# Polytope of Type {6,6,8}

Atlas Canonical Name : {6,6,8}*1728f
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 4
Schlafli Type : {6,6,8}
Number of vertices, edges, etc : 6, 54, 72, 24
Order of s0s1s2s3 : 24
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 : {6,6,4}*864j
3-fold quotients : {2,6,8}*576
6-fold quotients : {2,6,4}*288
9-fold quotients : {6,2,8}*192
12-fold quotients : {2,6,4}*144
18-fold quotients : {3,2,8}*96, {6,2,4}*96
27-fold quotients : {2,2,8}*64
36-fold quotients : {3,2,4}*48, {6,2,2}*48
54-fold quotients : {2,2,4}*32
72-fold quotients : {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,217)(  2,218)(  3,219)(  4,220)(  5,221)(  6,222)(  7,223)(  8,224)
(  9,225)( 10,235)( 11,236)( 12,237)( 13,238)( 14,239)( 15,240)( 16,241)
( 17,242)( 18,243)( 19,226)( 20,227)( 21,228)( 22,229)( 23,230)( 24,231)
( 25,232)( 26,233)( 27,234)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,262)( 38,263)( 39,264)( 40,265)
( 41,266)( 42,267)( 43,268)( 44,269)( 45,270)( 46,253)( 47,254)( 48,255)
( 49,256)( 50,257)( 51,258)( 52,259)( 53,260)( 54,261)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,289)
( 65,290)( 66,291)( 67,292)( 68,293)( 69,294)( 70,295)( 71,296)( 72,297)
( 73,280)( 74,281)( 75,282)( 76,283)( 77,284)( 78,285)( 79,286)( 80,287)
( 81,288)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,316)( 92,317)( 93,318)( 94,319)( 95,320)( 96,321)
( 97,322)( 98,323)( 99,324)(100,307)(101,308)(102,309)(103,310)(104,311)
(105,312)(106,313)(107,314)(108,315)(109,325)(110,326)(111,327)(112,328)
(113,329)(114,330)(115,331)(116,332)(117,333)(118,343)(119,344)(120,345)
(121,346)(122,347)(123,348)(124,349)(125,350)(126,351)(127,334)(128,335)
(129,336)(130,337)(131,338)(132,339)(133,340)(134,341)(135,342)(136,352)
(137,353)(138,354)(139,355)(140,356)(141,357)(142,358)(143,359)(144,360)
(145,370)(146,371)(147,372)(148,373)(149,374)(150,375)(151,376)(152,377)
(153,378)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)
(161,368)(162,369)(163,379)(164,380)(165,381)(166,382)(167,383)(168,384)
(169,385)(170,386)(171,387)(172,397)(173,398)(174,399)(175,400)(176,401)
(177,402)(178,403)(179,404)(180,405)(181,388)(182,389)(183,390)(184,391)
(185,392)(186,393)(187,394)(188,395)(189,396)(190,406)(191,407)(192,408)
(193,409)(194,410)(195,411)(196,412)(197,413)(198,414)(199,424)(200,425)
(201,426)(202,427)(203,428)(204,429)(205,430)(206,431)(207,432)(208,415)
(209,416)(210,417)(211,418)(212,419)(213,420)(214,421)(215,422)(216,423)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,667)(443,668)(444,669)(445,670)(446,671)(447,672)(448,673)
(449,674)(450,675)(451,658)(452,659)(453,660)(454,661)(455,662)(456,663)
(457,664)(458,665)(459,666)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,694)(470,695)(471,696)(472,697)
(473,698)(474,699)(475,700)(476,701)(477,702)(478,685)(479,686)(480,687)
(481,688)(482,689)(483,690)(484,691)(485,692)(486,693)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,721)
(497,722)(498,723)(499,724)(500,725)(501,726)(502,727)(503,728)(504,729)
(505,712)(506,713)(507,714)(508,715)(509,716)(510,717)(511,718)(512,719)
(513,720)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,748)(524,749)(525,750)(526,751)(527,752)(528,753)
(529,754)(530,755)(531,756)(532,739)(533,740)(534,741)(535,742)(536,743)
(537,744)(538,745)(539,746)(540,747)(541,757)(542,758)(543,759)(544,760)
(545,761)(546,762)(547,763)(548,764)(549,765)(550,775)(551,776)(552,777)
(553,778)(554,779)(555,780)(556,781)(557,782)(558,783)(559,766)(560,767)
(561,768)(562,769)(563,770)(564,771)(565,772)(566,773)(567,774)(568,784)
(569,785)(570,786)(571,787)(572,788)(573,789)(574,790)(575,791)(576,792)
(577,802)(578,803)(579,804)(580,805)(581,806)(582,807)(583,808)(584,809)
(585,810)(586,793)(587,794)(588,795)(589,796)(590,797)(591,798)(592,799)
(593,800)(594,801)(595,811)(596,812)(597,813)(598,814)(599,815)(600,816)
(601,817)(602,818)(603,819)(604,829)(605,830)(606,831)(607,832)(608,833)
(609,834)(610,835)(611,836)(612,837)(613,820)(614,821)(615,822)(616,823)
(617,824)(618,825)(619,826)(620,827)(621,828)(622,838)(623,839)(624,840)
(625,841)(626,842)(627,843)(628,844)(629,845)(630,846)(631,856)(632,857)
(633,858)(634,859)(635,860)(636,861)(637,862)(638,863)(639,864)(640,847)
(641,848)(642,849)(643,850)(644,851)(645,852)(646,853)(647,854)(648,855);;
s1 := (  1,388)(  2,390)(  3,389)(  4,394)(  5,396)(  6,395)(  7,391)(  8,393)
(  9,392)( 10,379)( 11,381)( 12,380)( 13,385)( 14,387)( 15,386)( 16,382)
( 17,384)( 18,383)( 19,397)( 20,399)( 21,398)( 22,403)( 23,405)( 24,404)
( 25,400)( 26,402)( 27,401)( 28,415)( 29,417)( 30,416)( 31,421)( 32,423)
( 33,422)( 34,418)( 35,420)( 36,419)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,424)( 47,426)( 48,425)
( 49,430)( 50,432)( 51,431)( 52,427)( 53,429)( 54,428)( 55,361)( 56,363)
( 57,362)( 58,367)( 59,369)( 60,368)( 61,364)( 62,366)( 63,365)( 64,352)
( 65,354)( 66,353)( 67,358)( 68,360)( 69,359)( 70,355)( 71,357)( 72,356)
( 73,370)( 74,372)( 75,371)( 76,376)( 77,378)( 78,377)( 79,373)( 80,375)
( 81,374)( 82,334)( 83,336)( 84,335)( 85,340)( 86,342)( 87,341)( 88,337)
( 89,339)( 90,338)( 91,325)( 92,327)( 93,326)( 94,331)( 95,333)( 96,332)
( 97,328)( 98,330)( 99,329)(100,343)(101,345)(102,344)(103,349)(104,351)
(105,350)(106,346)(107,348)(108,347)(109,307)(110,309)(111,308)(112,313)
(113,315)(114,314)(115,310)(116,312)(117,311)(118,298)(119,300)(120,299)
(121,304)(122,306)(123,305)(124,301)(125,303)(126,302)(127,316)(128,318)
(129,317)(130,322)(131,324)(132,323)(133,319)(134,321)(135,320)(136,280)
(137,282)(138,281)(139,286)(140,288)(141,287)(142,283)(143,285)(144,284)
(145,271)(146,273)(147,272)(148,277)(149,279)(150,278)(151,274)(152,276)
(153,275)(154,289)(155,291)(156,290)(157,295)(158,297)(159,296)(160,292)
(161,294)(162,293)(163,226)(164,228)(165,227)(166,232)(167,234)(168,233)
(169,229)(170,231)(171,230)(172,217)(173,219)(174,218)(175,223)(176,225)
(177,224)(178,220)(179,222)(180,221)(181,235)(182,237)(183,236)(184,241)
(185,243)(186,242)(187,238)(188,240)(189,239)(190,253)(191,255)(192,254)
(193,259)(194,261)(195,260)(196,256)(197,258)(198,257)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,262)
(209,264)(210,263)(211,268)(212,270)(213,269)(214,265)(215,267)(216,266)
(433,820)(434,822)(435,821)(436,826)(437,828)(438,827)(439,823)(440,825)
(441,824)(442,811)(443,813)(444,812)(445,817)(446,819)(447,818)(448,814)
(449,816)(450,815)(451,829)(452,831)(453,830)(454,835)(455,837)(456,836)
(457,832)(458,834)(459,833)(460,847)(461,849)(462,848)(463,853)(464,855)
(465,854)(466,850)(467,852)(468,851)(469,838)(470,840)(471,839)(472,844)
(473,846)(474,845)(475,841)(476,843)(477,842)(478,856)(479,858)(480,857)
(481,862)(482,864)(483,863)(484,859)(485,861)(486,860)(487,793)(488,795)
(489,794)(490,799)(491,801)(492,800)(493,796)(494,798)(495,797)(496,784)
(497,786)(498,785)(499,790)(500,792)(501,791)(502,787)(503,789)(504,788)
(505,802)(506,804)(507,803)(508,808)(509,810)(510,809)(511,805)(512,807)
(513,806)(514,766)(515,768)(516,767)(517,772)(518,774)(519,773)(520,769)
(521,771)(522,770)(523,757)(524,759)(525,758)(526,763)(527,765)(528,764)
(529,760)(530,762)(531,761)(532,775)(533,777)(534,776)(535,781)(536,783)
(537,782)(538,778)(539,780)(540,779)(541,739)(542,741)(543,740)(544,745)
(545,747)(546,746)(547,742)(548,744)(549,743)(550,730)(551,732)(552,731)
(553,736)(554,738)(555,737)(556,733)(557,735)(558,734)(559,748)(560,750)
(561,749)(562,754)(563,756)(564,755)(565,751)(566,753)(567,752)(568,712)
(569,714)(570,713)(571,718)(572,720)(573,719)(574,715)(575,717)(576,716)
(577,703)(578,705)(579,704)(580,709)(581,711)(582,710)(583,706)(584,708)
(585,707)(586,721)(587,723)(588,722)(589,727)(590,729)(591,728)(592,724)
(593,726)(594,725)(595,658)(596,660)(597,659)(598,664)(599,666)(600,665)
(601,661)(602,663)(603,662)(604,649)(605,651)(606,650)(607,655)(608,657)
(609,656)(610,652)(611,654)(612,653)(613,667)(614,669)(615,668)(616,673)
(617,675)(618,674)(619,670)(620,672)(621,671)(622,685)(623,687)(624,686)
(625,691)(626,693)(627,692)(628,688)(629,690)(630,689)(631,676)(632,678)
(633,677)(634,682)(635,684)(636,683)(637,679)(638,681)(639,680)(640,694)
(641,696)(642,695)(643,700)(644,702)(645,701)(646,697)(647,699)(648,698);;
s2 := (  1,  4)(  3,  9)(  5,  8)( 10, 13)( 12, 18)( 14, 17)( 19, 22)( 21, 27)
( 23, 26)( 28, 31)( 30, 36)( 32, 35)( 37, 40)( 39, 45)( 41, 44)( 46, 49)
( 48, 54)( 50, 53)( 55, 85)( 56, 83)( 57, 90)( 58, 82)( 59, 89)( 60, 87)
( 61, 88)( 62, 86)( 63, 84)( 64, 94)( 65, 92)( 66, 99)( 67, 91)( 68, 98)
( 69, 96)( 70, 97)( 71, 95)( 72, 93)( 73,103)( 74,101)( 75,108)( 76,100)
( 77,107)( 78,105)( 79,106)( 80,104)( 81,102)(109,139)(110,137)(111,144)
(112,136)(113,143)(114,141)(115,142)(116,140)(117,138)(118,148)(119,146)
(120,153)(121,145)(122,152)(123,150)(124,151)(125,149)(126,147)(127,157)
(128,155)(129,162)(130,154)(131,161)(132,159)(133,160)(134,158)(135,156)
(163,166)(165,171)(167,170)(172,175)(174,180)(176,179)(181,184)(183,189)
(185,188)(190,193)(192,198)(194,197)(199,202)(201,207)(203,206)(208,211)
(210,216)(212,215)(217,220)(219,225)(221,224)(226,229)(228,234)(230,233)
(235,238)(237,243)(239,242)(244,247)(246,252)(248,251)(253,256)(255,261)
(257,260)(262,265)(264,270)(266,269)(271,301)(272,299)(273,306)(274,298)
(275,305)(276,303)(277,304)(278,302)(279,300)(280,310)(281,308)(282,315)
(283,307)(284,314)(285,312)(286,313)(287,311)(288,309)(289,319)(290,317)
(291,324)(292,316)(293,323)(294,321)(295,322)(296,320)(297,318)(325,355)
(326,353)(327,360)(328,352)(329,359)(330,357)(331,358)(332,356)(333,354)
(334,364)(335,362)(336,369)(337,361)(338,368)(339,366)(340,367)(341,365)
(342,363)(343,373)(344,371)(345,378)(346,370)(347,377)(348,375)(349,376)
(350,374)(351,372)(379,382)(381,387)(383,386)(388,391)(390,396)(392,395)
(397,400)(399,405)(401,404)(406,409)(408,414)(410,413)(415,418)(417,423)
(419,422)(424,427)(426,432)(428,431)(433,490)(434,488)(435,495)(436,487)
(437,494)(438,492)(439,493)(440,491)(441,489)(442,499)(443,497)(444,504)
(445,496)(446,503)(447,501)(448,502)(449,500)(450,498)(451,508)(452,506)
(453,513)(454,505)(455,512)(456,510)(457,511)(458,509)(459,507)(460,517)
(461,515)(462,522)(463,514)(464,521)(465,519)(466,520)(467,518)(468,516)
(469,526)(470,524)(471,531)(472,523)(473,530)(474,528)(475,529)(476,527)
(477,525)(478,535)(479,533)(480,540)(481,532)(482,539)(483,537)(484,538)
(485,536)(486,534)(541,625)(542,623)(543,630)(544,622)(545,629)(546,627)
(547,628)(548,626)(549,624)(550,634)(551,632)(552,639)(553,631)(554,638)
(555,636)(556,637)(557,635)(558,633)(559,643)(560,641)(561,648)(562,640)
(563,647)(564,645)(565,646)(566,644)(567,642)(568,598)(569,596)(570,603)
(571,595)(572,602)(573,600)(574,601)(575,599)(576,597)(577,607)(578,605)
(579,612)(580,604)(581,611)(582,609)(583,610)(584,608)(585,606)(586,616)
(587,614)(588,621)(589,613)(590,620)(591,618)(592,619)(593,617)(594,615)
(649,706)(650,704)(651,711)(652,703)(653,710)(654,708)(655,709)(656,707)
(657,705)(658,715)(659,713)(660,720)(661,712)(662,719)(663,717)(664,718)
(665,716)(666,714)(667,724)(668,722)(669,729)(670,721)(671,728)(672,726)
(673,727)(674,725)(675,723)(676,733)(677,731)(678,738)(679,730)(680,737)
(681,735)(682,736)(683,734)(684,732)(685,742)(686,740)(687,747)(688,739)
(689,746)(690,744)(691,745)(692,743)(693,741)(694,751)(695,749)(696,756)
(697,748)(698,755)(699,753)(700,754)(701,752)(702,750)(757,841)(758,839)
(759,846)(760,838)(761,845)(762,843)(763,844)(764,842)(765,840)(766,850)
(767,848)(768,855)(769,847)(770,854)(771,852)(772,853)(773,851)(774,849)
(775,859)(776,857)(777,864)(778,856)(779,863)(780,861)(781,862)(782,860)
(783,858)(784,814)(785,812)(786,819)(787,811)(788,818)(789,816)(790,817)
(791,815)(792,813)(793,823)(794,821)(795,828)(796,820)(797,827)(798,825)
(799,826)(800,824)(801,822)(802,832)(803,830)(804,837)(805,829)(806,836)
(807,834)(808,835)(809,833)(810,831);;
s3 := (  1,433)(  2,434)(  3,435)(  4,441)(  5,439)(  6,440)(  7,437)(  8,438)
(  9,436)( 10,442)( 11,443)( 12,444)( 13,450)( 14,448)( 15,449)( 16,446)
( 17,447)( 18,445)( 19,451)( 20,452)( 21,453)( 22,459)( 23,457)( 24,458)
( 25,455)( 26,456)( 27,454)( 28,460)( 29,461)( 30,462)( 31,468)( 32,466)
( 33,467)( 34,464)( 35,465)( 36,463)( 37,469)( 38,470)( 39,471)( 40,477)
( 41,475)( 42,476)( 43,473)( 44,474)( 45,472)( 46,478)( 47,479)( 48,480)
( 49,486)( 50,484)( 51,485)( 52,482)( 53,483)( 54,481)( 55,514)( 56,515)
( 57,516)( 58,522)( 59,520)( 60,521)( 61,518)( 62,519)( 63,517)( 64,523)
( 65,524)( 66,525)( 67,531)( 68,529)( 69,530)( 70,527)( 71,528)( 72,526)
( 73,532)( 74,533)( 75,534)( 76,540)( 77,538)( 78,539)( 79,536)( 80,537)
( 81,535)( 82,487)( 83,488)( 84,489)( 85,495)( 86,493)( 87,494)( 88,491)
( 89,492)( 90,490)( 91,496)( 92,497)( 93,498)( 94,504)( 95,502)( 96,503)
( 97,500)( 98,501)( 99,499)(100,505)(101,506)(102,507)(103,513)(104,511)
(105,512)(106,509)(107,510)(108,508)(109,568)(110,569)(111,570)(112,576)
(113,574)(114,575)(115,572)(116,573)(117,571)(118,577)(119,578)(120,579)
(121,585)(122,583)(123,584)(124,581)(125,582)(126,580)(127,586)(128,587)
(129,588)(130,594)(131,592)(132,593)(133,590)(134,591)(135,589)(136,541)
(137,542)(138,543)(139,549)(140,547)(141,548)(142,545)(143,546)(144,544)
(145,550)(146,551)(147,552)(148,558)(149,556)(150,557)(151,554)(152,555)
(153,553)(154,559)(155,560)(156,561)(157,567)(158,565)(159,566)(160,563)
(161,564)(162,562)(163,595)(164,596)(165,597)(166,603)(167,601)(168,602)
(169,599)(170,600)(171,598)(172,604)(173,605)(174,606)(175,612)(176,610)
(177,611)(178,608)(179,609)(180,607)(181,613)(182,614)(183,615)(184,621)
(185,619)(186,620)(187,617)(188,618)(189,616)(190,622)(191,623)(192,624)
(193,630)(194,628)(195,629)(196,626)(197,627)(198,625)(199,631)(200,632)
(201,633)(202,639)(203,637)(204,638)(205,635)(206,636)(207,634)(208,640)
(209,641)(210,642)(211,648)(212,646)(213,647)(214,644)(215,645)(216,643)
(217,649)(218,650)(219,651)(220,657)(221,655)(222,656)(223,653)(224,654)
(225,652)(226,658)(227,659)(228,660)(229,666)(230,664)(231,665)(232,662)
(233,663)(234,661)(235,667)(236,668)(237,669)(238,675)(239,673)(240,674)
(241,671)(242,672)(243,670)(244,676)(245,677)(246,678)(247,684)(248,682)
(249,683)(250,680)(251,681)(252,679)(253,685)(254,686)(255,687)(256,693)
(257,691)(258,692)(259,689)(260,690)(261,688)(262,694)(263,695)(264,696)
(265,702)(266,700)(267,701)(268,698)(269,699)(270,697)(271,730)(272,731)
(273,732)(274,738)(275,736)(276,737)(277,734)(278,735)(279,733)(280,739)
(281,740)(282,741)(283,747)(284,745)(285,746)(286,743)(287,744)(288,742)
(289,748)(290,749)(291,750)(292,756)(293,754)(294,755)(295,752)(296,753)
(297,751)(298,703)(299,704)(300,705)(301,711)(302,709)(303,710)(304,707)
(305,708)(306,706)(307,712)(308,713)(309,714)(310,720)(311,718)(312,719)
(313,716)(314,717)(315,715)(316,721)(317,722)(318,723)(319,729)(320,727)
(321,728)(322,725)(323,726)(324,724)(325,784)(326,785)(327,786)(328,792)
(329,790)(330,791)(331,788)(332,789)(333,787)(334,793)(335,794)(336,795)
(337,801)(338,799)(339,800)(340,797)(341,798)(342,796)(343,802)(344,803)
(345,804)(346,810)(347,808)(348,809)(349,806)(350,807)(351,805)(352,757)
(353,758)(354,759)(355,765)(356,763)(357,764)(358,761)(359,762)(360,760)
(361,766)(362,767)(363,768)(364,774)(365,772)(366,773)(367,770)(368,771)
(369,769)(370,775)(371,776)(372,777)(373,783)(374,781)(375,782)(376,779)
(377,780)(378,778)(379,811)(380,812)(381,813)(382,819)(383,817)(384,818)
(385,815)(386,816)(387,814)(388,820)(389,821)(390,822)(391,828)(392,826)
(393,827)(394,824)(395,825)(396,823)(397,829)(398,830)(399,831)(400,837)
(401,835)(402,836)(403,833)(404,834)(405,832)(406,838)(407,839)(408,840)
(409,846)(410,844)(411,845)(412,842)(413,843)(414,841)(415,847)(416,848)
(417,849)(418,855)(419,853)(420,854)(421,851)(422,852)(423,850)(424,856)
(425,857)(426,858)(427,864)(428,862)(429,863)(430,860)(431,861)(432,859);;
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*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```
References : None.
to this polytope