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# Polytope of Type {6,12,12}

Atlas Canonical Name : {6,12,12}*1728g
if this polytope has a name.
Group : SmallGroup(1728,38778)
Rank : 4
Schlafli Type : {6,12,12}
Number of vertices, edges, etc : 6, 36, 72, 12
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 : {6,6,12}*864e, {6,12,6}*864f
3-fold quotients : {6,4,12}*576, {6,12,4}*576c
4-fold quotients : {3,6,12}*432b, {6,6,6}*432g
6-fold quotients : {6,2,12}*288, {6,4,6}*288, {6,6,4}*288c, {6,12,2}*288c
8-fold quotients : {3,6,6}*216b
9-fold quotients : {2,4,12}*192a, {6,4,4}*192
12-fold quotients : {3,2,12}*144, {3,6,4}*144, {6,2,6}*144, {6,6,2}*144c
18-fold quotients : {2,2,12}*96, {2,4,6}*96a, {6,2,4}*96, {6,4,2}*96a
24-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {3,2,4}*48, {2,2,6}*48, {6,2,2}*48
48-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,2,3}*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 := (  1,649)(  2,651)(  3,650)(  4,652)(  5,654)(  6,653)(  7,655)(  8,657)
(  9,656)( 10,667)( 11,669)( 12,668)( 13,670)( 14,672)( 15,671)( 16,673)
( 17,675)( 18,674)( 19,658)( 20,660)( 21,659)( 22,661)( 23,663)( 24,662)
( 25,664)( 26,666)( 27,665)( 28,676)( 29,678)( 30,677)( 31,679)( 32,681)
( 33,680)( 34,682)( 35,684)( 36,683)( 37,694)( 38,696)( 39,695)( 40,697)
( 41,699)( 42,698)( 43,700)( 44,702)( 45,701)( 46,685)( 47,687)( 48,686)
( 49,688)( 50,690)( 51,689)( 52,691)( 53,693)( 54,692)( 55,703)( 56,705)
( 57,704)( 58,706)( 59,708)( 60,707)( 61,709)( 62,711)( 63,710)( 64,721)
( 65,723)( 66,722)( 67,724)( 68,726)( 69,725)( 70,727)( 71,729)( 72,728)
( 73,712)( 74,714)( 75,713)( 76,715)( 77,717)( 78,716)( 79,718)( 80,720)
( 81,719)( 82,730)( 83,732)( 84,731)( 85,733)( 86,735)( 87,734)( 88,736)
( 89,738)( 90,737)( 91,748)( 92,750)( 93,749)( 94,751)( 95,753)( 96,752)
( 97,754)( 98,756)( 99,755)(100,739)(101,741)(102,740)(103,742)(104,744)
(105,743)(106,745)(107,747)(108,746)(109,757)(110,759)(111,758)(112,760)
(113,762)(114,761)(115,763)(116,765)(117,764)(118,775)(119,777)(120,776)
(121,778)(122,780)(123,779)(124,781)(125,783)(126,782)(127,766)(128,768)
(129,767)(130,769)(131,771)(132,770)(133,772)(134,774)(135,773)(136,784)
(137,786)(138,785)(139,787)(140,789)(141,788)(142,790)(143,792)(144,791)
(145,802)(146,804)(147,803)(148,805)(149,807)(150,806)(151,808)(152,810)
(153,809)(154,793)(155,795)(156,794)(157,796)(158,798)(159,797)(160,799)
(161,801)(162,800)(163,811)(164,813)(165,812)(166,814)(167,816)(168,815)
(169,817)(170,819)(171,818)(172,829)(173,831)(174,830)(175,832)(176,834)
(177,833)(178,835)(179,837)(180,836)(181,820)(182,822)(183,821)(184,823)
(185,825)(186,824)(187,826)(188,828)(189,827)(190,838)(191,840)(192,839)
(193,841)(194,843)(195,842)(196,844)(197,846)(198,845)(199,856)(200,858)
(201,857)(202,859)(203,861)(204,860)(205,862)(206,864)(207,863)(208,847)
(209,849)(210,848)(211,850)(212,852)(213,851)(214,853)(215,855)(216,854)
(217,460)(218,462)(219,461)(220,463)(221,465)(222,464)(223,466)(224,468)
(225,467)(226,478)(227,480)(228,479)(229,481)(230,483)(231,482)(232,484)
(233,486)(234,485)(235,469)(236,471)(237,470)(238,472)(239,474)(240,473)
(241,475)(242,477)(243,476)(244,433)(245,435)(246,434)(247,436)(248,438)
(249,437)(250,439)(251,441)(252,440)(253,451)(254,453)(255,452)(256,454)
(257,456)(258,455)(259,457)(260,459)(261,458)(262,442)(263,444)(264,443)
(265,445)(266,447)(267,446)(268,448)(269,450)(270,449)(271,514)(272,516)
(273,515)(274,517)(275,519)(276,518)(277,520)(278,522)(279,521)(280,532)
(281,534)(282,533)(283,535)(284,537)(285,536)(286,538)(287,540)(288,539)
(289,523)(290,525)(291,524)(292,526)(293,528)(294,527)(295,529)(296,531)
(297,530)(298,487)(299,489)(300,488)(301,490)(302,492)(303,491)(304,493)
(305,495)(306,494)(307,505)(308,507)(309,506)(310,508)(311,510)(312,509)
(313,511)(314,513)(315,512)(316,496)(317,498)(318,497)(319,499)(320,501)
(321,500)(322,502)(323,504)(324,503)(325,568)(326,570)(327,569)(328,571)
(329,573)(330,572)(331,574)(332,576)(333,575)(334,586)(335,588)(336,587)
(337,589)(338,591)(339,590)(340,592)(341,594)(342,593)(343,577)(344,579)
(345,578)(346,580)(347,582)(348,581)(349,583)(350,585)(351,584)(352,541)
(353,543)(354,542)(355,544)(356,546)(357,545)(358,547)(359,549)(360,548)
(361,559)(362,561)(363,560)(364,562)(365,564)(366,563)(367,565)(368,567)
(369,566)(370,550)(371,552)(372,551)(373,553)(374,555)(375,554)(376,556)
(377,558)(378,557)(379,622)(380,624)(381,623)(382,625)(383,627)(384,626)
(385,628)(386,630)(387,629)(388,640)(389,642)(390,641)(391,643)(392,645)
(393,644)(394,646)(395,648)(396,647)(397,631)(398,633)(399,632)(400,634)
(401,636)(402,635)(403,637)(404,639)(405,638)(406,595)(407,597)(408,596)
(409,598)(410,600)(411,599)(412,601)(413,603)(414,602)(415,613)(416,615)
(417,614)(418,616)(419,618)(420,617)(421,619)(422,621)(423,620)(424,604)
(425,606)(426,605)(427,607)(428,609)(429,608)(430,610)(431,612)(432,611);;
s1 := (  1,767)(  2,766)(  3,768)(  4,770)(  5,769)(  6,771)(  7,773)(  8,772)
(  9,774)( 10,758)( 11,757)( 12,759)( 13,761)( 14,760)( 15,762)( 16,764)
( 17,763)( 18,765)( 19,776)( 20,775)( 21,777)( 22,779)( 23,778)( 24,780)
( 25,782)( 26,781)( 27,783)( 28,794)( 29,793)( 30,795)( 31,797)( 32,796)
( 33,798)( 34,800)( 35,799)( 36,801)( 37,785)( 38,784)( 39,786)( 40,788)
( 41,787)( 42,789)( 43,791)( 44,790)( 45,792)( 46,803)( 47,802)( 48,804)
( 49,806)( 50,805)( 51,807)( 52,809)( 53,808)( 54,810)( 55,821)( 56,820)
( 57,822)( 58,824)( 59,823)( 60,825)( 61,827)( 62,826)( 63,828)( 64,812)
( 65,811)( 66,813)( 67,815)( 68,814)( 69,816)( 70,818)( 71,817)( 72,819)
( 73,830)( 74,829)( 75,831)( 76,833)( 77,832)( 78,834)( 79,836)( 80,835)
( 81,837)( 82,848)( 83,847)( 84,849)( 85,851)( 86,850)( 87,852)( 88,854)
( 89,853)( 90,855)( 91,839)( 92,838)( 93,840)( 94,842)( 95,841)( 96,843)
( 97,845)( 98,844)( 99,846)(100,857)(101,856)(102,858)(103,860)(104,859)
(105,861)(106,863)(107,862)(108,864)(109,659)(110,658)(111,660)(112,662)
(113,661)(114,663)(115,665)(116,664)(117,666)(118,650)(119,649)(120,651)
(121,653)(122,652)(123,654)(124,656)(125,655)(126,657)(127,668)(128,667)
(129,669)(130,671)(131,670)(132,672)(133,674)(134,673)(135,675)(136,686)
(137,685)(138,687)(139,689)(140,688)(141,690)(142,692)(143,691)(144,693)
(145,677)(146,676)(147,678)(148,680)(149,679)(150,681)(151,683)(152,682)
(153,684)(154,695)(155,694)(156,696)(157,698)(158,697)(159,699)(160,701)
(161,700)(162,702)(163,713)(164,712)(165,714)(166,716)(167,715)(168,717)
(169,719)(170,718)(171,720)(172,704)(173,703)(174,705)(175,707)(176,706)
(177,708)(178,710)(179,709)(180,711)(181,722)(182,721)(183,723)(184,725)
(185,724)(186,726)(187,728)(188,727)(189,729)(190,740)(191,739)(192,741)
(193,743)(194,742)(195,744)(196,746)(197,745)(198,747)(199,731)(200,730)
(201,732)(202,734)(203,733)(204,735)(205,737)(206,736)(207,738)(208,749)
(209,748)(210,750)(211,752)(212,751)(213,753)(214,755)(215,754)(216,756)
(217,578)(218,577)(219,579)(220,581)(221,580)(222,582)(223,584)(224,583)
(225,585)(226,569)(227,568)(228,570)(229,572)(230,571)(231,573)(232,575)
(233,574)(234,576)(235,587)(236,586)(237,588)(238,590)(239,589)(240,591)
(241,593)(242,592)(243,594)(244,551)(245,550)(246,552)(247,554)(248,553)
(249,555)(250,557)(251,556)(252,558)(253,542)(254,541)(255,543)(256,545)
(257,544)(258,546)(259,548)(260,547)(261,549)(262,560)(263,559)(264,561)
(265,563)(266,562)(267,564)(268,566)(269,565)(270,567)(271,632)(272,631)
(273,633)(274,635)(275,634)(276,636)(277,638)(278,637)(279,639)(280,623)
(281,622)(282,624)(283,626)(284,625)(285,627)(286,629)(287,628)(288,630)
(289,641)(290,640)(291,642)(292,644)(293,643)(294,645)(295,647)(296,646)
(297,648)(298,605)(299,604)(300,606)(301,608)(302,607)(303,609)(304,611)
(305,610)(306,612)(307,596)(308,595)(309,597)(310,599)(311,598)(312,600)
(313,602)(314,601)(315,603)(316,614)(317,613)(318,615)(319,617)(320,616)
(321,618)(322,620)(323,619)(324,621)(325,470)(326,469)(327,471)(328,473)
(329,472)(330,474)(331,476)(332,475)(333,477)(334,461)(335,460)(336,462)
(337,464)(338,463)(339,465)(340,467)(341,466)(342,468)(343,479)(344,478)
(345,480)(346,482)(347,481)(348,483)(349,485)(350,484)(351,486)(352,443)
(353,442)(354,444)(355,446)(356,445)(357,447)(358,449)(359,448)(360,450)
(361,434)(362,433)(363,435)(364,437)(365,436)(366,438)(367,440)(368,439)
(369,441)(370,452)(371,451)(372,453)(373,455)(374,454)(375,456)(376,458)
(377,457)(378,459)(379,524)(380,523)(381,525)(382,527)(383,526)(384,528)
(385,530)(386,529)(387,531)(388,515)(389,514)(390,516)(391,518)(392,517)
(393,519)(394,521)(395,520)(396,522)(397,533)(398,532)(399,534)(400,536)
(401,535)(402,537)(403,539)(404,538)(405,540)(406,497)(407,496)(408,498)
(409,500)(410,499)(411,501)(412,503)(413,502)(414,504)(415,488)(416,487)
(417,489)(418,491)(419,490)(420,492)(421,494)(422,493)(423,495)(424,506)
(425,505)(426,507)(427,509)(428,508)(429,510)(430,512)(431,511)(432,513);;
s2 := (  1,433)(  2,434)(  3,435)(  4,439)(  5,440)(  6,441)(  7,436)(  8,437)
(  9,438)( 10,451)( 11,452)( 12,453)( 13,457)( 14,458)( 15,459)( 16,454)
( 17,455)( 18,456)( 19,442)( 20,443)( 21,444)( 22,448)( 23,449)( 24,450)
( 25,445)( 26,446)( 27,447)( 28,460)( 29,461)( 30,462)( 31,466)( 32,467)
( 33,468)( 34,463)( 35,464)( 36,465)( 37,478)( 38,479)( 39,480)( 40,484)
( 41,485)( 42,486)( 43,481)( 44,482)( 45,483)( 46,469)( 47,470)( 48,471)
( 49,475)( 50,476)( 51,477)( 52,472)( 53,473)( 54,474)( 55,487)( 56,488)
( 57,489)( 58,493)( 59,494)( 60,495)( 61,490)( 62,491)( 63,492)( 64,505)
( 65,506)( 66,507)( 67,511)( 68,512)( 69,513)( 70,508)( 71,509)( 72,510)
( 73,496)( 74,497)( 75,498)( 76,502)( 77,503)( 78,504)( 79,499)( 80,500)
( 81,501)( 82,514)( 83,515)( 84,516)( 85,520)( 86,521)( 87,522)( 88,517)
( 89,518)( 90,519)( 91,532)( 92,533)( 93,534)( 94,538)( 95,539)( 96,540)
( 97,535)( 98,536)( 99,537)(100,523)(101,524)(102,525)(103,529)(104,530)
(105,531)(106,526)(107,527)(108,528)(109,595)(110,596)(111,597)(112,601)
(113,602)(114,603)(115,598)(116,599)(117,600)(118,613)(119,614)(120,615)
(121,619)(122,620)(123,621)(124,616)(125,617)(126,618)(127,604)(128,605)
(129,606)(130,610)(131,611)(132,612)(133,607)(134,608)(135,609)(136,622)
(137,623)(138,624)(139,628)(140,629)(141,630)(142,625)(143,626)(144,627)
(145,640)(146,641)(147,642)(148,646)(149,647)(150,648)(151,643)(152,644)
(153,645)(154,631)(155,632)(156,633)(157,637)(158,638)(159,639)(160,634)
(161,635)(162,636)(163,541)(164,542)(165,543)(166,547)(167,548)(168,549)
(169,544)(170,545)(171,546)(172,559)(173,560)(174,561)(175,565)(176,566)
(177,567)(178,562)(179,563)(180,564)(181,550)(182,551)(183,552)(184,556)
(185,557)(186,558)(187,553)(188,554)(189,555)(190,568)(191,569)(192,570)
(193,574)(194,575)(195,576)(196,571)(197,572)(198,573)(199,586)(200,587)
(201,588)(202,592)(203,593)(204,594)(205,589)(206,590)(207,591)(208,577)
(209,578)(210,579)(211,583)(212,584)(213,585)(214,580)(215,581)(216,582)
(217,676)(218,677)(219,678)(220,682)(221,683)(222,684)(223,679)(224,680)
(225,681)(226,694)(227,695)(228,696)(229,700)(230,701)(231,702)(232,697)
(233,698)(234,699)(235,685)(236,686)(237,687)(238,691)(239,692)(240,693)
(241,688)(242,689)(243,690)(244,649)(245,650)(246,651)(247,655)(248,656)
(249,657)(250,652)(251,653)(252,654)(253,667)(254,668)(255,669)(256,673)
(257,674)(258,675)(259,670)(260,671)(261,672)(262,658)(263,659)(264,660)
(265,664)(266,665)(267,666)(268,661)(269,662)(270,663)(271,730)(272,731)
(273,732)(274,736)(275,737)(276,738)(277,733)(278,734)(279,735)(280,748)
(281,749)(282,750)(283,754)(284,755)(285,756)(286,751)(287,752)(288,753)
(289,739)(290,740)(291,741)(292,745)(293,746)(294,747)(295,742)(296,743)
(297,744)(298,703)(299,704)(300,705)(301,709)(302,710)(303,711)(304,706)
(305,707)(306,708)(307,721)(308,722)(309,723)(310,727)(311,728)(312,729)
(313,724)(314,725)(315,726)(316,712)(317,713)(318,714)(319,718)(320,719)
(321,720)(322,715)(323,716)(324,717)(325,838)(326,839)(327,840)(328,844)
(329,845)(330,846)(331,841)(332,842)(333,843)(334,856)(335,857)(336,858)
(337,862)(338,863)(339,864)(340,859)(341,860)(342,861)(343,847)(344,848)
(345,849)(346,853)(347,854)(348,855)(349,850)(350,851)(351,852)(352,811)
(353,812)(354,813)(355,817)(356,818)(357,819)(358,814)(359,815)(360,816)
(361,829)(362,830)(363,831)(364,835)(365,836)(366,837)(367,832)(368,833)
(369,834)(370,820)(371,821)(372,822)(373,826)(374,827)(375,828)(376,823)
(377,824)(378,825)(379,784)(380,785)(381,786)(382,790)(383,791)(384,792)
(385,787)(386,788)(387,789)(388,802)(389,803)(390,804)(391,808)(392,809)
(393,810)(394,805)(395,806)(396,807)(397,793)(398,794)(399,795)(400,799)
(401,800)(402,801)(403,796)(404,797)(405,798)(406,757)(407,758)(408,759)
(409,763)(410,764)(411,765)(412,760)(413,761)(414,762)(415,775)(416,776)
(417,777)(418,781)(419,782)(420,783)(421,778)(422,779)(423,780)(424,766)
(425,767)(426,768)(427,772)(428,773)(429,774)(430,769)(431,770)(432,771);;
s3 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)(164,167)
(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)(190,193)
(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)(210,213)
(217,247)(218,248)(219,249)(220,244)(221,245)(222,246)(223,250)(224,251)
(225,252)(226,256)(227,257)(228,258)(229,253)(230,254)(231,255)(232,259)
(233,260)(234,261)(235,265)(236,266)(237,267)(238,262)(239,263)(240,264)
(241,268)(242,269)(243,270)(271,301)(272,302)(273,303)(274,298)(275,299)
(276,300)(277,304)(278,305)(279,306)(280,310)(281,311)(282,312)(283,307)
(284,308)(285,309)(286,313)(287,314)(288,315)(289,319)(290,320)(291,321)
(292,316)(293,317)(294,318)(295,322)(296,323)(297,324)(325,355)(326,356)
(327,357)(328,352)(329,353)(330,354)(331,358)(332,359)(333,360)(334,364)
(335,365)(336,366)(337,361)(338,362)(339,363)(340,367)(341,368)(342,369)
(343,373)(344,374)(345,375)(346,370)(347,371)(348,372)(349,376)(350,377)
(351,378)(379,409)(380,410)(381,411)(382,406)(383,407)(384,408)(385,412)
(386,413)(387,414)(388,418)(389,419)(390,420)(391,415)(392,416)(393,417)
(394,421)(395,422)(396,423)(397,427)(398,428)(399,429)(400,424)(401,425)
(402,426)(403,430)(404,431)(405,432)(433,463)(434,464)(435,465)(436,460)
(437,461)(438,462)(439,466)(440,467)(441,468)(442,472)(443,473)(444,474)
(445,469)(446,470)(447,471)(448,475)(449,476)(450,477)(451,481)(452,482)
(453,483)(454,478)(455,479)(456,480)(457,484)(458,485)(459,486)(487,517)
(488,518)(489,519)(490,514)(491,515)(492,516)(493,520)(494,521)(495,522)
(496,526)(497,527)(498,528)(499,523)(500,524)(501,525)(502,529)(503,530)
(504,531)(505,535)(506,536)(507,537)(508,532)(509,533)(510,534)(511,538)
(512,539)(513,540)(541,571)(542,572)(543,573)(544,568)(545,569)(546,570)
(547,574)(548,575)(549,576)(550,580)(551,581)(552,582)(553,577)(554,578)
(555,579)(556,583)(557,584)(558,585)(559,589)(560,590)(561,591)(562,586)
(563,587)(564,588)(565,592)(566,593)(567,594)(595,625)(596,626)(597,627)
(598,622)(599,623)(600,624)(601,628)(602,629)(603,630)(604,634)(605,635)
(606,636)(607,631)(608,632)(609,633)(610,637)(611,638)(612,639)(613,643)
(614,644)(615,645)(616,640)(617,641)(618,642)(619,646)(620,647)(621,648)
(649,652)(650,653)(651,654)(658,661)(659,662)(660,663)(667,670)(668,671)
(669,672)(676,679)(677,680)(678,681)(685,688)(686,689)(687,690)(694,697)
(695,698)(696,699)(703,706)(704,707)(705,708)(712,715)(713,716)(714,717)
(721,724)(722,725)(723,726)(730,733)(731,734)(732,735)(739,742)(740,743)
(741,744)(748,751)(749,752)(750,753)(757,760)(758,761)(759,762)(766,769)
(767,770)(768,771)(775,778)(776,779)(777,780)(784,787)(785,788)(786,789)
(793,796)(794,797)(795,798)(802,805)(803,806)(804,807)(811,814)(812,815)
(813,816)(820,823)(821,824)(822,825)(829,832)(830,833)(831,834)(838,841)
(839,842)(840,843)(847,850)(848,851)(849,852)(856,859)(857,860)(858,861);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```
References : None.
to this polytope