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Polytope of Type {4,12,6}

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