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Polytope of Type {12,6,3}

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