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# Polytope of Type {4,6,12}

Atlas Canonical Name : {4,6,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,17531)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 36, 108, 36
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 : {4,6,6}*864a, {2,6,12}*864c
3-fold quotients : {4,6,12}*576c
4-fold quotients : {2,6,6}*432c
6-fold quotients : {4,6,6}*288b, {2,6,12}*288c
8-fold quotients : {2,3,6}*216
9-fold quotients : {4,6,4}*192a
12-fold quotients : {2,6,6}*144c
18-fold quotients : {2,6,4}*96a, {4,6,2}*96a
24-fold quotients : {2,3,6}*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 := (  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);;
s1 := (  1,541)(  2,542)(  3,543)(  4,547)(  5,548)(  6,549)(  7,544)(  8,545)
(  9,546)( 10,559)( 11,560)( 12,561)( 13,565)( 14,566)( 15,567)( 16,562)
( 17,563)( 18,564)( 19,550)( 20,551)( 21,552)( 22,556)( 23,557)( 24,558)
( 25,553)( 26,554)( 27,555)( 28,568)( 29,569)( 30,570)( 31,574)( 32,575)
( 33,576)( 34,571)( 35,572)( 36,573)( 37,586)( 38,587)( 39,588)( 40,592)
( 41,593)( 42,594)( 43,589)( 44,590)( 45,591)( 46,577)( 47,578)( 48,579)
( 49,583)( 50,584)( 51,585)( 52,580)( 53,581)( 54,582)( 55,595)( 56,596)
( 57,597)( 58,601)( 59,602)( 60,603)( 61,598)( 62,599)( 63,600)( 64,613)
( 65,614)( 66,615)( 67,619)( 68,620)( 69,621)( 70,616)( 71,617)( 72,618)
( 73,604)( 74,605)( 75,606)( 76,610)( 77,611)( 78,612)( 79,607)( 80,608)
( 81,609)( 82,622)( 83,623)( 84,624)( 85,628)( 86,629)( 87,630)( 88,625)
( 89,626)( 90,627)( 91,640)( 92,641)( 93,642)( 94,646)( 95,647)( 96,648)
( 97,643)( 98,644)( 99,645)(100,631)(101,632)(102,633)(103,637)(104,638)
(105,639)(106,634)(107,635)(108,636)(109,433)(110,434)(111,435)(112,439)
(113,440)(114,441)(115,436)(116,437)(117,438)(118,451)(119,452)(120,453)
(121,457)(122,458)(123,459)(124,454)(125,455)(126,456)(127,442)(128,443)
(129,444)(130,448)(131,449)(132,450)(133,445)(134,446)(135,447)(136,460)
(137,461)(138,462)(139,466)(140,467)(141,468)(142,463)(143,464)(144,465)
(145,478)(146,479)(147,480)(148,484)(149,485)(150,486)(151,481)(152,482)
(153,483)(154,469)(155,470)(156,471)(157,475)(158,476)(159,477)(160,472)
(161,473)(162,474)(163,487)(164,488)(165,489)(166,493)(167,494)(168,495)
(169,490)(170,491)(171,492)(172,505)(173,506)(174,507)(175,511)(176,512)
(177,513)(178,508)(179,509)(180,510)(181,496)(182,497)(183,498)(184,502)
(185,503)(186,504)(187,499)(188,500)(189,501)(190,514)(191,515)(192,516)
(193,520)(194,521)(195,522)(196,517)(197,518)(198,519)(199,532)(200,533)
(201,534)(202,538)(203,539)(204,540)(205,535)(206,536)(207,537)(208,523)
(209,524)(210,525)(211,529)(212,530)(213,531)(214,526)(215,527)(216,528)
(217,757)(218,758)(219,759)(220,763)(221,764)(222,765)(223,760)(224,761)
(225,762)(226,775)(227,776)(228,777)(229,781)(230,782)(231,783)(232,778)
(233,779)(234,780)(235,766)(236,767)(237,768)(238,772)(239,773)(240,774)
(241,769)(242,770)(243,771)(244,784)(245,785)(246,786)(247,790)(248,791)
(249,792)(250,787)(251,788)(252,789)(253,802)(254,803)(255,804)(256,808)
(257,809)(258,810)(259,805)(260,806)(261,807)(262,793)(263,794)(264,795)
(265,799)(266,800)(267,801)(268,796)(269,797)(270,798)(271,811)(272,812)
(273,813)(274,817)(275,818)(276,819)(277,814)(278,815)(279,816)(280,829)
(281,830)(282,831)(283,835)(284,836)(285,837)(286,832)(287,833)(288,834)
(289,820)(290,821)(291,822)(292,826)(293,827)(294,828)(295,823)(296,824)
(297,825)(298,838)(299,839)(300,840)(301,844)(302,845)(303,846)(304,841)
(305,842)(306,843)(307,856)(308,857)(309,858)(310,862)(311,863)(312,864)
(313,859)(314,860)(315,861)(316,847)(317,848)(318,849)(319,853)(320,854)
(321,855)(322,850)(323,851)(324,852)(325,649)(326,650)(327,651)(328,655)
(329,656)(330,657)(331,652)(332,653)(333,654)(334,667)(335,668)(336,669)
(337,673)(338,674)(339,675)(340,670)(341,671)(342,672)(343,658)(344,659)
(345,660)(346,664)(347,665)(348,666)(349,661)(350,662)(351,663)(352,676)
(353,677)(354,678)(355,682)(356,683)(357,684)(358,679)(359,680)(360,681)
(361,694)(362,695)(363,696)(364,700)(365,701)(366,702)(367,697)(368,698)
(369,699)(370,685)(371,686)(372,687)(373,691)(374,692)(375,693)(376,688)
(377,689)(378,690)(379,703)(380,704)(381,705)(382,709)(383,710)(384,711)
(385,706)(386,707)(387,708)(388,721)(389,722)(390,723)(391,727)(392,728)
(393,729)(394,724)(395,725)(396,726)(397,712)(398,713)(399,714)(400,718)
(401,719)(402,720)(403,715)(404,716)(405,717)(406,730)(407,731)(408,732)
(409,736)(410,737)(411,738)(412,733)(413,734)(414,735)(415,748)(416,749)
(417,750)(418,754)(419,755)(420,756)(421,751)(422,752)(423,753)(424,739)
(425,740)(426,741)(427,745)(428,746)(429,747)(430,742)(431,743)(432,744);;
s2 := (  1,455)(  2,456)(  3,454)(  4,453)(  5,451)(  6,452)(  7,457)(  8,458)
(  9,459)( 10,446)( 11,447)( 12,445)( 13,444)( 14,442)( 15,443)( 16,448)
( 17,449)( 18,450)( 19,437)( 20,438)( 21,436)( 22,435)( 23,433)( 24,434)
( 25,439)( 26,440)( 27,441)( 28,482)( 29,483)( 30,481)( 31,480)( 32,478)
( 33,479)( 34,484)( 35,485)( 36,486)( 37,473)( 38,474)( 39,472)( 40,471)
( 41,469)( 42,470)( 43,475)( 44,476)( 45,477)( 46,464)( 47,465)( 48,463)
( 49,462)( 50,460)( 51,461)( 52,466)( 53,467)( 54,468)( 55,509)( 56,510)
( 57,508)( 58,507)( 59,505)( 60,506)( 61,511)( 62,512)( 63,513)( 64,500)
( 65,501)( 66,499)( 67,498)( 68,496)( 69,497)( 70,502)( 71,503)( 72,504)
( 73,491)( 74,492)( 75,490)( 76,489)( 77,487)( 78,488)( 79,493)( 80,494)
( 81,495)( 82,536)( 83,537)( 84,535)( 85,534)( 86,532)( 87,533)( 88,538)
( 89,539)( 90,540)( 91,527)( 92,528)( 93,526)( 94,525)( 95,523)( 96,524)
( 97,529)( 98,530)( 99,531)(100,518)(101,519)(102,517)(103,516)(104,514)
(105,515)(106,520)(107,521)(108,522)(109,563)(110,564)(111,562)(112,561)
(113,559)(114,560)(115,565)(116,566)(117,567)(118,554)(119,555)(120,553)
(121,552)(122,550)(123,551)(124,556)(125,557)(126,558)(127,545)(128,546)
(129,544)(130,543)(131,541)(132,542)(133,547)(134,548)(135,549)(136,590)
(137,591)(138,589)(139,588)(140,586)(141,587)(142,592)(143,593)(144,594)
(145,581)(146,582)(147,580)(148,579)(149,577)(150,578)(151,583)(152,584)
(153,585)(154,572)(155,573)(156,571)(157,570)(158,568)(159,569)(160,574)
(161,575)(162,576)(163,617)(164,618)(165,616)(166,615)(167,613)(168,614)
(169,619)(170,620)(171,621)(172,608)(173,609)(174,607)(175,606)(176,604)
(177,605)(178,610)(179,611)(180,612)(181,599)(182,600)(183,598)(184,597)
(185,595)(186,596)(187,601)(188,602)(189,603)(190,644)(191,645)(192,643)
(193,642)(194,640)(195,641)(196,646)(197,647)(198,648)(199,635)(200,636)
(201,634)(202,633)(203,631)(204,632)(205,637)(206,638)(207,639)(208,626)
(209,627)(210,625)(211,624)(212,622)(213,623)(214,628)(215,629)(216,630)
(217,671)(218,672)(219,670)(220,669)(221,667)(222,668)(223,673)(224,674)
(225,675)(226,662)(227,663)(228,661)(229,660)(230,658)(231,659)(232,664)
(233,665)(234,666)(235,653)(236,654)(237,652)(238,651)(239,649)(240,650)
(241,655)(242,656)(243,657)(244,698)(245,699)(246,697)(247,696)(248,694)
(249,695)(250,700)(251,701)(252,702)(253,689)(254,690)(255,688)(256,687)
(257,685)(258,686)(259,691)(260,692)(261,693)(262,680)(263,681)(264,679)
(265,678)(266,676)(267,677)(268,682)(269,683)(270,684)(271,725)(272,726)
(273,724)(274,723)(275,721)(276,722)(277,727)(278,728)(279,729)(280,716)
(281,717)(282,715)(283,714)(284,712)(285,713)(286,718)(287,719)(288,720)
(289,707)(290,708)(291,706)(292,705)(293,703)(294,704)(295,709)(296,710)
(297,711)(298,752)(299,753)(300,751)(301,750)(302,748)(303,749)(304,754)
(305,755)(306,756)(307,743)(308,744)(309,742)(310,741)(311,739)(312,740)
(313,745)(314,746)(315,747)(316,734)(317,735)(318,733)(319,732)(320,730)
(321,731)(322,736)(323,737)(324,738)(325,779)(326,780)(327,778)(328,777)
(329,775)(330,776)(331,781)(332,782)(333,783)(334,770)(335,771)(336,769)
(337,768)(338,766)(339,767)(340,772)(341,773)(342,774)(343,761)(344,762)
(345,760)(346,759)(347,757)(348,758)(349,763)(350,764)(351,765)(352,806)
(353,807)(354,805)(355,804)(356,802)(357,803)(358,808)(359,809)(360,810)
(361,797)(362,798)(363,796)(364,795)(365,793)(366,794)(367,799)(368,800)
(369,801)(370,788)(371,789)(372,787)(373,786)(374,784)(375,785)(376,790)
(377,791)(378,792)(379,833)(380,834)(381,832)(382,831)(383,829)(384,830)
(385,835)(386,836)(387,837)(388,824)(389,825)(390,823)(391,822)(392,820)
(393,821)(394,826)(395,827)(396,828)(397,815)(398,816)(399,814)(400,813)
(401,811)(402,812)(403,817)(404,818)(405,819)(406,860)(407,861)(408,859)
(409,858)(410,856)(411,857)(412,862)(413,863)(414,864)(415,851)(416,852)
(417,850)(418,849)(419,847)(420,848)(421,853)(422,854)(423,855)(424,842)
(425,843)(426,841)(427,840)(428,838)(429,839)(430,844)(431,845)(432,846);;
s3 := (  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);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2 >;

```
References : None.
to this polytope