Polytope of Type {12,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,17531)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 36, 108, 36, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,4}*864a, {12,6,2}*864c
3-fold quotients : {12,6,4}*576c
4-fold quotients : {6,6,2}*432a
6-fold quotients : {6,6,4}*288b, {12,6,2}*288c
8-fold quotients : {6,3,2}*216
9-fold quotients : {4,6,4}*192a
12-fold quotients : {6,6,2}*144b
18-fold quotients : {2,6,4}*96a, {4,6,2}*96a
24-fold quotients : {6,3,2}*72
27-fold quotients : {4,2,4}*64
36-fold quotients : {2,6,2}*48
54-fold quotients : {2,2,4}*32, {4,2,2}*32
72-fold quotients : {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)
(117,143)(118,154)(119,156)(120,155)(121,157)(122,159)(123,158)(124,160)
(125,162)(126,161)(127,145)(128,147)(129,146)(130,148)(131,150)(132,149)
(133,151)(134,153)(135,152)(163,190)(164,192)(165,191)(166,193)(167,195)
(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)(175,211)
(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)(183,200)
(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(218,219)(221,222)
(224,225)(226,235)(227,237)(228,236)(229,238)(230,240)(231,239)(232,241)
(233,243)(234,242)(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)
(256,265)(257,267)(258,266)(259,268)(260,270)(261,269)(272,273)(275,276)
(278,279)(280,289)(281,291)(282,290)(283,292)(284,294)(285,293)(286,295)
(287,297)(288,296)(299,300)(302,303)(305,306)(307,316)(308,318)(309,317)
(310,319)(311,321)(312,320)(313,322)(314,324)(315,323)(325,352)(326,354)
(327,353)(328,355)(329,357)(330,356)(331,358)(332,360)(333,359)(334,370)
(335,372)(336,371)(337,373)(338,375)(339,374)(340,376)(341,378)(342,377)
(343,361)(344,363)(345,362)(346,364)(347,366)(348,365)(349,367)(350,369)
(351,368)(379,406)(380,408)(381,407)(382,409)(383,411)(384,410)(385,412)
(386,414)(387,413)(388,424)(389,426)(390,425)(391,427)(392,429)(393,428)
(394,430)(395,432)(396,431)(397,415)(398,417)(399,416)(400,418)(401,420)
(402,419)(403,421)(404,423)(405,422)(433,460)(434,462)(435,461)(436,463)
(437,465)(438,464)(439,466)(440,468)(441,467)(442,478)(443,480)(444,479)
(445,481)(446,483)(447,482)(448,484)(449,486)(450,485)(451,469)(452,471)
(453,470)(454,472)(455,474)(456,473)(457,475)(458,477)(459,476)(487,514)
(488,516)(489,515)(490,517)(491,519)(492,518)(493,520)(494,522)(495,521)
(496,532)(497,534)(498,533)(499,535)(500,537)(501,536)(502,538)(503,540)
(504,539)(505,523)(506,525)(507,524)(508,526)(509,528)(510,527)(511,529)
(512,531)(513,530)(542,543)(545,546)(548,549)(550,559)(551,561)(552,560)
(553,562)(554,564)(555,563)(556,565)(557,567)(558,566)(569,570)(572,573)
(575,576)(577,586)(578,588)(579,587)(580,589)(581,591)(582,590)(583,592)
(584,594)(585,593)(596,597)(599,600)(602,603)(604,613)(605,615)(606,614)
(607,616)(608,618)(609,617)(610,619)(611,621)(612,620)(623,624)(626,627)
(629,630)(631,640)(632,642)(633,641)(634,643)(635,645)(636,644)(637,646)
(638,648)(639,647)(649,676)(650,678)(651,677)(652,679)(653,681)(654,680)
(655,682)(656,684)(657,683)(658,694)(659,696)(660,695)(661,697)(662,699)
(663,698)(664,700)(665,702)(666,701)(667,685)(668,687)(669,686)(670,688)
(671,690)(672,689)(673,691)(674,693)(675,692)(703,730)(704,732)(705,731)
(706,733)(707,735)(708,734)(709,736)(710,738)(711,737)(712,748)(713,750)
(714,749)(715,751)(716,753)(717,752)(718,754)(719,756)(720,755)(721,739)
(722,741)(723,740)(724,742)(725,744)(726,743)(727,745)(728,747)(729,746)
(758,759)(761,762)(764,765)(766,775)(767,777)(768,776)(769,778)(770,780)
(771,779)(772,781)(773,783)(774,782)(785,786)(788,789)(791,792)(793,802)
(794,804)(795,803)(796,805)(797,807)(798,806)(799,808)(800,810)(801,809)
(812,813)(815,816)(818,819)(820,829)(821,831)(822,830)(823,832)(824,834)
(825,833)(826,835)(827,837)(828,836)(839,840)(842,843)(845,846)(847,856)
(848,858)(849,857)(850,859)(851,861)(852,860)(853,862)(854,864)(855,863);;
s1 := ( 1,442)( 2,443)( 3,444)( 4,450)( 5,448)( 6,449)( 7,446)( 8,447)
( 9,445)( 10,433)( 11,434)( 12,435)( 13,441)( 14,439)( 15,440)( 16,437)
( 17,438)( 18,436)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)( 24,458)
( 25,455)( 26,456)( 27,454)( 28,469)( 29,470)( 30,471)( 31,477)( 32,475)
( 33,476)( 34,473)( 35,474)( 36,472)( 37,460)( 38,461)( 39,462)( 40,468)
( 41,466)( 42,467)( 43,464)( 44,465)( 45,463)( 46,478)( 47,479)( 48,480)
( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,496)( 56,497)
( 57,498)( 58,504)( 59,502)( 60,503)( 61,500)( 62,501)( 63,499)( 64,487)
( 65,488)( 66,489)( 67,495)( 68,493)( 69,494)( 70,491)( 71,492)( 72,490)
( 73,505)( 74,506)( 75,507)( 76,513)( 77,511)( 78,512)( 79,509)( 80,510)
( 81,508)( 82,523)( 83,524)( 84,525)( 85,531)( 86,529)( 87,530)( 88,527)
( 89,528)( 90,526)( 91,514)( 92,515)( 93,516)( 94,522)( 95,520)( 96,521)
( 97,518)( 98,519)( 99,517)(100,532)(101,533)(102,534)(103,540)(104,538)
(105,539)(106,536)(107,537)(108,535)(109,550)(110,551)(111,552)(112,558)
(113,556)(114,557)(115,554)(116,555)(117,553)(118,541)(119,542)(120,543)
(121,549)(122,547)(123,548)(124,545)(125,546)(126,544)(127,559)(128,560)
(129,561)(130,567)(131,565)(132,566)(133,563)(134,564)(135,562)(136,577)
(137,578)(138,579)(139,585)(140,583)(141,584)(142,581)(143,582)(144,580)
(145,568)(146,569)(147,570)(148,576)(149,574)(150,575)(151,572)(152,573)
(153,571)(154,586)(155,587)(156,588)(157,594)(158,592)(159,593)(160,590)
(161,591)(162,589)(163,604)(164,605)(165,606)(166,612)(167,610)(168,611)
(169,608)(170,609)(171,607)(172,595)(173,596)(174,597)(175,603)(176,601)
(177,602)(178,599)(179,600)(180,598)(181,613)(182,614)(183,615)(184,621)
(185,619)(186,620)(187,617)(188,618)(189,616)(190,631)(191,632)(192,633)
(193,639)(194,637)(195,638)(196,635)(197,636)(198,634)(199,622)(200,623)
(201,624)(202,630)(203,628)(204,629)(205,626)(206,627)(207,625)(208,640)
(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)(216,643)
(217,658)(218,659)(219,660)(220,666)(221,664)(222,665)(223,662)(224,663)
(225,661)(226,649)(227,650)(228,651)(229,657)(230,655)(231,656)(232,653)
(233,654)(234,652)(235,667)(236,668)(237,669)(238,675)(239,673)(240,674)
(241,671)(242,672)(243,670)(244,685)(245,686)(246,687)(247,693)(248,691)
(249,692)(250,689)(251,690)(252,688)(253,676)(254,677)(255,678)(256,684)
(257,682)(258,683)(259,680)(260,681)(261,679)(262,694)(263,695)(264,696)
(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,712)(272,713)
(273,714)(274,720)(275,718)(276,719)(277,716)(278,717)(279,715)(280,703)
(281,704)(282,705)(283,711)(284,709)(285,710)(286,707)(287,708)(288,706)
(289,721)(290,722)(291,723)(292,729)(293,727)(294,728)(295,725)(296,726)
(297,724)(298,739)(299,740)(300,741)(301,747)(302,745)(303,746)(304,743)
(305,744)(306,742)(307,730)(308,731)(309,732)(310,738)(311,736)(312,737)
(313,734)(314,735)(315,733)(316,748)(317,749)(318,750)(319,756)(320,754)
(321,755)(322,752)(323,753)(324,751)(325,766)(326,767)(327,768)(328,774)
(329,772)(330,773)(331,770)(332,771)(333,769)(334,757)(335,758)(336,759)
(337,765)(338,763)(339,764)(340,761)(341,762)(342,760)(343,775)(344,776)
(345,777)(346,783)(347,781)(348,782)(349,779)(350,780)(351,778)(352,793)
(353,794)(354,795)(355,801)(356,799)(357,800)(358,797)(359,798)(360,796)
(361,784)(362,785)(363,786)(364,792)(365,790)(366,791)(367,788)(368,789)
(369,787)(370,802)(371,803)(372,804)(373,810)(374,808)(375,809)(376,806)
(377,807)(378,805)(379,820)(380,821)(381,822)(382,828)(383,826)(384,827)
(385,824)(386,825)(387,823)(388,811)(389,812)(390,813)(391,819)(392,817)
(393,818)(394,815)(395,816)(396,814)(397,829)(398,830)(399,831)(400,837)
(401,835)(402,836)(403,833)(404,834)(405,832)(406,847)(407,848)(408,849)
(409,855)(410,853)(411,854)(412,851)(413,852)(414,850)(415,838)(416,839)
(417,840)(418,846)(419,844)(420,845)(421,842)(422,843)(423,841)(424,856)
(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)(432,859);;
s2 := ( 1,544)( 2,545)( 3,546)( 4,541)( 5,542)( 6,543)( 7,547)( 8,548)
( 9,549)( 10,562)( 11,563)( 12,564)( 13,559)( 14,560)( 15,561)( 16,565)
( 17,566)( 18,567)( 19,553)( 20,554)( 21,555)( 22,550)( 23,551)( 24,552)
( 25,556)( 26,557)( 27,558)( 28,571)( 29,572)( 30,573)( 31,568)( 32,569)
( 33,570)( 34,574)( 35,575)( 36,576)( 37,589)( 38,590)( 39,591)( 40,586)
( 41,587)( 42,588)( 43,592)( 44,593)( 45,594)( 46,580)( 47,581)( 48,582)
( 49,577)( 50,578)( 51,579)( 52,583)( 53,584)( 54,585)( 55,598)( 56,599)
( 57,600)( 58,595)( 59,596)( 60,597)( 61,601)( 62,602)( 63,603)( 64,616)
( 65,617)( 66,618)( 67,613)( 68,614)( 69,615)( 70,619)( 71,620)( 72,621)
( 73,607)( 74,608)( 75,609)( 76,604)( 77,605)( 78,606)( 79,610)( 80,611)
( 81,612)( 82,625)( 83,626)( 84,627)( 85,622)( 86,623)( 87,624)( 88,628)
( 89,629)( 90,630)( 91,643)( 92,644)( 93,645)( 94,640)( 95,641)( 96,642)
( 97,646)( 98,647)( 99,648)(100,634)(101,635)(102,636)(103,631)(104,632)
(105,633)(106,637)(107,638)(108,639)(109,436)(110,437)(111,438)(112,433)
(113,434)(114,435)(115,439)(116,440)(117,441)(118,454)(119,455)(120,456)
(121,451)(122,452)(123,453)(124,457)(125,458)(126,459)(127,445)(128,446)
(129,447)(130,442)(131,443)(132,444)(133,448)(134,449)(135,450)(136,463)
(137,464)(138,465)(139,460)(140,461)(141,462)(142,466)(143,467)(144,468)
(145,481)(146,482)(147,483)(148,478)(149,479)(150,480)(151,484)(152,485)
(153,486)(154,472)(155,473)(156,474)(157,469)(158,470)(159,471)(160,475)
(161,476)(162,477)(163,490)(164,491)(165,492)(166,487)(167,488)(168,489)
(169,493)(170,494)(171,495)(172,508)(173,509)(174,510)(175,505)(176,506)
(177,507)(178,511)(179,512)(180,513)(181,499)(182,500)(183,501)(184,496)
(185,497)(186,498)(187,502)(188,503)(189,504)(190,517)(191,518)(192,519)
(193,514)(194,515)(195,516)(196,520)(197,521)(198,522)(199,535)(200,536)
(201,537)(202,532)(203,533)(204,534)(205,538)(206,539)(207,540)(208,526)
(209,527)(210,528)(211,523)(212,524)(213,525)(214,529)(215,530)(216,531)
(217,760)(218,761)(219,762)(220,757)(221,758)(222,759)(223,763)(224,764)
(225,765)(226,778)(227,779)(228,780)(229,775)(230,776)(231,777)(232,781)
(233,782)(234,783)(235,769)(236,770)(237,771)(238,766)(239,767)(240,768)
(241,772)(242,773)(243,774)(244,787)(245,788)(246,789)(247,784)(248,785)
(249,786)(250,790)(251,791)(252,792)(253,805)(254,806)(255,807)(256,802)
(257,803)(258,804)(259,808)(260,809)(261,810)(262,796)(263,797)(264,798)
(265,793)(266,794)(267,795)(268,799)(269,800)(270,801)(271,814)(272,815)
(273,816)(274,811)(275,812)(276,813)(277,817)(278,818)(279,819)(280,832)
(281,833)(282,834)(283,829)(284,830)(285,831)(286,835)(287,836)(288,837)
(289,823)(290,824)(291,825)(292,820)(293,821)(294,822)(295,826)(296,827)
(297,828)(298,841)(299,842)(300,843)(301,838)(302,839)(303,840)(304,844)
(305,845)(306,846)(307,859)(308,860)(309,861)(310,856)(311,857)(312,858)
(313,862)(314,863)(315,864)(316,850)(317,851)(318,852)(319,847)(320,848)
(321,849)(322,853)(323,854)(324,855)(325,652)(326,653)(327,654)(328,649)
(329,650)(330,651)(331,655)(332,656)(333,657)(334,670)(335,671)(336,672)
(337,667)(338,668)(339,669)(340,673)(341,674)(342,675)(343,661)(344,662)
(345,663)(346,658)(347,659)(348,660)(349,664)(350,665)(351,666)(352,679)
(353,680)(354,681)(355,676)(356,677)(357,678)(358,682)(359,683)(360,684)
(361,697)(362,698)(363,699)(364,694)(365,695)(366,696)(367,700)(368,701)
(369,702)(370,688)(371,689)(372,690)(373,685)(374,686)(375,687)(376,691)
(377,692)(378,693)(379,706)(380,707)(381,708)(382,703)(383,704)(384,705)
(385,709)(386,710)(387,711)(388,724)(389,725)(390,726)(391,721)(392,722)
(393,723)(394,727)(395,728)(396,729)(397,715)(398,716)(399,717)(400,712)
(401,713)(402,714)(403,718)(404,719)(405,720)(406,733)(407,734)(408,735)
(409,730)(410,731)(411,732)(412,736)(413,737)(414,738)(415,751)(416,752)
(417,753)(418,748)(419,749)(420,750)(421,754)(422,755)(423,756)(424,742)
(425,743)(426,744)(427,739)(428,740)(429,741)(430,745)(431,746)(432,747);;
s3 := ( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)( 8,224)
( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)( 96,312)
( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)(448,664)
(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)(472,688)
(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,712)
(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)(504,720)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)(528,744)
(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)(544,814)
(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)(552,822)
(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)(560,830)
(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)(576,846)
(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)(584,854)
(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)(600,762)
(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)(608,770)
(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)(616,778)
(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)(632,794)
(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)(640,802)
(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)(648,810);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(864)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)
(116,144)(117,143)(118,154)(119,156)(120,155)(121,157)(122,159)(123,158)
(124,160)(125,162)(126,161)(127,145)(128,147)(129,146)(130,148)(131,150)
(132,149)(133,151)(134,153)(135,152)(163,190)(164,192)(165,191)(166,193)
(167,195)(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)
(175,211)(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)
(183,200)(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(218,219)
(221,222)(224,225)(226,235)(227,237)(228,236)(229,238)(230,240)(231,239)
(232,241)(233,243)(234,242)(245,246)(248,249)(251,252)(253,262)(254,264)
(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)(261,269)(272,273)
(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)(284,294)(285,293)
(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)(307,316)(308,318)
(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)(315,323)(325,352)
(326,354)(327,353)(328,355)(329,357)(330,356)(331,358)(332,360)(333,359)
(334,370)(335,372)(336,371)(337,373)(338,375)(339,374)(340,376)(341,378)
(342,377)(343,361)(344,363)(345,362)(346,364)(347,366)(348,365)(349,367)
(350,369)(351,368)(379,406)(380,408)(381,407)(382,409)(383,411)(384,410)
(385,412)(386,414)(387,413)(388,424)(389,426)(390,425)(391,427)(392,429)
(393,428)(394,430)(395,432)(396,431)(397,415)(398,417)(399,416)(400,418)
(401,420)(402,419)(403,421)(404,423)(405,422)(433,460)(434,462)(435,461)
(436,463)(437,465)(438,464)(439,466)(440,468)(441,467)(442,478)(443,480)
(444,479)(445,481)(446,483)(447,482)(448,484)(449,486)(450,485)(451,469)
(452,471)(453,470)(454,472)(455,474)(456,473)(457,475)(458,477)(459,476)
(487,514)(488,516)(489,515)(490,517)(491,519)(492,518)(493,520)(494,522)
(495,521)(496,532)(497,534)(498,533)(499,535)(500,537)(501,536)(502,538)
(503,540)(504,539)(505,523)(506,525)(507,524)(508,526)(509,528)(510,527)
(511,529)(512,531)(513,530)(542,543)(545,546)(548,549)(550,559)(551,561)
(552,560)(553,562)(554,564)(555,563)(556,565)(557,567)(558,566)(569,570)
(572,573)(575,576)(577,586)(578,588)(579,587)(580,589)(581,591)(582,590)
(583,592)(584,594)(585,593)(596,597)(599,600)(602,603)(604,613)(605,615)
(606,614)(607,616)(608,618)(609,617)(610,619)(611,621)(612,620)(623,624)
(626,627)(629,630)(631,640)(632,642)(633,641)(634,643)(635,645)(636,644)
(637,646)(638,648)(639,647)(649,676)(650,678)(651,677)(652,679)(653,681)
(654,680)(655,682)(656,684)(657,683)(658,694)(659,696)(660,695)(661,697)
(662,699)(663,698)(664,700)(665,702)(666,701)(667,685)(668,687)(669,686)
(670,688)(671,690)(672,689)(673,691)(674,693)(675,692)(703,730)(704,732)
(705,731)(706,733)(707,735)(708,734)(709,736)(710,738)(711,737)(712,748)
(713,750)(714,749)(715,751)(716,753)(717,752)(718,754)(719,756)(720,755)
(721,739)(722,741)(723,740)(724,742)(725,744)(726,743)(727,745)(728,747)
(729,746)(758,759)(761,762)(764,765)(766,775)(767,777)(768,776)(769,778)
(770,780)(771,779)(772,781)(773,783)(774,782)(785,786)(788,789)(791,792)
(793,802)(794,804)(795,803)(796,805)(797,807)(798,806)(799,808)(800,810)
(801,809)(812,813)(815,816)(818,819)(820,829)(821,831)(822,830)(823,832)
(824,834)(825,833)(826,835)(827,837)(828,836)(839,840)(842,843)(845,846)
(847,856)(848,858)(849,857)(850,859)(851,861)(852,860)(853,862)(854,864)
(855,863);
s1 := Sym(864)!( 1,442)( 2,443)( 3,444)( 4,450)( 5,448)( 6,449)( 7,446)
( 8,447)( 9,445)( 10,433)( 11,434)( 12,435)( 13,441)( 14,439)( 15,440)
( 16,437)( 17,438)( 18,436)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)
( 24,458)( 25,455)( 26,456)( 27,454)( 28,469)( 29,470)( 30,471)( 31,477)
( 32,475)( 33,476)( 34,473)( 35,474)( 36,472)( 37,460)( 38,461)( 39,462)
( 40,468)( 41,466)( 42,467)( 43,464)( 44,465)( 45,463)( 46,478)( 47,479)
( 48,480)( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,496)
( 56,497)( 57,498)( 58,504)( 59,502)( 60,503)( 61,500)( 62,501)( 63,499)
( 64,487)( 65,488)( 66,489)( 67,495)( 68,493)( 69,494)( 70,491)( 71,492)
( 72,490)( 73,505)( 74,506)( 75,507)( 76,513)( 77,511)( 78,512)( 79,509)
( 80,510)( 81,508)( 82,523)( 83,524)( 84,525)( 85,531)( 86,529)( 87,530)
( 88,527)( 89,528)( 90,526)( 91,514)( 92,515)( 93,516)( 94,522)( 95,520)
( 96,521)( 97,518)( 98,519)( 99,517)(100,532)(101,533)(102,534)(103,540)
(104,538)(105,539)(106,536)(107,537)(108,535)(109,550)(110,551)(111,552)
(112,558)(113,556)(114,557)(115,554)(116,555)(117,553)(118,541)(119,542)
(120,543)(121,549)(122,547)(123,548)(124,545)(125,546)(126,544)(127,559)
(128,560)(129,561)(130,567)(131,565)(132,566)(133,563)(134,564)(135,562)
(136,577)(137,578)(138,579)(139,585)(140,583)(141,584)(142,581)(143,582)
(144,580)(145,568)(146,569)(147,570)(148,576)(149,574)(150,575)(151,572)
(152,573)(153,571)(154,586)(155,587)(156,588)(157,594)(158,592)(159,593)
(160,590)(161,591)(162,589)(163,604)(164,605)(165,606)(166,612)(167,610)
(168,611)(169,608)(170,609)(171,607)(172,595)(173,596)(174,597)(175,603)
(176,601)(177,602)(178,599)(179,600)(180,598)(181,613)(182,614)(183,615)
(184,621)(185,619)(186,620)(187,617)(188,618)(189,616)(190,631)(191,632)
(192,633)(193,639)(194,637)(195,638)(196,635)(197,636)(198,634)(199,622)
(200,623)(201,624)(202,630)(203,628)(204,629)(205,626)(206,627)(207,625)
(208,640)(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)
(216,643)(217,658)(218,659)(219,660)(220,666)(221,664)(222,665)(223,662)
(224,663)(225,661)(226,649)(227,650)(228,651)(229,657)(230,655)(231,656)
(232,653)(233,654)(234,652)(235,667)(236,668)(237,669)(238,675)(239,673)
(240,674)(241,671)(242,672)(243,670)(244,685)(245,686)(246,687)(247,693)
(248,691)(249,692)(250,689)(251,690)(252,688)(253,676)(254,677)(255,678)
(256,684)(257,682)(258,683)(259,680)(260,681)(261,679)(262,694)(263,695)
(264,696)(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,712)
(272,713)(273,714)(274,720)(275,718)(276,719)(277,716)(278,717)(279,715)
(280,703)(281,704)(282,705)(283,711)(284,709)(285,710)(286,707)(287,708)
(288,706)(289,721)(290,722)(291,723)(292,729)(293,727)(294,728)(295,725)
(296,726)(297,724)(298,739)(299,740)(300,741)(301,747)(302,745)(303,746)
(304,743)(305,744)(306,742)(307,730)(308,731)(309,732)(310,738)(311,736)
(312,737)(313,734)(314,735)(315,733)(316,748)(317,749)(318,750)(319,756)
(320,754)(321,755)(322,752)(323,753)(324,751)(325,766)(326,767)(327,768)
(328,774)(329,772)(330,773)(331,770)(332,771)(333,769)(334,757)(335,758)
(336,759)(337,765)(338,763)(339,764)(340,761)(341,762)(342,760)(343,775)
(344,776)(345,777)(346,783)(347,781)(348,782)(349,779)(350,780)(351,778)
(352,793)(353,794)(354,795)(355,801)(356,799)(357,800)(358,797)(359,798)
(360,796)(361,784)(362,785)(363,786)(364,792)(365,790)(366,791)(367,788)
(368,789)(369,787)(370,802)(371,803)(372,804)(373,810)(374,808)(375,809)
(376,806)(377,807)(378,805)(379,820)(380,821)(381,822)(382,828)(383,826)
(384,827)(385,824)(386,825)(387,823)(388,811)(389,812)(390,813)(391,819)
(392,817)(393,818)(394,815)(395,816)(396,814)(397,829)(398,830)(399,831)
(400,837)(401,835)(402,836)(403,833)(404,834)(405,832)(406,847)(407,848)
(408,849)(409,855)(410,853)(411,854)(412,851)(413,852)(414,850)(415,838)
(416,839)(417,840)(418,846)(419,844)(420,845)(421,842)(422,843)(423,841)
(424,856)(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)
(432,859);
s2 := Sym(864)!( 1,544)( 2,545)( 3,546)( 4,541)( 5,542)( 6,543)( 7,547)
( 8,548)( 9,549)( 10,562)( 11,563)( 12,564)( 13,559)( 14,560)( 15,561)
( 16,565)( 17,566)( 18,567)( 19,553)( 20,554)( 21,555)( 22,550)( 23,551)
( 24,552)( 25,556)( 26,557)( 27,558)( 28,571)( 29,572)( 30,573)( 31,568)
( 32,569)( 33,570)( 34,574)( 35,575)( 36,576)( 37,589)( 38,590)( 39,591)
( 40,586)( 41,587)( 42,588)( 43,592)( 44,593)( 45,594)( 46,580)( 47,581)
( 48,582)( 49,577)( 50,578)( 51,579)( 52,583)( 53,584)( 54,585)( 55,598)
( 56,599)( 57,600)( 58,595)( 59,596)( 60,597)( 61,601)( 62,602)( 63,603)
( 64,616)( 65,617)( 66,618)( 67,613)( 68,614)( 69,615)( 70,619)( 71,620)
( 72,621)( 73,607)( 74,608)( 75,609)( 76,604)( 77,605)( 78,606)( 79,610)
( 80,611)( 81,612)( 82,625)( 83,626)( 84,627)( 85,622)( 86,623)( 87,624)
( 88,628)( 89,629)( 90,630)( 91,643)( 92,644)( 93,645)( 94,640)( 95,641)
( 96,642)( 97,646)( 98,647)( 99,648)(100,634)(101,635)(102,636)(103,631)
(104,632)(105,633)(106,637)(107,638)(108,639)(109,436)(110,437)(111,438)
(112,433)(113,434)(114,435)(115,439)(116,440)(117,441)(118,454)(119,455)
(120,456)(121,451)(122,452)(123,453)(124,457)(125,458)(126,459)(127,445)
(128,446)(129,447)(130,442)(131,443)(132,444)(133,448)(134,449)(135,450)
(136,463)(137,464)(138,465)(139,460)(140,461)(141,462)(142,466)(143,467)
(144,468)(145,481)(146,482)(147,483)(148,478)(149,479)(150,480)(151,484)
(152,485)(153,486)(154,472)(155,473)(156,474)(157,469)(158,470)(159,471)
(160,475)(161,476)(162,477)(163,490)(164,491)(165,492)(166,487)(167,488)
(168,489)(169,493)(170,494)(171,495)(172,508)(173,509)(174,510)(175,505)
(176,506)(177,507)(178,511)(179,512)(180,513)(181,499)(182,500)(183,501)
(184,496)(185,497)(186,498)(187,502)(188,503)(189,504)(190,517)(191,518)
(192,519)(193,514)(194,515)(195,516)(196,520)(197,521)(198,522)(199,535)
(200,536)(201,537)(202,532)(203,533)(204,534)(205,538)(206,539)(207,540)
(208,526)(209,527)(210,528)(211,523)(212,524)(213,525)(214,529)(215,530)
(216,531)(217,760)(218,761)(219,762)(220,757)(221,758)(222,759)(223,763)
(224,764)(225,765)(226,778)(227,779)(228,780)(229,775)(230,776)(231,777)
(232,781)(233,782)(234,783)(235,769)(236,770)(237,771)(238,766)(239,767)
(240,768)(241,772)(242,773)(243,774)(244,787)(245,788)(246,789)(247,784)
(248,785)(249,786)(250,790)(251,791)(252,792)(253,805)(254,806)(255,807)
(256,802)(257,803)(258,804)(259,808)(260,809)(261,810)(262,796)(263,797)
(264,798)(265,793)(266,794)(267,795)(268,799)(269,800)(270,801)(271,814)
(272,815)(273,816)(274,811)(275,812)(276,813)(277,817)(278,818)(279,819)
(280,832)(281,833)(282,834)(283,829)(284,830)(285,831)(286,835)(287,836)
(288,837)(289,823)(290,824)(291,825)(292,820)(293,821)(294,822)(295,826)
(296,827)(297,828)(298,841)(299,842)(300,843)(301,838)(302,839)(303,840)
(304,844)(305,845)(306,846)(307,859)(308,860)(309,861)(310,856)(311,857)
(312,858)(313,862)(314,863)(315,864)(316,850)(317,851)(318,852)(319,847)
(320,848)(321,849)(322,853)(323,854)(324,855)(325,652)(326,653)(327,654)
(328,649)(329,650)(330,651)(331,655)(332,656)(333,657)(334,670)(335,671)
(336,672)(337,667)(338,668)(339,669)(340,673)(341,674)(342,675)(343,661)
(344,662)(345,663)(346,658)(347,659)(348,660)(349,664)(350,665)(351,666)
(352,679)(353,680)(354,681)(355,676)(356,677)(357,678)(358,682)(359,683)
(360,684)(361,697)(362,698)(363,699)(364,694)(365,695)(366,696)(367,700)
(368,701)(369,702)(370,688)(371,689)(372,690)(373,685)(374,686)(375,687)
(376,691)(377,692)(378,693)(379,706)(380,707)(381,708)(382,703)(383,704)
(384,705)(385,709)(386,710)(387,711)(388,724)(389,725)(390,726)(391,721)
(392,722)(393,723)(394,727)(395,728)(396,729)(397,715)(398,716)(399,717)
(400,712)(401,713)(402,714)(403,718)(404,719)(405,720)(406,733)(407,734)
(408,735)(409,730)(410,731)(411,732)(412,736)(413,737)(414,738)(415,751)
(416,752)(417,753)(418,748)(419,749)(420,750)(421,754)(422,755)(423,756)
(424,742)(425,743)(426,744)(427,739)(428,740)(429,741)(430,745)(431,746)
(432,747);
s3 := Sym(864)!( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)
( 8,224)( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)
( 96,312)( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)
(448,664)(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)
(472,688)(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,712)(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)
(504,720)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)
(528,744)(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)
(544,814)(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)
(552,822)(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)
(560,830)(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)
(576,846)(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)
(584,854)(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)
(600,762)(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)
(608,770)(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)
(616,778)(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)
(632,794)(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)
(640,802)(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)
(648,810);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope