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Polytope of Type {4,54,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54,4}*1728a
Also Known As : {{4,54|2},{54,4|2}}. if this polytope has another name.
Group : SmallGroup(1728,2392)
Rank : 4
Schlafli Type : {4,54,4}
Number of vertices, edges, etc : 4, 108, 108, 4
Order of s0s1s2s3 : 108
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,54,4}*864a, {4,54,2}*864a
   3-fold quotients : {4,18,4}*576a
   4-fold quotients : {2,54,2}*432
   6-fold quotients : {2,18,4}*288a, {4,18,2}*288a
   8-fold quotients : {2,27,2}*216
   9-fold quotients : {4,6,4}*192a
   12-fold quotients : {2,18,2}*144
   18-fold quotients : {2,6,4}*96a, {4,6,2}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {4,2,4}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {4,2,2}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,433)(  2,434)(  3,435)(  4,436)(  5,437)(  6,438)(  7,439)(  8,440)
(  9,441)( 10,442)( 11,443)( 12,444)( 13,445)( 14,446)( 15,447)( 16,448)
( 17,449)( 18,450)( 19,451)( 20,452)( 21,453)( 22,454)( 23,455)( 24,456)
( 25,457)( 26,458)( 27,459)( 28,460)( 29,461)( 30,462)( 31,463)( 32,464)
( 33,465)( 34,466)( 35,467)( 36,468)( 37,469)( 38,470)( 39,471)( 40,472)
( 41,473)( 42,474)( 43,475)( 44,476)( 45,477)( 46,478)( 47,479)( 48,480)
( 49,481)( 50,482)( 51,483)( 52,484)( 53,485)( 54,486)( 55,487)( 56,488)
( 57,489)( 58,490)( 59,491)( 60,492)( 61,493)( 62,494)( 63,495)( 64,496)
( 65,497)( 66,498)( 67,499)( 68,500)( 69,501)( 70,502)( 71,503)( 72,504)
( 73,505)( 74,506)( 75,507)( 76,508)( 77,509)( 78,510)( 79,511)( 80,512)
( 81,513)( 82,514)( 83,515)( 84,516)( 85,517)( 86,518)( 87,519)( 88,520)
( 89,521)( 90,522)( 91,523)( 92,524)( 93,525)( 94,526)( 95,527)( 96,528)
( 97,529)( 98,530)( 99,531)(100,532)(101,533)(102,534)(103,535)(104,536)
(105,537)(106,538)(107,539)(108,540)(109,568)(110,569)(111,570)(112,571)
(113,572)(114,573)(115,574)(116,575)(117,576)(118,577)(119,578)(120,579)
(121,580)(122,581)(123,582)(124,583)(125,584)(126,585)(127,586)(128,587)
(129,588)(130,589)(131,590)(132,591)(133,592)(134,593)(135,594)(136,541)
(137,542)(138,543)(139,544)(140,545)(141,546)(142,547)(143,548)(144,549)
(145,550)(146,551)(147,552)(148,553)(149,554)(150,555)(151,556)(152,557)
(153,558)(154,559)(155,560)(156,561)(157,562)(158,563)(159,564)(160,565)
(161,566)(162,567)(163,622)(164,623)(165,624)(166,625)(167,626)(168,627)
(169,628)(170,629)(171,630)(172,631)(173,632)(174,633)(175,634)(176,635)
(177,636)(178,637)(179,638)(180,639)(181,640)(182,641)(183,642)(184,643)
(185,644)(186,645)(187,646)(188,647)(189,648)(190,595)(191,596)(192,597)
(193,598)(194,599)(195,600)(196,601)(197,602)(198,603)(199,604)(200,605)
(201,606)(202,607)(203,608)(204,609)(205,610)(206,611)(207,612)(208,613)
(209,614)(210,615)(211,616)(212,617)(213,618)(214,619)(215,620)(216,621)
(217,649)(218,650)(219,651)(220,652)(221,653)(222,654)(223,655)(224,656)
(225,657)(226,658)(227,659)(228,660)(229,661)(230,662)(231,663)(232,664)
(233,665)(234,666)(235,667)(236,668)(237,669)(238,670)(239,671)(240,672)
(241,673)(242,674)(243,675)(244,676)(245,677)(246,678)(247,679)(248,680)
(249,681)(250,682)(251,683)(252,684)(253,685)(254,686)(255,687)(256,688)
(257,689)(258,690)(259,691)(260,692)(261,693)(262,694)(263,695)(264,696)
(265,697)(266,698)(267,699)(268,700)(269,701)(270,702)(271,703)(272,704)
(273,705)(274,706)(275,707)(276,708)(277,709)(278,710)(279,711)(280,712)
(281,713)(282,714)(283,715)(284,716)(285,717)(286,718)(287,719)(288,720)
(289,721)(290,722)(291,723)(292,724)(293,725)(294,726)(295,727)(296,728)
(297,729)(298,730)(299,731)(300,732)(301,733)(302,734)(303,735)(304,736)
(305,737)(306,738)(307,739)(308,740)(309,741)(310,742)(311,743)(312,744)
(313,745)(314,746)(315,747)(316,748)(317,749)(318,750)(319,751)(320,752)
(321,753)(322,754)(323,755)(324,756)(325,784)(326,785)(327,786)(328,787)
(329,788)(330,789)(331,790)(332,791)(333,792)(334,793)(335,794)(336,795)
(337,796)(338,797)(339,798)(340,799)(341,800)(342,801)(343,802)(344,803)
(345,804)(346,805)(347,806)(348,807)(349,808)(350,809)(351,810)(352,757)
(353,758)(354,759)(355,760)(356,761)(357,762)(358,763)(359,764)(360,765)
(361,766)(362,767)(363,768)(364,769)(365,770)(366,771)(367,772)(368,773)
(369,774)(370,775)(371,776)(372,777)(373,778)(374,779)(375,780)(376,781)
(377,782)(378,783)(379,838)(380,839)(381,840)(382,841)(383,842)(384,843)
(385,844)(386,845)(387,846)(388,847)(389,848)(390,849)(391,850)(392,851)
(393,852)(394,853)(395,854)(396,855)(397,856)(398,857)(399,858)(400,859)
(401,860)(402,861)(403,862)(404,863)(405,864)(406,811)(407,812)(408,813)
(409,814)(410,815)(411,816)(412,817)(413,818)(414,819)(415,820)(416,821)
(417,822)(418,823)(419,824)(420,825)(421,826)(422,827)(423,828)(424,829)
(425,830)(426,831)(427,832)(428,833)(429,834)(430,835)(431,836)(432,837);;
s1 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)
( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)
( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)
( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)(114,115)
(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)
(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)
(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)(166,171)
(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)
(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)(199,216)
(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)
(218,219)(220,225)(221,224)(222,223)(226,243)(227,242)(228,241)(229,240)
(230,239)(231,238)(232,237)(233,236)(234,235)(245,246)(247,252)(248,251)
(249,250)(253,270)(254,269)(255,268)(256,267)(257,266)(258,265)(259,264)
(260,263)(261,262)(272,273)(274,279)(275,278)(276,277)(280,297)(281,296)
(282,295)(283,294)(284,293)(285,292)(286,291)(287,290)(288,289)(299,300)
(301,306)(302,305)(303,304)(307,324)(308,323)(309,322)(310,321)(311,320)
(312,319)(313,318)(314,317)(315,316)(326,327)(328,333)(329,332)(330,331)
(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)(341,344)
(342,343)(353,354)(355,360)(356,359)(357,358)(361,378)(362,377)(363,376)
(364,375)(365,374)(366,373)(367,372)(368,371)(369,370)(380,381)(382,387)
(383,386)(384,385)(388,405)(389,404)(390,403)(391,402)(392,401)(393,400)
(394,399)(395,398)(396,397)(407,408)(409,414)(410,413)(411,412)(415,432)
(416,431)(417,430)(418,429)(419,428)(420,427)(421,426)(422,425)(423,424)
(433,460)(434,462)(435,461)(436,468)(437,467)(438,466)(439,465)(440,464)
(441,463)(442,486)(443,485)(444,484)(445,483)(446,482)(447,481)(448,480)
(449,479)(450,478)(451,477)(452,476)(453,475)(454,474)(455,473)(456,472)
(457,471)(458,470)(459,469)(487,514)(488,516)(489,515)(490,522)(491,521)
(492,520)(493,519)(494,518)(495,517)(496,540)(497,539)(498,538)(499,537)
(500,536)(501,535)(502,534)(503,533)(504,532)(505,531)(506,530)(507,529)
(508,528)(509,527)(510,526)(511,525)(512,524)(513,523)(541,568)(542,570)
(543,569)(544,576)(545,575)(546,574)(547,573)(548,572)(549,571)(550,594)
(551,593)(552,592)(553,591)(554,590)(555,589)(556,588)(557,587)(558,586)
(559,585)(560,584)(561,583)(562,582)(563,581)(564,580)(565,579)(566,578)
(567,577)(595,622)(596,624)(597,623)(598,630)(599,629)(600,628)(601,627)
(602,626)(603,625)(604,648)(605,647)(606,646)(607,645)(608,644)(609,643)
(610,642)(611,641)(612,640)(613,639)(614,638)(615,637)(616,636)(617,635)
(618,634)(619,633)(620,632)(621,631)(649,676)(650,678)(651,677)(652,684)
(653,683)(654,682)(655,681)(656,680)(657,679)(658,702)(659,701)(660,700)
(661,699)(662,698)(663,697)(664,696)(665,695)(666,694)(667,693)(668,692)
(669,691)(670,690)(671,689)(672,688)(673,687)(674,686)(675,685)(703,730)
(704,732)(705,731)(706,738)(707,737)(708,736)(709,735)(710,734)(711,733)
(712,756)(713,755)(714,754)(715,753)(716,752)(717,751)(718,750)(719,749)
(720,748)(721,747)(722,746)(723,745)(724,744)(725,743)(726,742)(727,741)
(728,740)(729,739)(757,784)(758,786)(759,785)(760,792)(761,791)(762,790)
(763,789)(764,788)(765,787)(766,810)(767,809)(768,808)(769,807)(770,806)
(771,805)(772,804)(773,803)(774,802)(775,801)(776,800)(777,799)(778,798)
(779,797)(780,796)(781,795)(782,794)(783,793)(811,838)(812,840)(813,839)
(814,846)(815,845)(816,844)(817,843)(818,842)(819,841)(820,864)(821,863)
(822,862)(823,861)(824,860)(825,859)(826,858)(827,857)(828,856)(829,855)
(830,854)(831,853)(832,852)(833,851)(834,850)(835,849)(836,848)(837,847);;
s2 := (  1,118)(  2,120)(  3,119)(  4,126)(  5,125)(  6,124)(  7,123)(  8,122)
(  9,121)( 10,109)( 11,111)( 12,110)( 13,117)( 14,116)( 15,115)( 16,114)
( 17,113)( 18,112)( 19,135)( 20,134)( 21,133)( 22,132)( 23,131)( 24,130)
( 25,129)( 26,128)( 27,127)( 28,145)( 29,147)( 30,146)( 31,153)( 32,152)
( 33,151)( 34,150)( 35,149)( 36,148)( 37,136)( 38,138)( 39,137)( 40,144)
( 41,143)( 42,142)( 43,141)( 44,140)( 45,139)( 46,162)( 47,161)( 48,160)
( 49,159)( 50,158)( 51,157)( 52,156)( 53,155)( 54,154)( 55,172)( 56,174)
( 57,173)( 58,180)( 59,179)( 60,178)( 61,177)( 62,176)( 63,175)( 64,163)
( 65,165)( 66,164)( 67,171)( 68,170)( 69,169)( 70,168)( 71,167)( 72,166)
( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,183)( 80,182)
( 81,181)( 82,199)( 83,201)( 84,200)( 85,207)( 86,206)( 87,205)( 88,204)
( 89,203)( 90,202)( 91,190)( 92,192)( 93,191)( 94,198)( 95,197)( 96,196)
( 97,195)( 98,194)( 99,193)(100,216)(101,215)(102,214)(103,213)(104,212)
(105,211)(106,210)(107,209)(108,208)(217,334)(218,336)(219,335)(220,342)
(221,341)(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)(228,326)
(229,333)(230,332)(231,331)(232,330)(233,329)(234,328)(235,351)(236,350)
(237,349)(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)(244,361)
(245,363)(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)(252,364)
(253,352)(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)(260,356)
(261,355)(262,378)(263,377)(264,376)(265,375)(266,374)(267,373)(268,372)
(269,371)(270,370)(271,388)(272,390)(273,389)(274,396)(275,395)(276,394)
(277,393)(278,392)(279,391)(280,379)(281,381)(282,380)(283,387)(284,386)
(285,385)(286,384)(287,383)(288,382)(289,405)(290,404)(291,403)(292,402)
(293,401)(294,400)(295,399)(296,398)(297,397)(298,415)(299,417)(300,416)
(301,423)(302,422)(303,421)(304,420)(305,419)(306,418)(307,406)(308,408)
(309,407)(310,414)(311,413)(312,412)(313,411)(314,410)(315,409)(316,432)
(317,431)(318,430)(319,429)(320,428)(321,427)(322,426)(323,425)(324,424)
(433,577)(434,579)(435,578)(436,585)(437,584)(438,583)(439,582)(440,581)
(441,580)(442,568)(443,570)(444,569)(445,576)(446,575)(447,574)(448,573)
(449,572)(450,571)(451,594)(452,593)(453,592)(454,591)(455,590)(456,589)
(457,588)(458,587)(459,586)(460,550)(461,552)(462,551)(463,558)(464,557)
(465,556)(466,555)(467,554)(468,553)(469,541)(470,543)(471,542)(472,549)
(473,548)(474,547)(475,546)(476,545)(477,544)(478,567)(479,566)(480,565)
(481,564)(482,563)(483,562)(484,561)(485,560)(486,559)(487,631)(488,633)
(489,632)(490,639)(491,638)(492,637)(493,636)(494,635)(495,634)(496,622)
(497,624)(498,623)(499,630)(500,629)(501,628)(502,627)(503,626)(504,625)
(505,648)(506,647)(507,646)(508,645)(509,644)(510,643)(511,642)(512,641)
(513,640)(514,604)(515,606)(516,605)(517,612)(518,611)(519,610)(520,609)
(521,608)(522,607)(523,595)(524,597)(525,596)(526,603)(527,602)(528,601)
(529,600)(530,599)(531,598)(532,621)(533,620)(534,619)(535,618)(536,617)
(537,616)(538,615)(539,614)(540,613)(649,793)(650,795)(651,794)(652,801)
(653,800)(654,799)(655,798)(656,797)(657,796)(658,784)(659,786)(660,785)
(661,792)(662,791)(663,790)(664,789)(665,788)(666,787)(667,810)(668,809)
(669,808)(670,807)(671,806)(672,805)(673,804)(674,803)(675,802)(676,766)
(677,768)(678,767)(679,774)(680,773)(681,772)(682,771)(683,770)(684,769)
(685,757)(686,759)(687,758)(688,765)(689,764)(690,763)(691,762)(692,761)
(693,760)(694,783)(695,782)(696,781)(697,780)(698,779)(699,778)(700,777)
(701,776)(702,775)(703,847)(704,849)(705,848)(706,855)(707,854)(708,853)
(709,852)(710,851)(711,850)(712,838)(713,840)(714,839)(715,846)(716,845)
(717,844)(718,843)(719,842)(720,841)(721,864)(722,863)(723,862)(724,861)
(725,860)(726,859)(727,858)(728,857)(729,856)(730,820)(731,822)(732,821)
(733,828)(734,827)(735,826)(736,825)(737,824)(738,823)(739,811)(740,813)
(741,812)(742,819)(743,818)(744,817)(745,816)(746,815)(747,814)(748,837)
(749,836)(750,835)(751,834)(752,833)(753,832)(754,831)(755,830)(756,829);;
s3 := (  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)(  8,224)
(  9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)( 96,312)
( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)(448,664)
(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)(472,688)
(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,712)
(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)(504,720)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)(528,744)
(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)(544,814)
(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)(552,822)
(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)(560,830)
(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)(576,846)
(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)(584,854)
(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)(600,762)
(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)(608,770)
(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)(616,778)
(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)(632,794)
(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)(640,802)
(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)(648,810);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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*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*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*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*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(864)!(  1,433)(  2,434)(  3,435)(  4,436)(  5,437)(  6,438)(  7,439)
(  8,440)(  9,441)( 10,442)( 11,443)( 12,444)( 13,445)( 14,446)( 15,447)
( 16,448)( 17,449)( 18,450)( 19,451)( 20,452)( 21,453)( 22,454)( 23,455)
( 24,456)( 25,457)( 26,458)( 27,459)( 28,460)( 29,461)( 30,462)( 31,463)
( 32,464)( 33,465)( 34,466)( 35,467)( 36,468)( 37,469)( 38,470)( 39,471)
( 40,472)( 41,473)( 42,474)( 43,475)( 44,476)( 45,477)( 46,478)( 47,479)
( 48,480)( 49,481)( 50,482)( 51,483)( 52,484)( 53,485)( 54,486)( 55,487)
( 56,488)( 57,489)( 58,490)( 59,491)( 60,492)( 61,493)( 62,494)( 63,495)
( 64,496)( 65,497)( 66,498)( 67,499)( 68,500)( 69,501)( 70,502)( 71,503)
( 72,504)( 73,505)( 74,506)( 75,507)( 76,508)( 77,509)( 78,510)( 79,511)
( 80,512)( 81,513)( 82,514)( 83,515)( 84,516)( 85,517)( 86,518)( 87,519)
( 88,520)( 89,521)( 90,522)( 91,523)( 92,524)( 93,525)( 94,526)( 95,527)
( 96,528)( 97,529)( 98,530)( 99,531)(100,532)(101,533)(102,534)(103,535)
(104,536)(105,537)(106,538)(107,539)(108,540)(109,568)(110,569)(111,570)
(112,571)(113,572)(114,573)(115,574)(116,575)(117,576)(118,577)(119,578)
(120,579)(121,580)(122,581)(123,582)(124,583)(125,584)(126,585)(127,586)
(128,587)(129,588)(130,589)(131,590)(132,591)(133,592)(134,593)(135,594)
(136,541)(137,542)(138,543)(139,544)(140,545)(141,546)(142,547)(143,548)
(144,549)(145,550)(146,551)(147,552)(148,553)(149,554)(150,555)(151,556)
(152,557)(153,558)(154,559)(155,560)(156,561)(157,562)(158,563)(159,564)
(160,565)(161,566)(162,567)(163,622)(164,623)(165,624)(166,625)(167,626)
(168,627)(169,628)(170,629)(171,630)(172,631)(173,632)(174,633)(175,634)
(176,635)(177,636)(178,637)(179,638)(180,639)(181,640)(182,641)(183,642)
(184,643)(185,644)(186,645)(187,646)(188,647)(189,648)(190,595)(191,596)
(192,597)(193,598)(194,599)(195,600)(196,601)(197,602)(198,603)(199,604)
(200,605)(201,606)(202,607)(203,608)(204,609)(205,610)(206,611)(207,612)
(208,613)(209,614)(210,615)(211,616)(212,617)(213,618)(214,619)(215,620)
(216,621)(217,649)(218,650)(219,651)(220,652)(221,653)(222,654)(223,655)
(224,656)(225,657)(226,658)(227,659)(228,660)(229,661)(230,662)(231,663)
(232,664)(233,665)(234,666)(235,667)(236,668)(237,669)(238,670)(239,671)
(240,672)(241,673)(242,674)(243,675)(244,676)(245,677)(246,678)(247,679)
(248,680)(249,681)(250,682)(251,683)(252,684)(253,685)(254,686)(255,687)
(256,688)(257,689)(258,690)(259,691)(260,692)(261,693)(262,694)(263,695)
(264,696)(265,697)(266,698)(267,699)(268,700)(269,701)(270,702)(271,703)
(272,704)(273,705)(274,706)(275,707)(276,708)(277,709)(278,710)(279,711)
(280,712)(281,713)(282,714)(283,715)(284,716)(285,717)(286,718)(287,719)
(288,720)(289,721)(290,722)(291,723)(292,724)(293,725)(294,726)(295,727)
(296,728)(297,729)(298,730)(299,731)(300,732)(301,733)(302,734)(303,735)
(304,736)(305,737)(306,738)(307,739)(308,740)(309,741)(310,742)(311,743)
(312,744)(313,745)(314,746)(315,747)(316,748)(317,749)(318,750)(319,751)
(320,752)(321,753)(322,754)(323,755)(324,756)(325,784)(326,785)(327,786)
(328,787)(329,788)(330,789)(331,790)(332,791)(333,792)(334,793)(335,794)
(336,795)(337,796)(338,797)(339,798)(340,799)(341,800)(342,801)(343,802)
(344,803)(345,804)(346,805)(347,806)(348,807)(349,808)(350,809)(351,810)
(352,757)(353,758)(354,759)(355,760)(356,761)(357,762)(358,763)(359,764)
(360,765)(361,766)(362,767)(363,768)(364,769)(365,770)(366,771)(367,772)
(368,773)(369,774)(370,775)(371,776)(372,777)(373,778)(374,779)(375,780)
(376,781)(377,782)(378,783)(379,838)(380,839)(381,840)(382,841)(383,842)
(384,843)(385,844)(386,845)(387,846)(388,847)(389,848)(390,849)(391,850)
(392,851)(393,852)(394,853)(395,854)(396,855)(397,856)(398,857)(399,858)
(400,859)(401,860)(402,861)(403,862)(404,863)(405,864)(406,811)(407,812)
(408,813)(409,814)(410,815)(411,816)(412,817)(413,818)(414,819)(415,820)
(416,821)(417,822)(418,823)(419,824)(420,825)(421,826)(422,827)(423,828)
(424,829)(425,830)(426,831)(427,832)(428,833)(429,834)(430,835)(431,836)
(432,837);
s1 := Sym(864)!(  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)
( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)
( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)
(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)
(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)
(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)
(177,184)(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)
(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)
(207,208)(218,219)(220,225)(221,224)(222,223)(226,243)(227,242)(228,241)
(229,240)(230,239)(231,238)(232,237)(233,236)(234,235)(245,246)(247,252)
(248,251)(249,250)(253,270)(254,269)(255,268)(256,267)(257,266)(258,265)
(259,264)(260,263)(261,262)(272,273)(274,279)(275,278)(276,277)(280,297)
(281,296)(282,295)(283,294)(284,293)(285,292)(286,291)(287,290)(288,289)
(299,300)(301,306)(302,305)(303,304)(307,324)(308,323)(309,322)(310,321)
(311,320)(312,319)(313,318)(314,317)(315,316)(326,327)(328,333)(329,332)
(330,331)(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)
(341,344)(342,343)(353,354)(355,360)(356,359)(357,358)(361,378)(362,377)
(363,376)(364,375)(365,374)(366,373)(367,372)(368,371)(369,370)(380,381)
(382,387)(383,386)(384,385)(388,405)(389,404)(390,403)(391,402)(392,401)
(393,400)(394,399)(395,398)(396,397)(407,408)(409,414)(410,413)(411,412)
(415,432)(416,431)(417,430)(418,429)(419,428)(420,427)(421,426)(422,425)
(423,424)(433,460)(434,462)(435,461)(436,468)(437,467)(438,466)(439,465)
(440,464)(441,463)(442,486)(443,485)(444,484)(445,483)(446,482)(447,481)
(448,480)(449,479)(450,478)(451,477)(452,476)(453,475)(454,474)(455,473)
(456,472)(457,471)(458,470)(459,469)(487,514)(488,516)(489,515)(490,522)
(491,521)(492,520)(493,519)(494,518)(495,517)(496,540)(497,539)(498,538)
(499,537)(500,536)(501,535)(502,534)(503,533)(504,532)(505,531)(506,530)
(507,529)(508,528)(509,527)(510,526)(511,525)(512,524)(513,523)(541,568)
(542,570)(543,569)(544,576)(545,575)(546,574)(547,573)(548,572)(549,571)
(550,594)(551,593)(552,592)(553,591)(554,590)(555,589)(556,588)(557,587)
(558,586)(559,585)(560,584)(561,583)(562,582)(563,581)(564,580)(565,579)
(566,578)(567,577)(595,622)(596,624)(597,623)(598,630)(599,629)(600,628)
(601,627)(602,626)(603,625)(604,648)(605,647)(606,646)(607,645)(608,644)
(609,643)(610,642)(611,641)(612,640)(613,639)(614,638)(615,637)(616,636)
(617,635)(618,634)(619,633)(620,632)(621,631)(649,676)(650,678)(651,677)
(652,684)(653,683)(654,682)(655,681)(656,680)(657,679)(658,702)(659,701)
(660,700)(661,699)(662,698)(663,697)(664,696)(665,695)(666,694)(667,693)
(668,692)(669,691)(670,690)(671,689)(672,688)(673,687)(674,686)(675,685)
(703,730)(704,732)(705,731)(706,738)(707,737)(708,736)(709,735)(710,734)
(711,733)(712,756)(713,755)(714,754)(715,753)(716,752)(717,751)(718,750)
(719,749)(720,748)(721,747)(722,746)(723,745)(724,744)(725,743)(726,742)
(727,741)(728,740)(729,739)(757,784)(758,786)(759,785)(760,792)(761,791)
(762,790)(763,789)(764,788)(765,787)(766,810)(767,809)(768,808)(769,807)
(770,806)(771,805)(772,804)(773,803)(774,802)(775,801)(776,800)(777,799)
(778,798)(779,797)(780,796)(781,795)(782,794)(783,793)(811,838)(812,840)
(813,839)(814,846)(815,845)(816,844)(817,843)(818,842)(819,841)(820,864)
(821,863)(822,862)(823,861)(824,860)(825,859)(826,858)(827,857)(828,856)
(829,855)(830,854)(831,853)(832,852)(833,851)(834,850)(835,849)(836,848)
(837,847);
s2 := Sym(864)!(  1,118)(  2,120)(  3,119)(  4,126)(  5,125)(  6,124)(  7,123)
(  8,122)(  9,121)( 10,109)( 11,111)( 12,110)( 13,117)( 14,116)( 15,115)
( 16,114)( 17,113)( 18,112)( 19,135)( 20,134)( 21,133)( 22,132)( 23,131)
( 24,130)( 25,129)( 26,128)( 27,127)( 28,145)( 29,147)( 30,146)( 31,153)
( 32,152)( 33,151)( 34,150)( 35,149)( 36,148)( 37,136)( 38,138)( 39,137)
( 40,144)( 41,143)( 42,142)( 43,141)( 44,140)( 45,139)( 46,162)( 47,161)
( 48,160)( 49,159)( 50,158)( 51,157)( 52,156)( 53,155)( 54,154)( 55,172)
( 56,174)( 57,173)( 58,180)( 59,179)( 60,178)( 61,177)( 62,176)( 63,175)
( 64,163)( 65,165)( 66,164)( 67,171)( 68,170)( 69,169)( 70,168)( 71,167)
( 72,166)( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,183)
( 80,182)( 81,181)( 82,199)( 83,201)( 84,200)( 85,207)( 86,206)( 87,205)
( 88,204)( 89,203)( 90,202)( 91,190)( 92,192)( 93,191)( 94,198)( 95,197)
( 96,196)( 97,195)( 98,194)( 99,193)(100,216)(101,215)(102,214)(103,213)
(104,212)(105,211)(106,210)(107,209)(108,208)(217,334)(218,336)(219,335)
(220,342)(221,341)(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)
(228,326)(229,333)(230,332)(231,331)(232,330)(233,329)(234,328)(235,351)
(236,350)(237,349)(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)
(244,361)(245,363)(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)
(252,364)(253,352)(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)
(260,356)(261,355)(262,378)(263,377)(264,376)(265,375)(266,374)(267,373)
(268,372)(269,371)(270,370)(271,388)(272,390)(273,389)(274,396)(275,395)
(276,394)(277,393)(278,392)(279,391)(280,379)(281,381)(282,380)(283,387)
(284,386)(285,385)(286,384)(287,383)(288,382)(289,405)(290,404)(291,403)
(292,402)(293,401)(294,400)(295,399)(296,398)(297,397)(298,415)(299,417)
(300,416)(301,423)(302,422)(303,421)(304,420)(305,419)(306,418)(307,406)
(308,408)(309,407)(310,414)(311,413)(312,412)(313,411)(314,410)(315,409)
(316,432)(317,431)(318,430)(319,429)(320,428)(321,427)(322,426)(323,425)
(324,424)(433,577)(434,579)(435,578)(436,585)(437,584)(438,583)(439,582)
(440,581)(441,580)(442,568)(443,570)(444,569)(445,576)(446,575)(447,574)
(448,573)(449,572)(450,571)(451,594)(452,593)(453,592)(454,591)(455,590)
(456,589)(457,588)(458,587)(459,586)(460,550)(461,552)(462,551)(463,558)
(464,557)(465,556)(466,555)(467,554)(468,553)(469,541)(470,543)(471,542)
(472,549)(473,548)(474,547)(475,546)(476,545)(477,544)(478,567)(479,566)
(480,565)(481,564)(482,563)(483,562)(484,561)(485,560)(486,559)(487,631)
(488,633)(489,632)(490,639)(491,638)(492,637)(493,636)(494,635)(495,634)
(496,622)(497,624)(498,623)(499,630)(500,629)(501,628)(502,627)(503,626)
(504,625)(505,648)(506,647)(507,646)(508,645)(509,644)(510,643)(511,642)
(512,641)(513,640)(514,604)(515,606)(516,605)(517,612)(518,611)(519,610)
(520,609)(521,608)(522,607)(523,595)(524,597)(525,596)(526,603)(527,602)
(528,601)(529,600)(530,599)(531,598)(532,621)(533,620)(534,619)(535,618)
(536,617)(537,616)(538,615)(539,614)(540,613)(649,793)(650,795)(651,794)
(652,801)(653,800)(654,799)(655,798)(656,797)(657,796)(658,784)(659,786)
(660,785)(661,792)(662,791)(663,790)(664,789)(665,788)(666,787)(667,810)
(668,809)(669,808)(670,807)(671,806)(672,805)(673,804)(674,803)(675,802)
(676,766)(677,768)(678,767)(679,774)(680,773)(681,772)(682,771)(683,770)
(684,769)(685,757)(686,759)(687,758)(688,765)(689,764)(690,763)(691,762)
(692,761)(693,760)(694,783)(695,782)(696,781)(697,780)(698,779)(699,778)
(700,777)(701,776)(702,775)(703,847)(704,849)(705,848)(706,855)(707,854)
(708,853)(709,852)(710,851)(711,850)(712,838)(713,840)(714,839)(715,846)
(716,845)(717,844)(718,843)(719,842)(720,841)(721,864)(722,863)(723,862)
(724,861)(725,860)(726,859)(727,858)(728,857)(729,856)(730,820)(731,822)
(732,821)(733,828)(734,827)(735,826)(736,825)(737,824)(738,823)(739,811)
(740,813)(741,812)(742,819)(743,818)(744,817)(745,816)(746,815)(747,814)
(748,837)(749,836)(750,835)(751,834)(752,833)(753,832)(754,831)(755,830)
(756,829);
s3 := Sym(864)!(  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)
(  8,224)(  9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)
( 96,312)( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)
(448,664)(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)
(472,688)(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,712)(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)
(504,720)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)
(528,744)(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)
(544,814)(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)
(552,822)(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)
(560,830)(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)
(576,846)(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)
(584,854)(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)
(600,762)(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)
(608,770)(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)
(616,778)(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)
(632,794)(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)
(640,802)(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)
(648,810);
poly := sub<Sym(864)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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*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*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*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*s1*s2*s1*s2 >; 
 
References : None.
to this polytope