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# Polytope of Type {4,24}

Atlas Canonical Name : {4,24}*1728h
if this polytope has a name.
Group : SmallGroup(1728,31293)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 36, 432, 216
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {4,12}*864d
3-fold quotients : {4,8}*576b
4-fold quotients : {4,12}*432b
6-fold quotients : {4,4}*288
9-fold quotients : {4,24}*192b
12-fold quotients : {4,4}*144
18-fold quotients : {4,12}*96a
24-fold quotients : {4,4}*72
27-fold quotients : {4,8}*64b
36-fold quotients : {2,12}*48, {4,6}*48a
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  8)(  3,  6)(  4,  7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)
( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)
( 48, 51)( 49, 52)( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)
( 61, 85)( 62, 83)( 63, 90)( 64, 91)( 65, 98)( 66, 96)( 67, 97)( 68, 95)
( 69, 93)( 70, 94)( 71, 92)( 72, 99)( 73,100)( 74,107)( 75,105)( 76,106)
( 77,104)( 78,102)( 79,103)( 80,101)( 81,108)(110,116)(111,114)(112,115)
(119,125)(120,123)(121,124)(128,134)(129,132)(130,133)(137,143)(138,141)
(139,142)(146,152)(147,150)(148,151)(155,161)(156,159)(157,160)(163,190)
(164,197)(165,195)(166,196)(167,194)(168,192)(169,193)(170,191)(171,198)
(172,199)(173,206)(174,204)(175,205)(176,203)(177,201)(178,202)(179,200)
(180,207)(181,208)(182,215)(183,213)(184,214)(185,212)(186,210)(187,211)
(188,209)(189,216)(218,224)(219,222)(220,223)(227,233)(228,231)(229,232)
(236,242)(237,240)(238,241)(245,251)(246,249)(247,250)(254,260)(255,258)
(256,259)(263,269)(264,267)(265,268)(271,298)(272,305)(273,303)(274,304)
(275,302)(276,300)(277,301)(278,299)(279,306)(280,307)(281,314)(282,312)
(283,313)(284,311)(285,309)(286,310)(287,308)(288,315)(289,316)(290,323)
(291,321)(292,322)(293,320)(294,318)(295,319)(296,317)(297,324)(326,332)
(327,330)(328,331)(335,341)(336,339)(337,340)(344,350)(345,348)(346,349)
(353,359)(354,357)(355,358)(362,368)(363,366)(364,367)(371,377)(372,375)
(373,376)(379,406)(380,413)(381,411)(382,412)(383,410)(384,408)(385,409)
(386,407)(387,414)(388,415)(389,422)(390,420)(391,421)(392,419)(393,417)
(394,418)(395,416)(396,423)(397,424)(398,431)(399,429)(400,430)(401,428)
(402,426)(403,427)(404,425)(405,432)(433,541)(434,548)(435,546)(436,547)
(437,545)(438,543)(439,544)(440,542)(441,549)(442,550)(443,557)(444,555)
(445,556)(446,554)(447,552)(448,553)(449,551)(450,558)(451,559)(452,566)
(453,564)(454,565)(455,563)(456,561)(457,562)(458,560)(459,567)(460,568)
(461,575)(462,573)(463,574)(464,572)(465,570)(466,571)(467,569)(468,576)
(469,577)(470,584)(471,582)(472,583)(473,581)(474,579)(475,580)(476,578)
(477,585)(478,586)(479,593)(480,591)(481,592)(482,590)(483,588)(484,589)
(485,587)(486,594)(487,622)(488,629)(489,627)(490,628)(491,626)(492,624)
(493,625)(494,623)(495,630)(496,631)(497,638)(498,636)(499,637)(500,635)
(501,633)(502,634)(503,632)(504,639)(505,640)(506,647)(507,645)(508,646)
(509,644)(510,642)(511,643)(512,641)(513,648)(514,595)(515,602)(516,600)
(517,601)(518,599)(519,597)(520,598)(521,596)(522,603)(523,604)(524,611)
(525,609)(526,610)(527,608)(528,606)(529,607)(530,605)(531,612)(532,613)
(533,620)(534,618)(535,619)(536,617)(537,615)(538,616)(539,614)(540,621)
(649,784)(650,791)(651,789)(652,790)(653,788)(654,786)(655,787)(656,785)
(657,792)(658,793)(659,800)(660,798)(661,799)(662,797)(663,795)(664,796)
(665,794)(666,801)(667,802)(668,809)(669,807)(670,808)(671,806)(672,804)
(673,805)(674,803)(675,810)(676,757)(677,764)(678,762)(679,763)(680,761)
(681,759)(682,760)(683,758)(684,765)(685,766)(686,773)(687,771)(688,772)
(689,770)(690,768)(691,769)(692,767)(693,774)(694,775)(695,782)(696,780)
(697,781)(698,779)(699,777)(700,778)(701,776)(702,783)(703,811)(704,818)
(705,816)(706,817)(707,815)(708,813)(709,814)(710,812)(711,819)(712,820)
(713,827)(714,825)(715,826)(716,824)(717,822)(718,823)(719,821)(720,828)
(721,829)(722,836)(723,834)(724,835)(725,833)(726,831)(727,832)(728,830)
(729,837)(730,838)(731,845)(732,843)(733,844)(734,842)(735,840)(736,841)
(737,839)(738,846)(739,847)(740,854)(741,852)(742,853)(743,851)(744,849)
(745,850)(746,848)(747,855)(748,856)(749,863)(750,861)(751,862)(752,860)
(753,858)(754,859)(755,857)(756,864);;
s1 := (  1,649)(  2,650)(  3,651)(  4,657)(  5,655)(  6,656)(  7,653)(  8,654)
(  9,652)( 10,667)( 11,668)( 12,669)( 13,675)( 14,673)( 15,674)( 16,671)
( 17,672)( 18,670)( 19,658)( 20,659)( 21,660)( 22,666)( 23,664)( 24,665)
( 25,662)( 26,663)( 27,661)( 28,676)( 29,677)( 30,678)( 31,684)( 32,682)
( 33,683)( 34,680)( 35,681)( 36,679)( 37,694)( 38,695)( 39,696)( 40,702)
( 41,700)( 42,701)( 43,698)( 44,699)( 45,697)( 46,685)( 47,686)( 48,687)
( 49,693)( 50,691)( 51,692)( 52,689)( 53,690)( 54,688)( 55,730)( 56,731)
( 57,732)( 58,738)( 59,736)( 60,737)( 61,734)( 62,735)( 63,733)( 64,748)
( 65,749)( 66,750)( 67,756)( 68,754)( 69,755)( 70,752)( 71,753)( 72,751)
( 73,739)( 74,740)( 75,741)( 76,747)( 77,745)( 78,746)( 79,743)( 80,744)
( 81,742)( 82,703)( 83,704)( 84,705)( 85,711)( 86,709)( 87,710)( 88,707)
( 89,708)( 90,706)( 91,721)( 92,722)( 93,723)( 94,729)( 95,727)( 96,728)
( 97,725)( 98,726)( 99,724)(100,712)(101,713)(102,714)(103,720)(104,718)
(105,719)(106,716)(107,717)(108,715)(109,757)(110,758)(111,759)(112,765)
(113,763)(114,764)(115,761)(116,762)(117,760)(118,775)(119,776)(120,777)
(121,783)(122,781)(123,782)(124,779)(125,780)(126,778)(127,766)(128,767)
(129,768)(130,774)(131,772)(132,773)(133,770)(134,771)(135,769)(136,784)
(137,785)(138,786)(139,792)(140,790)(141,791)(142,788)(143,789)(144,787)
(145,802)(146,803)(147,804)(148,810)(149,808)(150,809)(151,806)(152,807)
(153,805)(154,793)(155,794)(156,795)(157,801)(158,799)(159,800)(160,797)
(161,798)(162,796)(163,838)(164,839)(165,840)(166,846)(167,844)(168,845)
(169,842)(170,843)(171,841)(172,856)(173,857)(174,858)(175,864)(176,862)
(177,863)(178,860)(179,861)(180,859)(181,847)(182,848)(183,849)(184,855)
(185,853)(186,854)(187,851)(188,852)(189,850)(190,811)(191,812)(192,813)
(193,819)(194,817)(195,818)(196,815)(197,816)(198,814)(199,829)(200,830)
(201,831)(202,837)(203,835)(204,836)(205,833)(206,834)(207,832)(208,820)
(209,821)(210,822)(211,828)(212,826)(213,827)(214,824)(215,825)(216,823)
(217,514)(218,515)(219,516)(220,522)(221,520)(222,521)(223,518)(224,519)
(225,517)(226,532)(227,533)(228,534)(229,540)(230,538)(231,539)(232,536)
(233,537)(234,535)(235,523)(236,524)(237,525)(238,531)(239,529)(240,530)
(241,527)(242,528)(243,526)(244,487)(245,488)(246,489)(247,495)(248,493)
(249,494)(250,491)(251,492)(252,490)(253,505)(254,506)(255,507)(256,513)
(257,511)(258,512)(259,509)(260,510)(261,508)(262,496)(263,497)(264,498)
(265,504)(266,502)(267,503)(268,500)(269,501)(270,499)(271,460)(272,461)
(273,462)(274,468)(275,466)(276,467)(277,464)(278,465)(279,463)(280,478)
(281,479)(282,480)(283,486)(284,484)(285,485)(286,482)(287,483)(288,481)
(289,469)(290,470)(291,471)(292,477)(293,475)(294,476)(295,473)(296,474)
(297,472)(298,433)(299,434)(300,435)(301,441)(302,439)(303,440)(304,437)
(305,438)(306,436)(307,451)(308,452)(309,453)(310,459)(311,457)(312,458)
(313,455)(314,456)(315,454)(316,442)(317,443)(318,444)(319,450)(320,448)
(321,449)(322,446)(323,447)(324,445)(325,622)(326,623)(327,624)(328,630)
(329,628)(330,629)(331,626)(332,627)(333,625)(334,640)(335,641)(336,642)
(337,648)(338,646)(339,647)(340,644)(341,645)(342,643)(343,631)(344,632)
(345,633)(346,639)(347,637)(348,638)(349,635)(350,636)(351,634)(352,595)
(353,596)(354,597)(355,603)(356,601)(357,602)(358,599)(359,600)(360,598)
(361,613)(362,614)(363,615)(364,621)(365,619)(366,620)(367,617)(368,618)
(369,616)(370,604)(371,605)(372,606)(373,612)(374,610)(375,611)(376,608)
(377,609)(378,607)(379,568)(380,569)(381,570)(382,576)(383,574)(384,575)
(385,572)(386,573)(387,571)(388,586)(389,587)(390,588)(391,594)(392,592)
(393,593)(394,590)(395,591)(396,589)(397,577)(398,578)(399,579)(400,585)
(401,583)(402,584)(403,581)(404,582)(405,580)(406,541)(407,542)(408,543)
(409,549)(410,547)(411,548)(412,545)(413,546)(414,544)(415,559)(416,560)
(417,561)(418,567)(419,565)(420,566)(421,563)(422,564)(423,562)(424,550)
(425,551)(426,552)(427,558)(428,556)(429,557)(430,554)(431,555)(432,553);;
s2 := (  1,338)(  2,340)(  3,336)(  4,341)(  5,334)(  6,339)(  7,335)(  8,337)
(  9,342)( 10,329)( 11,331)( 12,327)( 13,332)( 14,325)( 15,330)( 16,326)
( 17,328)( 18,333)( 19,347)( 20,349)( 21,345)( 22,350)( 23,343)( 24,348)
( 25,344)( 26,346)( 27,351)( 28,365)( 29,367)( 30,363)( 31,368)( 32,361)
( 33,366)( 34,362)( 35,364)( 36,369)( 37,356)( 38,358)( 39,354)( 40,359)
( 41,352)( 42,357)( 43,353)( 44,355)( 45,360)( 46,374)( 47,376)( 48,372)
( 49,377)( 50,370)( 51,375)( 52,371)( 53,373)( 54,378)( 55,392)( 56,394)
( 57,390)( 58,395)( 59,388)( 60,393)( 61,389)( 62,391)( 63,396)( 64,383)
( 65,385)( 66,381)( 67,386)( 68,379)( 69,384)( 70,380)( 71,382)( 72,387)
( 73,401)( 74,403)( 75,399)( 76,404)( 77,397)( 78,402)( 79,398)( 80,400)
( 81,405)( 82,419)( 83,421)( 84,417)( 85,422)( 86,415)( 87,420)( 88,416)
( 89,418)( 90,423)( 91,410)( 92,412)( 93,408)( 94,413)( 95,406)( 96,411)
( 97,407)( 98,409)( 99,414)(100,428)(101,430)(102,426)(103,431)(104,424)
(105,429)(106,425)(107,427)(108,432)(109,257)(110,259)(111,255)(112,260)
(113,253)(114,258)(115,254)(116,256)(117,261)(118,248)(119,250)(120,246)
(121,251)(122,244)(123,249)(124,245)(125,247)(126,252)(127,266)(128,268)
(129,264)(130,269)(131,262)(132,267)(133,263)(134,265)(135,270)(136,230)
(137,232)(138,228)(139,233)(140,226)(141,231)(142,227)(143,229)(144,234)
(145,221)(146,223)(147,219)(148,224)(149,217)(150,222)(151,218)(152,220)
(153,225)(154,239)(155,241)(156,237)(157,242)(158,235)(159,240)(160,236)
(161,238)(162,243)(163,311)(164,313)(165,309)(166,314)(167,307)(168,312)
(169,308)(170,310)(171,315)(172,302)(173,304)(174,300)(175,305)(176,298)
(177,303)(178,299)(179,301)(180,306)(181,320)(182,322)(183,318)(184,323)
(185,316)(186,321)(187,317)(188,319)(189,324)(190,284)(191,286)(192,282)
(193,287)(194,280)(195,285)(196,281)(197,283)(198,288)(199,275)(200,277)
(201,273)(202,278)(203,271)(204,276)(205,272)(206,274)(207,279)(208,293)
(209,295)(210,291)(211,296)(212,289)(213,294)(214,290)(215,292)(216,297)
(433,689)(434,691)(435,687)(436,692)(437,685)(438,690)(439,686)(440,688)
(441,693)(442,680)(443,682)(444,678)(445,683)(446,676)(447,681)(448,677)
(449,679)(450,684)(451,698)(452,700)(453,696)(454,701)(455,694)(456,699)
(457,695)(458,697)(459,702)(460,662)(461,664)(462,660)(463,665)(464,658)
(465,663)(466,659)(467,661)(468,666)(469,653)(470,655)(471,651)(472,656)
(473,649)(474,654)(475,650)(476,652)(477,657)(478,671)(479,673)(480,669)
(481,674)(482,667)(483,672)(484,668)(485,670)(486,675)(487,743)(488,745)
(489,741)(490,746)(491,739)(492,744)(493,740)(494,742)(495,747)(496,734)
(497,736)(498,732)(499,737)(500,730)(501,735)(502,731)(503,733)(504,738)
(505,752)(506,754)(507,750)(508,755)(509,748)(510,753)(511,749)(512,751)
(513,756)(514,716)(515,718)(516,714)(517,719)(518,712)(519,717)(520,713)
(521,715)(522,720)(523,707)(524,709)(525,705)(526,710)(527,703)(528,708)
(529,704)(530,706)(531,711)(532,725)(533,727)(534,723)(535,728)(536,721)
(537,726)(538,722)(539,724)(540,729)(541,770)(542,772)(543,768)(544,773)
(545,766)(546,771)(547,767)(548,769)(549,774)(550,761)(551,763)(552,759)
(553,764)(554,757)(555,762)(556,758)(557,760)(558,765)(559,779)(560,781)
(561,777)(562,782)(563,775)(564,780)(565,776)(566,778)(567,783)(568,797)
(569,799)(570,795)(571,800)(572,793)(573,798)(574,794)(575,796)(576,801)
(577,788)(578,790)(579,786)(580,791)(581,784)(582,789)(583,785)(584,787)
(585,792)(586,806)(587,808)(588,804)(589,809)(590,802)(591,807)(592,803)
(593,805)(594,810)(595,824)(596,826)(597,822)(598,827)(599,820)(600,825)
(601,821)(602,823)(603,828)(604,815)(605,817)(606,813)(607,818)(608,811)
(609,816)(610,812)(611,814)(612,819)(613,833)(614,835)(615,831)(616,836)
(617,829)(618,834)(619,830)(620,832)(621,837)(622,851)(623,853)(624,849)
(625,854)(626,847)(627,852)(628,848)(629,850)(630,855)(631,842)(632,844)
(633,840)(634,845)(635,838)(636,843)(637,839)(638,841)(639,846)(640,860)
(641,862)(642,858)(643,863)(644,856)(645,861)(646,857)(647,859)(648,864);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

```
References : None.
to this polytope