Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,4,6}

Atlas Canonical Name {12,4,6}*1728a

Overview

Group
SmallGroup(1728,46611)
Rank
4
Schläfli Type
{12,4,6}
Vertices, edges, …
12, 72, 36, 18
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
{{12,4|2},{4,6}4}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

72-fold

108-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s2*s1*(s3*s2*s1*s2)^2*s3> of order 2

9 facets

12 vertex figures

  • 12 of 2-fold non-regular quotient of {4,6}*144
P/N, where N=<(s2*s3)^2> of order 3

6 facets

12 vertex figures

  • 12 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s2*s1*s2*s3)^2> of order 3

6 facets

12 vertex figures

  • 12 of 3-fold non-regular quotient of {4,6}*144
P/N, where N=<(s2*s3)^2, s1*(s2*s3)^2*s2*s1*s3> of order 6

3 facets

12 vertex figures

  • 12 of 6-fold non-regular quotient of {4,6}*144

Representations

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

References

None.

to this polytope.