,487)
(281,500)(282,499)(283,498)(284,497)(285,501)(286,502)(287,504)(288,503)
(289,588)(290,587)(291,586)(292,585)(293,589)(294,590)(295,592)(296,591)
(297,580)(298,579)(299,578)(300,577)(301,581)(302,582)(303,584)(304,583)
(305,596)(306,595)(307,594)(308,593)(309,597)(310,598)(311,600)(312,599)
(313,612)(314,611)(315,610)(316,609)(317,613)(318,614)(319,616)(320,615)
(321,604)(322,603)(323,602)(324,601)(325,605)(326,606)(327,608)(328,607)
(329,620)(330,619)(331,618)(332,617)(333,621)(334,622)(335,624)(336,623)
(337,636)(338,635)(339,634)(340,633)(341,637)(342,638)(343,640)(344,639)
(345,628)(346,627)(347,626)(348,625)(349,629)(350,630)(351,632)(352,631)
(353,644)(354,643)(355,642)(356,641)(357,645)(358,646)(359,648)(360,647)
(361,516)(362,515)(363,514)(364,513)(365,517)(366,518)(367,520)(368,519)
(369,508)(370,507)(371,506)(372,505)(373,509)(374,510)(375,512)(376,511)
(377,524)(378,523)(379,522)(380,521)(381,525)(382,526)(383,528)(384,527)
(385,540)(386,539)(387,538)(388,537)(389,541)(390,542)(391,544)(392,543)
(393,532)(394,531)(395,530)(396,529)(397,533)(398,534)(399,536)(400,535)
(401,548)(402,547)(403,546)(404,545)(405,549)(406,550)(407,552)(408,551)
(409,564)(410,563)(411,562)(412,561)(413,565)(414,566)(415,568)(416,567)
(417,556)(418,555)(419,554)(420,553)(421,557)(422,558)(423,560)(424,559)
(425,572)(426,571)(427,570)(428,569)(429,573)(430,574)(431,576)(432,575);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(864)!( 25, 49)( 26, 50)( 27, 51)( 28, 52)( 29, 53)( 30, 54)( 31, 55)
( 32, 56)( 33, 57)( 34, 58)( 35, 59)( 36, 60)( 37, 61)( 38, 62)( 39, 63)
( 40, 64)( 41, 65)( 42, 66)( 43, 67)( 44, 68)( 45, 69)( 46, 70)( 47, 71)
( 48, 72)( 97,121)( 98,122)( 99,123)(100,124)(101,125)(102,126)(103,127)
(104,128)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)
(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)
(120,144)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)
(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)
(184,208)(185,209)(186,210)(187,211)(188,212)(189,213)(190,214)(191,215)
(192,216)(241,265)(242,266)(243,267)(244,268)(245,269)(246,270)(247,271)
(248,272)(249,273)(250,274)(251,275)(252,276)(253,277)(254,278)(255,279)
(256,280)(257,281)(258,282)(259,283)(260,284)(261,285)(262,286)(263,287)
(264,288)(313,337)(314,338)(315,339)(316,340)(317,341)(318,342)(319,343)
(320,344)(321,345)(322,346)(323,347)(324,348)(325,349)(326,350)(327,351)
(328,352)(329,353)(330,354)(331,355)(332,356)(333,357)(334,358)(335,359)
(336,360)(385,409)(386,410)(387,411)(388,412)(389,413)(390,414)(391,415)
(392,416)(393,417)(394,418)(395,419)(396,420)(397,421)(398,422)(399,423)
(400,424)(401,425)(402,426)(403,427)(404,428)(405,429)(406,430)(407,431)
(408,432)(457,481)(458,482)(459,483)(460,484)(461,485)(462,486)(463,487)
(464,488)(465,489)(466,490)(467,491)(468,492)(469,493)(470,494)(471,495)
(472,496)(473,497)(474,498)(475,499)(476,500)(477,501)(478,502)(479,503)
(480,504)(529,553)(530,554)(531,555)(532,556)(533,557)(534,558)(535,559)
(536,560)(537,561)(538,562)(539,563)(540,564)(541,565)(542,566)(543,567)
(544,568)(545,569)(546,570)(547,571)(548,572)(549,573)(550,574)(551,575)
(552,576)(601,625)(602,626)(603,627)(604,628)(605,629)(606,630)(607,631)
(608,632)(609,633)(610,634)(611,635)(612,636)(613,637)(614,638)(615,639)
(616,640)(617,641)(618,642)(619,643)(620,644)(621,645)(622,646)(623,647)
(624,648)(673,697)(674,698)(675,699)(676,700)(677,701)(678,702)(679,703)
(680,704)(681,705)(682,706)(683,707)(684,708)(685,709)(686,710)(687,711)
(688,712)(689,713)(690,714)(691,715)(692,716)(693,717)(694,718)(695,719)
(696,720)(745,769)(746,770)(747,771)(748,772)(749,773)(750,774)(751,775)
(752,776)(753,777)(754,778)(755,779)(756,780)(757,781)(758,782)(759,783)
(760,784)(761,785)(762,786)(763,787)(764,788)(765,789)(766,790)(767,791)
(768,792)(817,841)(818,842)(819,843)(820,844)(821,845)(822,846)(823,847)
(824,848)(825,849)(826,850)(827,851)(828,852)(829,853)(830,854)(831,855)
(832,856)(833,857)(834,858)(835,859)(836,860)(837,861)(838,862)(839,863)
(840,864);
s1 := Sym(864)!( 1,241)( 2,242)( 3,244)( 4,243)( 5,247)( 6,248)( 7,245)
( 8,246)( 9,249)( 10,250)( 11,252)( 12,251)( 13,255)( 14,256)( 15,253)
( 16,254)( 17,257)( 18,258)( 19,260)( 20,259)( 21,263)( 22,264)( 23,261)
( 24,262)( 25,217)( 26,218)( 27,220)( 28,219)( 29,223)( 30,224)( 31,221)
( 32,222)( 33,225)( 34,226)( 35,228)( 36,227)( 37,231)( 38,232)( 39,229)
( 40,230)( 41,233)( 42,234)( 43,236)( 44,235)( 45,239)( 46,240)( 47,237)
( 48,238)( 49,265)( 50,266)( 51,268)( 52,267)( 53,271)( 54,272)( 55,269)
( 56,270)( 57,273)( 58,274)( 59,276)( 60,275)( 61,279)( 62,280)( 63,277)
( 64,278)( 65,281)( 66,282)( 67,284)( 68,283)( 69,287)( 70,288)( 71,285)
( 72,286)( 73,385)( 74,386)( 75,388)( 76,387)( 77,391)( 78,392)( 79,389)
( 80,390)( 81,393)( 82,394)( 83,396)( 84,395)( 85,399)( 86,400)( 87,397)
( 88,398)( 89,401)( 90,402)( 91,404)( 92,403)( 93,407)( 94,408)( 95,405)
( 96,406)( 97,361)( 98,362)( 99,364)(100,363)(101,367)(102,368)(103,365)
(104,366)(105,369)(106,370)(107,372)(108,371)(109,375)(110,376)(111,373)
(112,374)(113,377)(114,378)(115,380)(116,379)(117,383)(118,384)(119,381)
(120,382)(121,409)(122,410)(123,412)(124,411)(125,415)(126,416)(127,413)
(128,414)(129,417)(130,418)(131,420)(132,419)(133,423)(134,424)(135,421)
(136,422)(137,425)(138,426)(139,428)(140,427)(141,431)(142,432)(143,429)
(144,430)(145,313)(146,314)(147,316)(148,315)(149,319)(150,320)(151,317)
(152,318)(153,321)(154,322)(155,324)(156,323)(157,327)(158,328)(159,325)
(160,326)(161,329)(162,330)(163,332)(164,331)(165,335)(166,336)(167,333)
(168,334)(169,289)(170,290)(171,292)(172,291)(173,295)(174,296)(175,293)
(176,294)(177,297)(178,298)(179,300)(180,299)(181,303)(182,304)(183,301)
(184,302)(185,305)(186,306)(187,308)(188,307)(189,311)(190,312)(191,309)
(192,310)(193,337)(194,338)(195,340)(196,339)(197,343)(198,344)(199,341)
(200,342)(201,345)(202,346)(203,348)(204,347)(205,351)(206,352)(207,349)
(208,350)(209,353)(210,354)(211,356)(212,355)(213,359)(214,360)(215,357)
(216,358)(433,673)(434,674)(435,676)(436,675)(437,679)(438,680)(439,677)
(440,678)(441,681)(442,682)(443,684)(444,683)(445,687)(446,688)(447,685)
(448,686)(449,689)(450,690)(451,692)(452,691)(453,695)(454,696)(455,693)
(456,694)(457,649)(458,650)(459,652)(460,651)(461,655)(462,656)(463,653)
(464,654)(465,657)(466,658)(467,660)(468,659)(469,663)(470,664)(471,661)
(472,662)(473,665)(474,666)(475,668)(476,667)(477,671)(478,672)(479,669)
(480,670)(481,697)(482,698)(483,700)(484,699)(485,703)(486,704)(487,701)
(488,702)(489,705)(490,706)(491,708)(492,707)(493,711)(494,712)(495,709)
(496,710)(497,713)(498,714)(499,716)(500,715)(501,719)(502,720)(503,717)
(504,718)(505,817)(506,818)(507,820)(508,819)(509,823)(510,824)(511,821)
(512,822)(513,825)(514,826)(515,828)(516,827)(517,831)(518,832)(519,829)
(520,830)(521,833)(522,834)(523,836)(524,835)(525,839)(526,840)(527,837)
(528,838)(529,793)(530,794)(531,796)(532,795)(533,799)(534,800)(535,797)
(536,798)(537,801)(538,802)(539,804)(540,803)(541,807)(542,808)(543,805)
(544,806)(545,809)(546,810)(547,812)(548,811)(549,815)(550,816)(551,813)
(552,814)(553,841)(554,842)(555,844)(556,843)(557,847)(558,848)(559,845)
(560,846)(561,849)(562,850)(563,852)(564,851)(565,855)(566,856)(567,853)
(568,854)(569,857)(570,858)(571,860)(572,859)(573,863)(574,864)(575,861)
(576,862)(577,745)(578,746)(579,748)(580,747)(581,751)(582,752)(583,749)
(584,750)(585,753)(586,754)(587,756)(588,755)(589,759)(590,760)(591,757)
(592,758)(593,761)(594,762)(595,764)(596,763)(597,767)(598,768)(599,765)
(600,766)(601,721)(602,722)(603,724)(604,723)(605,727)(606,728)(607,725)
(608,726)(609,729)(610,730)(611,732)(612,731)(613,735)(614,736)(615,733)
(616,734)(617,737)(618,738)(619,740)(620,739)(621,743)(622,744)(623,741)
(624,742)(625,769)(626,770)(627,772)(628,771)(629,775)(630,776)(631,773)
(632,774)(633,777)(634,778)(635,780)(636,779)(637,783)(638,784)(639,781)
(640,782)(641,785)(642,786)(643,788)(644,787)(645,791)(646,792)(647,789)
(648,790);
s2 := Sym(864)!( 1,721)( 2,722)( 3,727)( 4,728)( 5,726)( 6,725)( 7,723)
( 8,724)( 9,737)( 10,738)( 11,743)( 12,744)( 13,742)( 14,741)( 15,739)
( 16,740)( 17,729)( 18,730)( 19,735)( 20,736)( 21,734)( 22,733)( 23,731)
( 24,732)( 25,745)( 26,746)( 27,751)( 28,752)( 29,750)( 30,749)( 31,747)
( 32,748)( 33,761)( 34,762)( 35,767)( 36,768)( 37,766)( 38,765)( 39,763)
( 40,764)( 41,753)( 42,754)( 43,759)( 44,760)( 45,758)( 46,757)( 47,755)
( 48,756)( 49,769)( 50,770)( 51,775)( 52,776)( 53,774)( 54,773)( 55,771)
( 56,772)( 57,785)( 58,786)( 59,791)( 60,792)( 61,790)( 62,789)( 63,787)
( 64,788)( 65,777)( 66,778)( 67,783)( 68,784)( 69,782)( 70,781)( 71,779)
( 72,780)( 73,649)( 74,650)( 75,655)( 76,656)( 77,654)( 78,653)( 79,651)
( 80,652)( 81,665)( 82,666)( 83,671)( 84,672)( 85,670)( 86,669)( 87,667)
( 88,668)( 89,657)( 90,658)( 91,663)( 92,664)( 93,662)( 94,661)( 95,659)
( 96,660)( 97,673)( 98,674)( 99,679)(100,680)(101,678)(102,677)(103,675)
(104,676)(105,689)(106,690)(107,695)(108,696)(109,694)(110,693)(111,691)
(112,692)(113,681)(114,682)(115,687)(116,688)(117,686)(118,685)(119,683)
(120,684)(121,697)(122,698)(123,703)(124,704)(125,702)(126,701)(127,699)
(128,700)(129,713)(130,714)(131,719)(132,720)(133,718)(134,717)(135,715)
(136,716)(137,705)(138,706)(139,711)(140,712)(141,710)(142,709)(143,707)
(144,708)(145,793)(146,794)(147,799)(148,800)(149,798)(150,797)(151,795)
(152,796)(153,809)(154,810)(155,815)(156,816)(157,814)(158,813)(159,811)
(160,812)(161,801)(162,802)(163,807)(164,808)(165,806)(166,805)(167,803)
(168,804)(169,817)(170,818)(171,823)(172,824)(173,822)(174,821)(175,819)
(176,820)(177,833)(178,834)(179,839)(180,840)(181,838)(182,837)(183,835)
(184,836)(185,825)(186,826)(187,831)(188,832)(189,830)(190,829)(191,827)
(192,828)(193,841)(194,842)(195,847)(196,848)(197,846)(198,845)(199,843)
(200,844)(201,857)(202,858)(203,863)(204,864)(205,862)(206,861)(207,859)
(208,860)(209,849)(210,850)(211,855)(212,856)(213,854)(214,853)(215,851)
(216,852)(217,506)(218,505)(219,512)(220,511)(221,509)(222,510)(223,508)
(224,507)(225,522)(226,521)(227,528)(228,527)(229,525)(230,526)(231,524)
(232,523)(233,514)(234,513)(235,520)(236,519)(237,517)(238,518)(239,516)
(240,515)(241,530)(242,529)(243,536)(244,535)(245,533)(246,534)(247,532)
(248,531)(249,546)(250,545)(251,552)(252,551)(253,549)(254,550)(255,548)
(256,547)(257,538)(258,537)(259,544)(260,543)(261,541)(262,542)(263,540)
(264,539)(265,554)(266,553)(267,560)(268,559)(269,557)(270,558)(271,556)
(272,555)(273,570)(274,569)(275,576)(276,575)(277,573)(278,574)(279,572)
(280,571)(281,562)(282,561)(283,568)(284,567)(285,565)(286,566)(287,564)
(288,563)(289,434)(290,433)(291,440)(292,439)(293,437)(294,438)(295,436)
(296,435)(297,450)(298,449)(299,456)(300,455)(301,453)(302,454)(303,452)
(304,451)(305,442)(306,441)(307,448)(308,447)(309,445)(310,446)(311,444)
(312,443)(313,458)(314,457)(315,464)(316,463)(317,461)(318,462)(319,460)
(320,459)(321,474)(322,473)(323,480)(324,479)(325,477)(326,478)(327,476)
(328,475)(329,466)(330,465)(331,472)(332,471)(333,469)(334,470)(335,468)
(336,467)(337,482)(338,481)(339,488)(340,487)(341,485)(342,486)(343,484)
(344,483)(345,498)(346,497)(347,504)(348,503)(349,501)(350,502)(351,500)
(352,499)(353,490)(354,489)(355,496)(356,495)(357,493)(358,494)(359,492)
(360,491)(361,578)(362,577)(363,584)(364,583)(365,581)(366,582)(367,580)
(368,579)(369,594)(370,593)(371,600)(372,599)(373,597)(374,598)(375,596)
(376,595)(377,586)(378,585)(379,592)(380,591)(381,589)(382,590)(383,588)
(384,587)(385,602)(386,601)(387,608)(388,607)(389,605)(390,606)(391,604)
(392,603)(393,618)(394,617)(395,624)(396,623)(397,621)(398,622)(399,620)
(400,619)(401,610)(402,609)(403,616)(404,615)(405,613)(406,614)(407,612)
(408,611)(409,626)(410,625)(411,632)(412,631)(413,629)(414,630)(415,628)
(416,627)(417,642)(418,641)(419,648)(420,647)(421,645)(422,646)(423,644)
(424,643)(425,634)(426,633)(427,640)(428,639)(429,637)(430,638)(431,636)
(432,635);
s3 := Sym(864)!( 1,659)( 2,660)( 3,657)( 4,658)( 5,662)( 6,661)( 7,663)
( 8,664)( 9,651)( 10,652)( 11,649)( 12,650)( 13,654)( 14,653)( 15,655)
( 16,656)( 17,667)( 18,668)( 19,665)( 20,666)( 21,670)( 22,669)( 23,671)
( 24,672)( 25,683)( 26,684)( 27,681)( 28,682)( 29,686)( 30,685)( 31,687)
( 32,688)( 33,675)( 34,676)( 35,673)( 36,674)( 37,678)( 38,677)( 39,679)
( 40,680)( 41,691)( 42,692)( 43,689)( 44,690)( 45,694)( 46,693)( 47,695)
( 48,696)( 49,707)( 50,708)( 51,705)( 52,706)( 53,710)( 54,709)( 55,711)
( 56,712)( 57,699)( 58,700)( 59,697)( 60,698)( 61,702)( 62,701)( 63,703)
( 64,704)( 65,715)( 66,716)( 67,713)( 68,714)( 69,718)( 70,717)( 71,719)
( 72,720)( 73,803)( 74,804)( 75,801)( 76,802)( 77,806)( 78,805)( 79,807)
( 80,808)( 81,795)( 82,796)( 83,793)( 84,794)( 85,798)( 86,797)( 87,799)
( 88,800)( 89,811)( 90,812)( 91,809)( 92,810)( 93,814)( 94,813)( 95,815)
( 96,816)( 97,827)( 98,828)( 99,825)(100,826)(101,830)(102,829)(103,831)
(104,832)(105,819)(106,820)(107,817)(108,818)(109,822)(110,821)(111,823)
(112,824)(113,835)(114,836)(115,833)(116,834)(117,838)(118,837)(119,839)
(120,840)(121,851)(122,852)(123,849)(124,850)(125,854)(126,853)(127,855)
(128,856)(129,843)(130,844)(131,841)(132,842)(133,846)(134,845)(135,847)
(136,848)(137,859)(138,860)(139,857)(140,858)(141,862)(142,861)(143,863)
(144,864)(145,731)(146,732)(147,729)(148,730)(149,734)(150,733)(151,735)
(152,736)(153,723)(154,724)(155,721)(156,722)(157,726)(158,725)(159,727)
(160,728)(161,739)(162,740)(163,737)(164,738)(165,742)(166,741)(167,743)
(168,744)(169,755)(170,756)(171,753)(172,754)(173,758)(174,757)(175,759)
(176,760)(177,747)(178,748)(179,745)(180,746)(181,750)(182,749)(183,751)
(184,752)(185,763)(186,764)(187,761)(188,762)(189,766)(190,765)(191,767)
(192,768)(193,779)(194,780)(195,777)(196,778)(197,782)(198,781)(199,783)
(200,784)(201,771)(202,772)(203,769)(204,770)(205,774)(206,773)(207,775)
(208,776)(209,787)(210,788)(211,785)(212,786)(213,790)(214,789)(215,791)
(216,792)(217,444)(218,443)(219,442)(220,441)(221,445)(222,446)(223,448)
(224,447)(225,436)(226,435)(227,434)(228,433)(229,437)(230,438)(231,440)
(232,439)(233,452)(234,451)(235,450)(236,449)(237,453)(238,454)(239,456)
(240,455)(241,468)(242,467)(243,466)(244,465)(245,469)(246,470)(247,472)
(248,471)(249,460)(250,459)(251,458)(252,457)(253,461)(254,462)(255,464)
(256,463)(257,476)(258,475)(259,474)(260,473)(261,477)(262,478)(263,480)
(264,479)(265,492)(266,491)(267,490)(268,489)(269,493)(270,494)(271,496)
(272,495)(273,484)(274,483)(275,482)(276,481)(277,485)(278,486)(279,488)
(280,487)(281,500)(282,499)(283,498)(284,497)(285,501)(286,502)(287,504)
(288,503)(289,588)(290,587)(291,586)(292,585)(293,589)(294,590)(295,592)
(296,591)(297,580)(298,579)(299,578)(300,577)(301,581)(302,582)(303,584)
(304,583)(305,596)(306,595)(307,594)(308,593)(309,597)(310,598)(311,600)
(312,599)(313,612)(314,611)(315,610)(316,609)(317,613)(318,614)(319,616)
(320,615)(321,604)(322,603)(323,602)(324,601)(325,605)(326,606)(327,608)
(328,607)(329,620)(330,619)(331,618)(332,617)(333,621)(334,622)(335,624)
(336,623)(337,636)(338,635)(339,634)(340,633)(341,637)(342,638)(343,640)
(344,639)(345,628)(346,627)(347,626)(348,625)(349,629)(350,630)(351,632)
(352,631)(353,644)(354,643)(355,642)(356,641)(357,645)(358,646)(359,648)
(360,647)(361,516)(362,515)(363,514)(364,513)(365,517)(366,518)(367,520)
(368,519)(369,508)(370,507)(371,506)(372,505)(373,509)(374,510)(375,512)
(376,511)(377,524)(378,523)(379,522)(380,521)(381,525)(382,526)(383,528)
(384,527)(385,540)(386,539)(387,538)(388,537)(389,541)(390,542)(391,544)
(392,543)(393,532)(394,531)(395,530)(396,529)(397,533)(398,534)(399,536)
(400,535)(401,548)(402,547)(403,546)(404,545)(405,549)(406,550)(407,552)
(408,551)(409,564)(410,563)(411,562)(412,561)(413,565)(414,566)(415,568)
(416,567)(417,556)(418,555)(419,554)(420,553)(421,557)(422,558)(423,560)
(424,559)(425,572)(426,571)(427,570)(428,569)(429,573)(430,574)(431,576)
(432,575);
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,
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2 >;
References : None.
to this polytope