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Polytope of Type {12,6,4}

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