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Polytope of Type {12,12,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12,3}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46356)
Rank : 4
Schlafli Type : {12,12,3}
Number of vertices, edges, etc : 12, 144, 36, 6
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 : {6,12,3}*864b
   3-fold quotients : {12,4,3}*576, {4,12,3}*576
   4-fold quotients : {12,6,3}*432b
   6-fold quotients : {6,4,3}*288, {2,12,3}*288
   8-fold quotients : {6,6,3}*216b
   9-fold quotients : {4,4,3}*192b
   12-fold quotients : {12,2,3}*144, {4,6,3}*144
   18-fold quotients : {2,4,3}*96
   24-fold quotients : {2,6,3}*72, {6,2,3}*72
   36-fold quotients : {4,2,3}*48, {2,4,3}*48
   48-fold quotients : {3,2,3}*36
   72-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)
(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)(224,332)
(225,333)(226,334)(227,335)(228,336)(229,349)(230,350)(231,351)(232,352)
(233,353)(234,354)(235,355)(236,356)(237,357)(238,358)(239,359)(240,360)
(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)(248,344)
(249,345)(250,346)(251,347)(252,348)(253,361)(254,362)(255,363)(256,364)
(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)(263,371)(264,372)
(265,385)(266,386)(267,387)(268,388)(269,389)(270,390)(271,391)(272,392)
(273,393)(274,394)(275,395)(276,396)(277,373)(278,374)(279,375)(280,376)
(281,377)(282,378)(283,379)(284,380)(285,381)(286,382)(287,383)(288,384)
(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)(296,404)
(297,405)(298,406)(299,407)(300,408)(301,421)(302,422)(303,423)(304,424)
(305,425)(306,426)(307,427)(308,428)(309,429)(310,430)(311,431)(312,432)
(313,409)(314,410)(315,411)(316,412)(317,413)(318,414)(319,415)(320,416)
(321,417)(322,418)(323,419)(324,420)(433,541)(434,542)(435,543)(436,544)
(437,545)(438,546)(439,547)(440,548)(441,549)(442,550)(443,551)(444,552)
(445,565)(446,566)(447,567)(448,568)(449,569)(450,570)(451,571)(452,572)
(453,573)(454,574)(455,575)(456,576)(457,553)(458,554)(459,555)(460,556)
(461,557)(462,558)(463,559)(464,560)(465,561)(466,562)(467,563)(468,564)
(469,577)(470,578)(471,579)(472,580)(473,581)(474,582)(475,583)(476,584)
(477,585)(478,586)(479,587)(480,588)(481,601)(482,602)(483,603)(484,604)
(485,605)(486,606)(487,607)(488,608)(489,609)(490,610)(491,611)(492,612)
(493,589)(494,590)(495,591)(496,592)(497,593)(498,594)(499,595)(500,596)
(501,597)(502,598)(503,599)(504,600)(505,613)(506,614)(507,615)(508,616)
(509,617)(510,618)(511,619)(512,620)(513,621)(514,622)(515,623)(516,624)
(517,637)(518,638)(519,639)(520,640)(521,641)(522,642)(523,643)(524,644)
(525,645)(526,646)(527,647)(528,648)(529,625)(530,626)(531,627)(532,628)
(533,629)(534,630)(535,631)(536,632)(537,633)(538,634)(539,635)(540,636)
(661,673)(662,674)(663,675)(664,676)(665,677)(666,678)(667,679)(668,680)
(669,681)(670,682)(671,683)(672,684)(697,709)(698,710)(699,711)(700,712)
(701,713)(702,714)(703,715)(704,716)(705,717)(706,718)(707,719)(708,720)
(733,745)(734,746)(735,747)(736,748)(737,749)(738,750)(739,751)(740,752)
(741,753)(742,754)(743,755)(744,756)(769,781)(770,782)(771,783)(772,784)
(773,785)(774,786)(775,787)(776,788)(777,789)(778,790)(779,791)(780,792)
(805,817)(806,818)(807,819)(808,820)(809,821)(810,822)(811,823)(812,824)
(813,825)(814,826)(815,827)(816,828)(841,853)(842,854)(843,855)(844,856)
(845,857)(846,858)(847,859)(848,860)(849,861)(850,862)(851,863)(852,864);;
s1 := (  1,447)(  2,448)(  3,445)(  4,446)(  5,455)(  6,456)(  7,453)(  8,454)
(  9,451)( 10,452)( 11,449)( 12,450)( 13,435)( 14,436)( 15,433)( 16,434)
( 17,443)( 18,444)( 19,441)( 20,442)( 21,439)( 22,440)( 23,437)( 24,438)
( 25,459)( 26,460)( 27,457)( 28,458)( 29,467)( 30,468)( 31,465)( 32,466)
( 33,463)( 34,464)( 35,461)( 36,462)( 37,483)( 38,484)( 39,481)( 40,482)
( 41,491)( 42,492)( 43,489)( 44,490)( 45,487)( 46,488)( 47,485)( 48,486)
( 49,471)( 50,472)( 51,469)( 52,470)( 53,479)( 54,480)( 55,477)( 56,478)
( 57,475)( 58,476)( 59,473)( 60,474)( 61,495)( 62,496)( 63,493)( 64,494)
( 65,503)( 66,504)( 67,501)( 68,502)( 69,499)( 70,500)( 71,497)( 72,498)
( 73,519)( 74,520)( 75,517)( 76,518)( 77,527)( 78,528)( 79,525)( 80,526)
( 81,523)( 82,524)( 83,521)( 84,522)( 85,507)( 86,508)( 87,505)( 88,506)
( 89,515)( 90,516)( 91,513)( 92,514)( 93,511)( 94,512)( 95,509)( 96,510)
( 97,531)( 98,532)( 99,529)(100,530)(101,539)(102,540)(103,537)(104,538)
(105,535)(106,536)(107,533)(108,534)(109,555)(110,556)(111,553)(112,554)
(113,563)(114,564)(115,561)(116,562)(117,559)(118,560)(119,557)(120,558)
(121,543)(122,544)(123,541)(124,542)(125,551)(126,552)(127,549)(128,550)
(129,547)(130,548)(131,545)(132,546)(133,567)(134,568)(135,565)(136,566)
(137,575)(138,576)(139,573)(140,574)(141,571)(142,572)(143,569)(144,570)
(145,591)(146,592)(147,589)(148,590)(149,599)(150,600)(151,597)(152,598)
(153,595)(154,596)(155,593)(156,594)(157,579)(158,580)(159,577)(160,578)
(161,587)(162,588)(163,585)(164,586)(165,583)(166,584)(167,581)(168,582)
(169,603)(170,604)(171,601)(172,602)(173,611)(174,612)(175,609)(176,610)
(177,607)(178,608)(179,605)(180,606)(181,627)(182,628)(183,625)(184,626)
(185,635)(186,636)(187,633)(188,634)(189,631)(190,632)(191,629)(192,630)
(193,615)(194,616)(195,613)(196,614)(197,623)(198,624)(199,621)(200,622)
(201,619)(202,620)(203,617)(204,618)(205,639)(206,640)(207,637)(208,638)
(209,647)(210,648)(211,645)(212,646)(213,643)(214,644)(215,641)(216,642)
(217,771)(218,772)(219,769)(220,770)(221,779)(222,780)(223,777)(224,778)
(225,775)(226,776)(227,773)(228,774)(229,759)(230,760)(231,757)(232,758)
(233,767)(234,768)(235,765)(236,766)(237,763)(238,764)(239,761)(240,762)
(241,783)(242,784)(243,781)(244,782)(245,791)(246,792)(247,789)(248,790)
(249,787)(250,788)(251,785)(252,786)(253,807)(254,808)(255,805)(256,806)
(257,815)(258,816)(259,813)(260,814)(261,811)(262,812)(263,809)(264,810)
(265,795)(266,796)(267,793)(268,794)(269,803)(270,804)(271,801)(272,802)
(273,799)(274,800)(275,797)(276,798)(277,819)(278,820)(279,817)(280,818)
(281,827)(282,828)(283,825)(284,826)(285,823)(286,824)(287,821)(288,822)
(289,843)(290,844)(291,841)(292,842)(293,851)(294,852)(295,849)(296,850)
(297,847)(298,848)(299,845)(300,846)(301,831)(302,832)(303,829)(304,830)
(305,839)(306,840)(307,837)(308,838)(309,835)(310,836)(311,833)(312,834)
(313,855)(314,856)(315,853)(316,854)(317,863)(318,864)(319,861)(320,862)
(321,859)(322,860)(323,857)(324,858)(325,663)(326,664)(327,661)(328,662)
(329,671)(330,672)(331,669)(332,670)(333,667)(334,668)(335,665)(336,666)
(337,651)(338,652)(339,649)(340,650)(341,659)(342,660)(343,657)(344,658)
(345,655)(346,656)(347,653)(348,654)(349,675)(350,676)(351,673)(352,674)
(353,683)(354,684)(355,681)(356,682)(357,679)(358,680)(359,677)(360,678)
(361,699)(362,700)(363,697)(364,698)(365,707)(366,708)(367,705)(368,706)
(369,703)(370,704)(371,701)(372,702)(373,687)(374,688)(375,685)(376,686)
(377,695)(378,696)(379,693)(380,694)(381,691)(382,692)(383,689)(384,690)
(385,711)(386,712)(387,709)(388,710)(389,719)(390,720)(391,717)(392,718)
(393,715)(394,716)(395,713)(396,714)(397,735)(398,736)(399,733)(400,734)
(401,743)(402,744)(403,741)(404,742)(405,739)(406,740)(407,737)(408,738)
(409,723)(410,724)(411,721)(412,722)(413,731)(414,732)(415,729)(416,730)
(417,727)(418,728)(419,725)(420,726)(421,747)(422,748)(423,745)(424,746)
(425,755)(426,756)(427,753)(428,754)(429,751)(430,752)(431,749)(432,750);;
s2 := (  1,653)(  2,654)(  3,656)(  4,655)(  5,649)(  6,650)(  7,652)(  8,651)
(  9,657)( 10,658)( 11,660)( 12,659)( 13,665)( 14,666)( 15,668)( 16,667)
( 17,661)( 18,662)( 19,664)( 20,663)( 21,669)( 22,670)( 23,672)( 24,671)
( 25,677)( 26,678)( 27,680)( 28,679)( 29,673)( 30,674)( 31,676)( 32,675)
( 33,681)( 34,682)( 35,684)( 36,683)( 37,725)( 38,726)( 39,728)( 40,727)
( 41,721)( 42,722)( 43,724)( 44,723)( 45,729)( 46,730)( 47,732)( 48,731)
( 49,737)( 50,738)( 51,740)( 52,739)( 53,733)( 54,734)( 55,736)( 56,735)
( 57,741)( 58,742)( 59,744)( 60,743)( 61,749)( 62,750)( 63,752)( 64,751)
( 65,745)( 66,746)( 67,748)( 68,747)( 69,753)( 70,754)( 71,756)( 72,755)
( 73,689)( 74,690)( 75,692)( 76,691)( 77,685)( 78,686)( 79,688)( 80,687)
( 81,693)( 82,694)( 83,696)( 84,695)( 85,701)( 86,702)( 87,704)( 88,703)
( 89,697)( 90,698)( 91,700)( 92,699)( 93,705)( 94,706)( 95,708)( 96,707)
( 97,713)( 98,714)( 99,716)(100,715)(101,709)(102,710)(103,712)(104,711)
(105,717)(106,718)(107,720)(108,719)(109,761)(110,762)(111,764)(112,763)
(113,757)(114,758)(115,760)(116,759)(117,765)(118,766)(119,768)(120,767)
(121,773)(122,774)(123,776)(124,775)(125,769)(126,770)(127,772)(128,771)
(129,777)(130,778)(131,780)(132,779)(133,785)(134,786)(135,788)(136,787)
(137,781)(138,782)(139,784)(140,783)(141,789)(142,790)(143,792)(144,791)
(145,833)(146,834)(147,836)(148,835)(149,829)(150,830)(151,832)(152,831)
(153,837)(154,838)(155,840)(156,839)(157,845)(158,846)(159,848)(160,847)
(161,841)(162,842)(163,844)(164,843)(165,849)(166,850)(167,852)(168,851)
(169,857)(170,858)(171,860)(172,859)(173,853)(174,854)(175,856)(176,855)
(177,861)(178,862)(179,864)(180,863)(181,797)(182,798)(183,800)(184,799)
(185,793)(186,794)(187,796)(188,795)(189,801)(190,802)(191,804)(192,803)
(193,809)(194,810)(195,812)(196,811)(197,805)(198,806)(199,808)(200,807)
(201,813)(202,814)(203,816)(204,815)(205,821)(206,822)(207,824)(208,823)
(209,817)(210,818)(211,820)(212,819)(213,825)(214,826)(215,828)(216,827)
(217,545)(218,546)(219,548)(220,547)(221,541)(222,542)(223,544)(224,543)
(225,549)(226,550)(227,552)(228,551)(229,557)(230,558)(231,560)(232,559)
(233,553)(234,554)(235,556)(236,555)(237,561)(238,562)(239,564)(240,563)
(241,569)(242,570)(243,572)(244,571)(245,565)(246,566)(247,568)(248,567)
(249,573)(250,574)(251,576)(252,575)(253,617)(254,618)(255,620)(256,619)
(257,613)(258,614)(259,616)(260,615)(261,621)(262,622)(263,624)(264,623)
(265,629)(266,630)(267,632)(268,631)(269,625)(270,626)(271,628)(272,627)
(273,633)(274,634)(275,636)(276,635)(277,641)(278,642)(279,644)(280,643)
(281,637)(282,638)(283,640)(284,639)(285,645)(286,646)(287,648)(288,647)
(289,581)(290,582)(291,584)(292,583)(293,577)(294,578)(295,580)(296,579)
(297,585)(298,586)(299,588)(300,587)(301,593)(302,594)(303,596)(304,595)
(305,589)(306,590)(307,592)(308,591)(309,597)(310,598)(311,600)(312,599)
(313,605)(314,606)(315,608)(316,607)(317,601)(318,602)(319,604)(320,603)
(321,609)(322,610)(323,612)(324,611)(325,437)(326,438)(327,440)(328,439)
(329,433)(330,434)(331,436)(332,435)(333,441)(334,442)(335,444)(336,443)
(337,449)(338,450)(339,452)(340,451)(341,445)(342,446)(343,448)(344,447)
(345,453)(346,454)(347,456)(348,455)(349,461)(350,462)(351,464)(352,463)
(353,457)(354,458)(355,460)(356,459)(357,465)(358,466)(359,468)(360,467)
(361,509)(362,510)(363,512)(364,511)(365,505)(366,506)(367,508)(368,507)
(369,513)(370,514)(371,516)(372,515)(373,521)(374,522)(375,524)(376,523)
(377,517)(378,518)(379,520)(380,519)(381,525)(382,526)(383,528)(384,527)
(385,533)(386,534)(387,536)(388,535)(389,529)(390,530)(391,532)(392,531)
(393,537)(394,538)(395,540)(396,539)(397,473)(398,474)(399,476)(400,475)
(401,469)(402,470)(403,472)(404,471)(405,477)(406,478)(407,480)(408,479)
(409,485)(410,486)(411,488)(412,487)(413,481)(414,482)(415,484)(416,483)
(417,489)(418,490)(419,492)(420,491)(421,497)(422,498)(423,500)(424,499)
(425,493)(426,494)(427,496)(428,495)(429,501)(430,502)(431,504)(432,503);;
s3 := (  1,685)(  2,688)(  3,687)(  4,686)(  5,693)(  6,696)(  7,695)(  8,694)
(  9,689)( 10,692)( 11,691)( 12,690)( 13,697)( 14,700)( 15,699)( 16,698)
( 17,705)( 18,708)( 19,707)( 20,706)( 21,701)( 22,704)( 23,703)( 24,702)
( 25,709)( 26,712)( 27,711)( 28,710)( 29,717)( 30,720)( 31,719)( 32,718)
( 33,713)( 34,716)( 35,715)( 36,714)( 37,649)( 38,652)( 39,651)( 40,650)
( 41,657)( 42,660)( 43,659)( 44,658)( 45,653)( 46,656)( 47,655)( 48,654)
( 49,661)( 50,664)( 51,663)( 52,662)( 53,669)( 54,672)( 55,671)( 56,670)
( 57,665)( 58,668)( 59,667)( 60,666)( 61,673)( 62,676)( 63,675)( 64,674)
( 65,681)( 66,684)( 67,683)( 68,682)( 69,677)( 70,680)( 71,679)( 72,678)
( 73,721)( 74,724)( 75,723)( 76,722)( 77,729)( 78,732)( 79,731)( 80,730)
( 81,725)( 82,728)( 83,727)( 84,726)( 85,733)( 86,736)( 87,735)( 88,734)
( 89,741)( 90,744)( 91,743)( 92,742)( 93,737)( 94,740)( 95,739)( 96,738)
( 97,745)( 98,748)( 99,747)(100,746)(101,753)(102,756)(103,755)(104,754)
(105,749)(106,752)(107,751)(108,750)(109,793)(110,796)(111,795)(112,794)
(113,801)(114,804)(115,803)(116,802)(117,797)(118,800)(119,799)(120,798)
(121,805)(122,808)(123,807)(124,806)(125,813)(126,816)(127,815)(128,814)
(129,809)(130,812)(131,811)(132,810)(133,817)(134,820)(135,819)(136,818)
(137,825)(138,828)(139,827)(140,826)(141,821)(142,824)(143,823)(144,822)
(145,757)(146,760)(147,759)(148,758)(149,765)(150,768)(151,767)(152,766)
(153,761)(154,764)(155,763)(156,762)(157,769)(158,772)(159,771)(160,770)
(161,777)(162,780)(163,779)(164,778)(165,773)(166,776)(167,775)(168,774)
(169,781)(170,784)(171,783)(172,782)(173,789)(174,792)(175,791)(176,790)
(177,785)(178,788)(179,787)(180,786)(181,829)(182,832)(183,831)(184,830)
(185,837)(186,840)(187,839)(188,838)(189,833)(190,836)(191,835)(192,834)
(193,841)(194,844)(195,843)(196,842)(197,849)(198,852)(199,851)(200,850)
(201,845)(202,848)(203,847)(204,846)(205,853)(206,856)(207,855)(208,854)
(209,861)(210,864)(211,863)(212,862)(213,857)(214,860)(215,859)(216,858)
(217,577)(218,580)(219,579)(220,578)(221,585)(222,588)(223,587)(224,586)
(225,581)(226,584)(227,583)(228,582)(229,589)(230,592)(231,591)(232,590)
(233,597)(234,600)(235,599)(236,598)(237,593)(238,596)(239,595)(240,594)
(241,601)(242,604)(243,603)(244,602)(245,609)(246,612)(247,611)(248,610)
(249,605)(250,608)(251,607)(252,606)(253,541)(254,544)(255,543)(256,542)
(257,549)(258,552)(259,551)(260,550)(261,545)(262,548)(263,547)(264,546)
(265,553)(266,556)(267,555)(268,554)(269,561)(270,564)(271,563)(272,562)
(273,557)(274,560)(275,559)(276,558)(277,565)(278,568)(279,567)(280,566)
(281,573)(282,576)(283,575)(284,574)(285,569)(286,572)(287,571)(288,570)
(289,613)(290,616)(291,615)(292,614)(293,621)(294,624)(295,623)(296,622)
(297,617)(298,620)(299,619)(300,618)(301,625)(302,628)(303,627)(304,626)
(305,633)(306,636)(307,635)(308,634)(309,629)(310,632)(311,631)(312,630)
(313,637)(314,640)(315,639)(316,638)(317,645)(318,648)(319,647)(320,646)
(321,641)(322,644)(323,643)(324,642)(325,469)(326,472)(327,471)(328,470)
(329,477)(330,480)(331,479)(332,478)(333,473)(334,476)(335,475)(336,474)
(337,481)(338,484)(339,483)(340,482)(341,489)(342,492)(343,491)(344,490)
(345,485)(346,488)(347,487)(348,486)(349,493)(350,496)(351,495)(352,494)
(353,501)(354,504)(355,503)(356,502)(357,497)(358,500)(359,499)(360,498)
(361,433)(362,436)(363,435)(364,434)(365,441)(366,444)(367,443)(368,442)
(369,437)(370,440)(371,439)(372,438)(373,445)(374,448)(375,447)(376,446)
(377,453)(378,456)(379,455)(380,454)(381,449)(382,452)(383,451)(384,450)
(385,457)(386,460)(387,459)(388,458)(389,465)(390,468)(391,467)(392,466)
(393,461)(394,464)(395,463)(396,462)(397,505)(398,508)(399,507)(400,506)
(401,513)(402,516)(403,515)(404,514)(405,509)(406,512)(407,511)(408,510)
(409,517)(410,520)(411,519)(412,518)(413,525)(414,528)(415,527)(416,526)
(417,521)(418,524)(419,523)(420,522)(421,529)(422,532)(423,531)(424,530)
(425,537)(426,540)(427,539)(428,538)(429,533)(430,536)(431,535)(432,534);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
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)!( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)
( 20, 32)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)
( 92,104)( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)
(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)
(132,144)(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)
(164,176)(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)
(196,208)(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)
(204,216)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)
(224,332)(225,333)(226,334)(227,335)(228,336)(229,349)(230,350)(231,351)
(232,352)(233,353)(234,354)(235,355)(236,356)(237,357)(238,358)(239,359)
(240,360)(241,337)(242,338)(243,339)(244,340)(245,341)(246,342)(247,343)
(248,344)(249,345)(250,346)(251,347)(252,348)(253,361)(254,362)(255,363)
(256,364)(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)(263,371)
(264,372)(265,385)(266,386)(267,387)(268,388)(269,389)(270,390)(271,391)
(272,392)(273,393)(274,394)(275,395)(276,396)(277,373)(278,374)(279,375)
(280,376)(281,377)(282,378)(283,379)(284,380)(285,381)(286,382)(287,383)
(288,384)(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)
(296,404)(297,405)(298,406)(299,407)(300,408)(301,421)(302,422)(303,423)
(304,424)(305,425)(306,426)(307,427)(308,428)(309,429)(310,430)(311,431)
(312,432)(313,409)(314,410)(315,411)(316,412)(317,413)(318,414)(319,415)
(320,416)(321,417)(322,418)(323,419)(324,420)(433,541)(434,542)(435,543)
(436,544)(437,545)(438,546)(439,547)(440,548)(441,549)(442,550)(443,551)
(444,552)(445,565)(446,566)(447,567)(448,568)(449,569)(450,570)(451,571)
(452,572)(453,573)(454,574)(455,575)(456,576)(457,553)(458,554)(459,555)
(460,556)(461,557)(462,558)(463,559)(464,560)(465,561)(466,562)(467,563)
(468,564)(469,577)(470,578)(471,579)(472,580)(473,581)(474,582)(475,583)
(476,584)(477,585)(478,586)(479,587)(480,588)(481,601)(482,602)(483,603)
(484,604)(485,605)(486,606)(487,607)(488,608)(489,609)(490,610)(491,611)
(492,612)(493,589)(494,590)(495,591)(496,592)(497,593)(498,594)(499,595)
(500,596)(501,597)(502,598)(503,599)(504,600)(505,613)(506,614)(507,615)
(508,616)(509,617)(510,618)(511,619)(512,620)(513,621)(514,622)(515,623)
(516,624)(517,637)(518,638)(519,639)(520,640)(521,641)(522,642)(523,643)
(524,644)(525,645)(526,646)(527,647)(528,648)(529,625)(530,626)(531,627)
(532,628)(533,629)(534,630)(535,631)(536,632)(537,633)(538,634)(539,635)
(540,636)(661,673)(662,674)(663,675)(664,676)(665,677)(666,678)(667,679)
(668,680)(669,681)(670,682)(671,683)(672,684)(697,709)(698,710)(699,711)
(700,712)(701,713)(702,714)(703,715)(704,716)(705,717)(706,718)(707,719)
(708,720)(733,745)(734,746)(735,747)(736,748)(737,749)(738,750)(739,751)
(740,752)(741,753)(742,754)(743,755)(744,756)(769,781)(770,782)(771,783)
(772,784)(773,785)(774,786)(775,787)(776,788)(777,789)(778,790)(779,791)
(780,792)(805,817)(806,818)(807,819)(808,820)(809,821)(810,822)(811,823)
(812,824)(813,825)(814,826)(815,827)(816,828)(841,853)(842,854)(843,855)
(844,856)(845,857)(846,858)(847,859)(848,860)(849,861)(850,862)(851,863)
(852,864);
s1 := Sym(864)!(  1,447)(  2,448)(  3,445)(  4,446)(  5,455)(  6,456)(  7,453)
(  8,454)(  9,451)( 10,452)( 11,449)( 12,450)( 13,435)( 14,436)( 15,433)
( 16,434)( 17,443)( 18,444)( 19,441)( 20,442)( 21,439)( 22,440)( 23,437)
( 24,438)( 25,459)( 26,460)( 27,457)( 28,458)( 29,467)( 30,468)( 31,465)
( 32,466)( 33,463)( 34,464)( 35,461)( 36,462)( 37,483)( 38,484)( 39,481)
( 40,482)( 41,491)( 42,492)( 43,489)( 44,490)( 45,487)( 46,488)( 47,485)
( 48,486)( 49,471)( 50,472)( 51,469)( 52,470)( 53,479)( 54,480)( 55,477)
( 56,478)( 57,475)( 58,476)( 59,473)( 60,474)( 61,495)( 62,496)( 63,493)
( 64,494)( 65,503)( 66,504)( 67,501)( 68,502)( 69,499)( 70,500)( 71,497)
( 72,498)( 73,519)( 74,520)( 75,517)( 76,518)( 77,527)( 78,528)( 79,525)
( 80,526)( 81,523)( 82,524)( 83,521)( 84,522)( 85,507)( 86,508)( 87,505)
( 88,506)( 89,515)( 90,516)( 91,513)( 92,514)( 93,511)( 94,512)( 95,509)
( 96,510)( 97,531)( 98,532)( 99,529)(100,530)(101,539)(102,540)(103,537)
(104,538)(105,535)(106,536)(107,533)(108,534)(109,555)(110,556)(111,553)
(112,554)(113,563)(114,564)(115,561)(116,562)(117,559)(118,560)(119,557)
(120,558)(121,543)(122,544)(123,541)(124,542)(125,551)(126,552)(127,549)
(128,550)(129,547)(130,548)(131,545)(132,546)(133,567)(134,568)(135,565)
(136,566)(137,575)(138,576)(139,573)(140,574)(141,571)(142,572)(143,569)
(144,570)(145,591)(146,592)(147,589)(148,590)(149,599)(150,600)(151,597)
(152,598)(153,595)(154,596)(155,593)(156,594)(157,579)(158,580)(159,577)
(160,578)(161,587)(162,588)(163,585)(164,586)(165,583)(166,584)(167,581)
(168,582)(169,603)(170,604)(171,601)(172,602)(173,611)(174,612)(175,609)
(176,610)(177,607)(178,608)(179,605)(180,606)(181,627)(182,628)(183,625)
(184,626)(185,635)(186,636)(187,633)(188,634)(189,631)(190,632)(191,629)
(192,630)(193,615)(194,616)(195,613)(196,614)(197,623)(198,624)(199,621)
(200,622)(201,619)(202,620)(203,617)(204,618)(205,639)(206,640)(207,637)
(208,638)(209,647)(210,648)(211,645)(212,646)(213,643)(214,644)(215,641)
(216,642)(217,771)(218,772)(219,769)(220,770)(221,779)(222,780)(223,777)
(224,778)(225,775)(226,776)(227,773)(228,774)(229,759)(230,760)(231,757)
(232,758)(233,767)(234,768)(235,765)(236,766)(237,763)(238,764)(239,761)
(240,762)(241,783)(242,784)(243,781)(244,782)(245,791)(246,792)(247,789)
(248,790)(249,787)(250,788)(251,785)(252,786)(253,807)(254,808)(255,805)
(256,806)(257,815)(258,816)(259,813)(260,814)(261,811)(262,812)(263,809)
(264,810)(265,795)(266,796)(267,793)(268,794)(269,803)(270,804)(271,801)
(272,802)(273,799)(274,800)(275,797)(276,798)(277,819)(278,820)(279,817)
(280,818)(281,827)(282,828)(283,825)(284,826)(285,823)(286,824)(287,821)
(288,822)(289,843)(290,844)(291,841)(292,842)(293,851)(294,852)(295,849)
(296,850)(297,847)(298,848)(299,845)(300,846)(301,831)(302,832)(303,829)
(304,830)(305,839)(306,840)(307,837)(308,838)(309,835)(310,836)(311,833)
(312,834)(313,855)(314,856)(315,853)(316,854)(317,863)(318,864)(319,861)
(320,862)(321,859)(322,860)(323,857)(324,858)(325,663)(326,664)(327,661)
(328,662)(329,671)(330,672)(331,669)(332,670)(333,667)(334,668)(335,665)
(336,666)(337,651)(338,652)(339,649)(340,650)(341,659)(342,660)(343,657)
(344,658)(345,655)(346,656)(347,653)(348,654)(349,675)(350,676)(351,673)
(352,674)(353,683)(354,684)(355,681)(356,682)(357,679)(358,680)(359,677)
(360,678)(361,699)(362,700)(363,697)(364,698)(365,707)(366,708)(367,705)
(368,706)(369,703)(370,704)(371,701)(372,702)(373,687)(374,688)(375,685)
(376,686)(377,695)(378,696)(379,693)(380,694)(381,691)(382,692)(383,689)
(384,690)(385,711)(386,712)(387,709)(388,710)(389,719)(390,720)(391,717)
(392,718)(393,715)(394,716)(395,713)(396,714)(397,735)(398,736)(399,733)
(400,734)(401,743)(402,744)(403,741)(404,742)(405,739)(406,740)(407,737)
(408,738)(409,723)(410,724)(411,721)(412,722)(413,731)(414,732)(415,729)
(416,730)(417,727)(418,728)(419,725)(420,726)(421,747)(422,748)(423,745)
(424,746)(425,755)(426,756)(427,753)(428,754)(429,751)(430,752)(431,749)
(432,750);
s2 := Sym(864)!(  1,653)(  2,654)(  3,656)(  4,655)(  5,649)(  6,650)(  7,652)
(  8,651)(  9,657)( 10,658)( 11,660)( 12,659)( 13,665)( 14,666)( 15,668)
( 16,667)( 17,661)( 18,662)( 19,664)( 20,663)( 21,669)( 22,670)( 23,672)
( 24,671)( 25,677)( 26,678)( 27,680)( 28,679)( 29,673)( 30,674)( 31,676)
( 32,675)( 33,681)( 34,682)( 35,684)( 36,683)( 37,725)( 38,726)( 39,728)
( 40,727)( 41,721)( 42,722)( 43,724)( 44,723)( 45,729)( 46,730)( 47,732)
( 48,731)( 49,737)( 50,738)( 51,740)( 52,739)( 53,733)( 54,734)( 55,736)
( 56,735)( 57,741)( 58,742)( 59,744)( 60,743)( 61,749)( 62,750)( 63,752)
( 64,751)( 65,745)( 66,746)( 67,748)( 68,747)( 69,753)( 70,754)( 71,756)
( 72,755)( 73,689)( 74,690)( 75,692)( 76,691)( 77,685)( 78,686)( 79,688)
( 80,687)( 81,693)( 82,694)( 83,696)( 84,695)( 85,701)( 86,702)( 87,704)
( 88,703)( 89,697)( 90,698)( 91,700)( 92,699)( 93,705)( 94,706)( 95,708)
( 96,707)( 97,713)( 98,714)( 99,716)(100,715)(101,709)(102,710)(103,712)
(104,711)(105,717)(106,718)(107,720)(108,719)(109,761)(110,762)(111,764)
(112,763)(113,757)(114,758)(115,760)(116,759)(117,765)(118,766)(119,768)
(120,767)(121,773)(122,774)(123,776)(124,775)(125,769)(126,770)(127,772)
(128,771)(129,777)(130,778)(131,780)(132,779)(133,785)(134,786)(135,788)
(136,787)(137,781)(138,782)(139,784)(140,783)(141,789)(142,790)(143,792)
(144,791)(145,833)(146,834)(147,836)(148,835)(149,829)(150,830)(151,832)
(152,831)(153,837)(154,838)(155,840)(156,839)(157,845)(158,846)(159,848)
(160,847)(161,841)(162,842)(163,844)(164,843)(165,849)(166,850)(167,852)
(168,851)(169,857)(170,858)(171,860)(172,859)(173,853)(174,854)(175,856)
(176,855)(177,861)(178,862)(179,864)(180,863)(181,797)(182,798)(183,800)
(184,799)(185,793)(186,794)(187,796)(188,795)(189,801)(190,802)(191,804)
(192,803)(193,809)(194,810)(195,812)(196,811)(197,805)(198,806)(199,808)
(200,807)(201,813)(202,814)(203,816)(204,815)(205,821)(206,822)(207,824)
(208,823)(209,817)(210,818)(211,820)(212,819)(213,825)(214,826)(215,828)
(216,827)(217,545)(218,546)(219,548)(220,547)(221,541)(222,542)(223,544)
(224,543)(225,549)(226,550)(227,552)(228,551)(229,557)(230,558)(231,560)
(232,559)(233,553)(234,554)(235,556)(236,555)(237,561)(238,562)(239,564)
(240,563)(241,569)(242,570)(243,572)(244,571)(245,565)(246,566)(247,568)
(248,567)(249,573)(250,574)(251,576)(252,575)(253,617)(254,618)(255,620)
(256,619)(257,613)(258,614)(259,616)(260,615)(261,621)(262,622)(263,624)
(264,623)(265,629)(266,630)(267,632)(268,631)(269,625)(270,626)(271,628)
(272,627)(273,633)(274,634)(275,636)(276,635)(277,641)(278,642)(279,644)
(280,643)(281,637)(282,638)(283,640)(284,639)(285,645)(286,646)(287,648)
(288,647)(289,581)(290,582)(291,584)(292,583)(293,577)(294,578)(295,580)
(296,579)(297,585)(298,586)(299,588)(300,587)(301,593)(302,594)(303,596)
(304,595)(305,589)(306,590)(307,592)(308,591)(309,597)(310,598)(311,600)
(312,599)(313,605)(314,606)(315,608)(316,607)(317,601)(318,602)(319,604)
(320,603)(321,609)(322,610)(323,612)(324,611)(325,437)(326,438)(327,440)
(328,439)(329,433)(330,434)(331,436)(332,435)(333,441)(334,442)(335,444)
(336,443)(337,449)(338,450)(339,452)(340,451)(341,445)(342,446)(343,448)
(344,447)(345,453)(346,454)(347,456)(348,455)(349,461)(350,462)(351,464)
(352,463)(353,457)(354,458)(355,460)(356,459)(357,465)(358,466)(359,468)
(360,467)(361,509)(362,510)(363,512)(364,511)(365,505)(366,506)(367,508)
(368,507)(369,513)(370,514)(371,516)(372,515)(373,521)(374,522)(375,524)
(376,523)(377,517)(378,518)(379,520)(380,519)(381,525)(382,526)(383,528)
(384,527)(385,533)(386,534)(387,536)(388,535)(389,529)(390,530)(391,532)
(392,531)(393,537)(394,538)(395,540)(396,539)(397,473)(398,474)(399,476)
(400,475)(401,469)(402,470)(403,472)(404,471)(405,477)(406,478)(407,480)
(408,479)(409,485)(410,486)(411,488)(412,487)(413,481)(414,482)(415,484)
(416,483)(417,489)(418,490)(419,492)(420,491)(421,497)(422,498)(423,500)
(424,499)(425,493)(426,494)(427,496)(428,495)(429,501)(430,502)(431,504)
(432,503);
s3 := Sym(864)!(  1,685)(  2,688)(  3,687)(  4,686)(  5,693)(  6,696)(  7,695)
(  8,694)(  9,689)( 10,692)( 11,691)( 12,690)( 13,697)( 14,700)( 15,699)
( 16,698)( 17,705)( 18,708)( 19,707)( 20,706)( 21,701)( 22,704)( 23,703)
( 24,702)( 25,709)( 26,712)( 27,711)( 28,710)( 29,717)( 30,720)( 31,719)
( 32,718)( 33,713)( 34,716)( 35,715)( 36,714)( 37,649)( 38,652)( 39,651)
( 40,650)( 41,657)( 42,660)( 43,659)( 44,658)( 45,653)( 46,656)( 47,655)
( 48,654)( 49,661)( 50,664)( 51,663)( 52,662)( 53,669)( 54,672)( 55,671)
( 56,670)( 57,665)( 58,668)( 59,667)( 60,666)( 61,673)( 62,676)( 63,675)
( 64,674)( 65,681)( 66,684)( 67,683)( 68,682)( 69,677)( 70,680)( 71,679)
( 72,678)( 73,721)( 74,724)( 75,723)( 76,722)( 77,729)( 78,732)( 79,731)
( 80,730)( 81,725)( 82,728)( 83,727)( 84,726)( 85,733)( 86,736)( 87,735)
( 88,734)( 89,741)( 90,744)( 91,743)( 92,742)( 93,737)( 94,740)( 95,739)
( 96,738)( 97,745)( 98,748)( 99,747)(100,746)(101,753)(102,756)(103,755)
(104,754)(105,749)(106,752)(107,751)(108,750)(109,793)(110,796)(111,795)
(112,794)(113,801)(114,804)(115,803)(116,802)(117,797)(118,800)(119,799)
(120,798)(121,805)(122,808)(123,807)(124,806)(125,813)(126,816)(127,815)
(128,814)(129,809)(130,812)(131,811)(132,810)(133,817)(134,820)(135,819)
(136,818)(137,825)(138,828)(139,827)(140,826)(141,821)(142,824)(143,823)
(144,822)(145,757)(146,760)(147,759)(148,758)(149,765)(150,768)(151,767)
(152,766)(153,761)(154,764)(155,763)(156,762)(157,769)(158,772)(159,771)
(160,770)(161,777)(162,780)(163,779)(164,778)(165,773)(166,776)(167,775)
(168,774)(169,781)(170,784)(171,783)(172,782)(173,789)(174,792)(175,791)
(176,790)(177,785)(178,788)(179,787)(180,786)(181,829)(182,832)(183,831)
(184,830)(185,837)(186,840)(187,839)(188,838)(189,833)(190,836)(191,835)
(192,834)(193,841)(194,844)(195,843)(196,842)(197,849)(198,852)(199,851)
(200,850)(201,845)(202,848)(203,847)(204,846)(205,853)(206,856)(207,855)
(208,854)(209,861)(210,864)(211,863)(212,862)(213,857)(214,860)(215,859)
(216,858)(217,577)(218,580)(219,579)(220,578)(221,585)(222,588)(223,587)
(224,586)(225,581)(226,584)(227,583)(228,582)(229,589)(230,592)(231,591)
(232,590)(233,597)(234,600)(235,599)(236,598)(237,593)(238,596)(239,595)
(240,594)(241,601)(242,604)(243,603)(244,602)(245,609)(246,612)(247,611)
(248,610)(249,605)(250,608)(251,607)(252,606)(253,541)(254,544)(255,543)
(256,542)(257,549)(258,552)(259,551)(260,550)(261,545)(262,548)(263,547)
(264,546)(265,553)(266,556)(267,555)(268,554)(269,561)(270,564)(271,563)
(272,562)(273,557)(274,560)(275,559)(276,558)(277,565)(278,568)(279,567)
(280,566)(281,573)(282,576)(283,575)(284,574)(285,569)(286,572)(287,571)
(288,570)(289,613)(290,616)(291,615)(292,614)(293,621)(294,624)(295,623)
(296,622)(297,617)(298,620)(299,619)(300,618)(301,625)(302,628)(303,627)
(304,626)(305,633)(306,636)(307,635)(308,634)(309,629)(310,632)(311,631)
(312,630)(313,637)(314,640)(315,639)(316,638)(317,645)(318,648)(319,647)
(320,646)(321,641)(322,644)(323,643)(324,642)(325,469)(326,472)(327,471)
(328,470)(329,477)(330,480)(331,479)(332,478)(333,473)(334,476)(335,475)
(336,474)(337,481)(338,484)(339,483)(340,482)(341,489)(342,492)(343,491)
(344,490)(345,485)(346,488)(347,487)(348,486)(349,493)(350,496)(351,495)
(352,494)(353,501)(354,504)(355,503)(356,502)(357,497)(358,500)(359,499)
(360,498)(361,433)(362,436)(363,435)(364,434)(365,441)(366,444)(367,443)
(368,442)(369,437)(370,440)(371,439)(372,438)(373,445)(374,448)(375,447)
(376,446)(377,453)(378,456)(379,455)(380,454)(381,449)(382,452)(383,451)
(384,450)(385,457)(386,460)(387,459)(388,458)(389,465)(390,468)(391,467)
(392,466)(393,461)(394,464)(395,463)(396,462)(397,505)(398,508)(399,507)
(400,506)(401,513)(402,516)(403,515)(404,514)(405,509)(406,512)(407,511)
(408,510)(409,517)(410,520)(411,519)(412,518)(413,525)(414,528)(415,527)
(416,526)(417,521)(418,524)(419,523)(420,522)(421,529)(422,532)(423,531)
(424,530)(425,537)(426,540)(427,539)(428,538)(429,533)(430,536)(431,535)
(432,534);
poly := sub<Sym(864)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
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