Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,8}

Atlas Canonical Name {6,6,8}*1728g

Overview

Group
SmallGroup(1728,46634)
Rank
4
Schläfli Type
{6,6,8}
Vertices, edges, …
6, 54, 72, 24
Order of s0s1s2s3
24
Order of s0s1s2s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

9-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*s3*s2)^2> of order 3

8 facets

6 vertex figures

  • 6 of 3-fold non-regular quotient of {6,8}*288
P/N, where N=<s1*s3*s2*s1*s2*s3> of order 3

16 facets

6 vertex figures

  • 6 of 3-fold non-regular quotient of {6,8}*288

Representations

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

References

None.

to this polytope.