Polytope of Type {4,108}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,108}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11356)
Rank : 3
Schlafli Type : {4,108}
Number of vertices, edges, etc : 8, 432, 216
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 4
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 : {4,108}*864b, {4,108}*864c, {4,54}*864
   3-fold quotients : {4,36}*576b
   4-fold quotients : {2,108}*432, {4,27}*432, {4,54}*432b, {4,54}*432c
   6-fold quotients : {4,36}*288b, {4,36}*288c, {4,18}*288
   8-fold quotients : {4,27}*216, {2,54}*216
   9-fold quotients : {4,12}*192b
   12-fold quotients : {2,36}*144, {4,9}*144, {4,18}*144b, {4,18}*144c
   16-fold quotients : {2,27}*108
   18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   24-fold quotients : {4,9}*72, {2,18}*72
   36-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {2,9}*36
   72-fold quotients : {4,3}*24, {2,6}*24
   108-fold quotients : {2,4}*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 := (  1,435)(  2,436)(  3,433)(  4,434)(  5,439)(  6,440)(  7,437)(  8,438)
(  9,443)( 10,444)( 11,441)( 12,442)( 13,447)( 14,448)( 15,445)( 16,446)
( 17,451)( 18,452)( 19,449)( 20,450)( 21,455)( 22,456)( 23,453)( 24,454)
( 25,459)( 26,460)( 27,457)( 28,458)( 29,463)( 30,464)( 31,461)( 32,462)
( 33,467)( 34,468)( 35,465)( 36,466)( 37,471)( 38,472)( 39,469)( 40,470)
( 41,475)( 42,476)( 43,473)( 44,474)( 45,479)( 46,480)( 47,477)( 48,478)
( 49,483)( 50,484)( 51,481)( 52,482)( 53,487)( 54,488)( 55,485)( 56,486)
( 57,491)( 58,492)( 59,489)( 60,490)( 61,495)( 62,496)( 63,493)( 64,494)
( 65,499)( 66,500)( 67,497)( 68,498)( 69,503)( 70,504)( 71,501)( 72,502)
( 73,507)( 74,508)( 75,505)( 76,506)( 77,511)( 78,512)( 79,509)( 80,510)
( 81,515)( 82,516)( 83,513)( 84,514)( 85,519)( 86,520)( 87,517)( 88,518)
( 89,523)( 90,524)( 91,521)( 92,522)( 93,527)( 94,528)( 95,525)( 96,526)
( 97,531)( 98,532)( 99,529)(100,530)(101,535)(102,536)(103,533)(104,534)
(105,539)(106,540)(107,537)(108,538)(109,543)(110,544)(111,541)(112,542)
(113,547)(114,548)(115,545)(116,546)(117,551)(118,552)(119,549)(120,550)
(121,555)(122,556)(123,553)(124,554)(125,559)(126,560)(127,557)(128,558)
(129,563)(130,564)(131,561)(132,562)(133,567)(134,568)(135,565)(136,566)
(137,571)(138,572)(139,569)(140,570)(141,575)(142,576)(143,573)(144,574)
(145,579)(146,580)(147,577)(148,578)(149,583)(150,584)(151,581)(152,582)
(153,587)(154,588)(155,585)(156,586)(157,591)(158,592)(159,589)(160,590)
(161,595)(162,596)(163,593)(164,594)(165,599)(166,600)(167,597)(168,598)
(169,603)(170,604)(171,601)(172,602)(173,607)(174,608)(175,605)(176,606)
(177,611)(178,612)(179,609)(180,610)(181,615)(182,616)(183,613)(184,614)
(185,619)(186,620)(187,617)(188,618)(189,623)(190,624)(191,621)(192,622)
(193,627)(194,628)(195,625)(196,626)(197,631)(198,632)(199,629)(200,630)
(201,635)(202,636)(203,633)(204,634)(205,639)(206,640)(207,637)(208,638)
(209,643)(210,644)(211,641)(212,642)(213,647)(214,648)(215,645)(216,646)
(217,651)(218,652)(219,649)(220,650)(221,655)(222,656)(223,653)(224,654)
(225,659)(226,660)(227,657)(228,658)(229,663)(230,664)(231,661)(232,662)
(233,667)(234,668)(235,665)(236,666)(237,671)(238,672)(239,669)(240,670)
(241,675)(242,676)(243,673)(244,674)(245,679)(246,680)(247,677)(248,678)
(249,683)(250,684)(251,681)(252,682)(253,687)(254,688)(255,685)(256,686)
(257,691)(258,692)(259,689)(260,690)(261,695)(262,696)(263,693)(264,694)
(265,699)(266,700)(267,697)(268,698)(269,703)(270,704)(271,701)(272,702)
(273,707)(274,708)(275,705)(276,706)(277,711)(278,712)(279,709)(280,710)
(281,715)(282,716)(283,713)(284,714)(285,719)(286,720)(287,717)(288,718)
(289,723)(290,724)(291,721)(292,722)(293,727)(294,728)(295,725)(296,726)
(297,731)(298,732)(299,729)(300,730)(301,735)(302,736)(303,733)(304,734)
(305,739)(306,740)(307,737)(308,738)(309,743)(310,744)(311,741)(312,742)
(313,747)(314,748)(315,745)(316,746)(317,751)(318,752)(319,749)(320,750)
(321,755)(322,756)(323,753)(324,754)(325,759)(326,760)(327,757)(328,758)
(329,763)(330,764)(331,761)(332,762)(333,767)(334,768)(335,765)(336,766)
(337,771)(338,772)(339,769)(340,770)(341,775)(342,776)(343,773)(344,774)
(345,779)(346,780)(347,777)(348,778)(349,783)(350,784)(351,781)(352,782)
(353,787)(354,788)(355,785)(356,786)(357,791)(358,792)(359,789)(360,790)
(361,795)(362,796)(363,793)(364,794)(365,799)(366,800)(367,797)(368,798)
(369,803)(370,804)(371,801)(372,802)(373,807)(374,808)(375,805)(376,806)
(377,811)(378,812)(379,809)(380,810)(381,815)(382,816)(383,813)(384,814)
(385,819)(386,820)(387,817)(388,818)(389,823)(390,824)(391,821)(392,822)
(393,827)(394,828)(395,825)(396,826)(397,831)(398,832)(399,829)(400,830)
(401,835)(402,836)(403,833)(404,834)(405,839)(406,840)(407,837)(408,838)
(409,843)(410,844)(411,841)(412,842)(413,847)(414,848)(415,845)(416,846)
(417,851)(418,852)(419,849)(420,850)(421,855)(422,856)(423,853)(424,854)
(425,859)(426,860)(427,857)(428,858)(429,863)(430,864)(431,861)(432,862);;
s1 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)( 43,104)
( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)( 51, 96)
( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)
( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)( 67, 80)
( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)(114,118)
(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)(146,214)
(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)
(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)
(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)(170,190)
(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)(178,182)
(179,184)(180,183)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)
(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)(230,358)
(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)(238,350)
(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)(246,342)
(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,429)(254,430)
(255,432)(256,431)(257,425)(258,426)(259,428)(260,427)(261,421)(262,422)
(263,424)(264,423)(265,417)(266,418)(267,420)(268,419)(269,413)(270,414)
(271,416)(272,415)(273,409)(274,410)(275,412)(276,411)(277,405)(278,406)
(279,408)(280,407)(281,401)(282,402)(283,404)(284,403)(285,397)(286,398)
(287,400)(288,399)(289,393)(290,394)(291,396)(292,395)(293,389)(294,390)
(295,392)(296,391)(297,385)(298,386)(299,388)(300,387)(301,381)(302,382)
(303,384)(304,383)(305,377)(306,378)(307,380)(308,379)(309,373)(310,374)
(311,376)(312,375)(313,369)(314,370)(315,372)(316,371)(317,365)(318,366)
(319,368)(320,367)(321,361)(322,362)(323,364)(324,363)(435,436)(437,441)
(438,442)(439,444)(440,443)(445,465)(446,466)(447,468)(448,467)(449,461)
(450,462)(451,464)(452,463)(453,457)(454,458)(455,460)(456,459)(469,537)
(470,538)(471,540)(472,539)(473,533)(474,534)(475,536)(476,535)(477,529)
(478,530)(479,532)(480,531)(481,525)(482,526)(483,528)(484,527)(485,521)
(486,522)(487,524)(488,523)(489,517)(490,518)(491,520)(492,519)(493,513)
(494,514)(495,516)(496,515)(497,509)(498,510)(499,512)(500,511)(501,505)
(502,506)(503,508)(504,507)(543,544)(545,549)(546,550)(547,552)(548,551)
(553,573)(554,574)(555,576)(556,575)(557,569)(558,570)(559,572)(560,571)
(561,565)(562,566)(563,568)(564,567)(577,645)(578,646)(579,648)(580,647)
(581,641)(582,642)(583,644)(584,643)(585,637)(586,638)(587,640)(588,639)
(589,633)(590,634)(591,636)(592,635)(593,629)(594,630)(595,632)(596,631)
(597,625)(598,626)(599,628)(600,627)(601,621)(602,622)(603,624)(604,623)
(605,617)(606,618)(607,620)(608,619)(609,613)(610,614)(611,616)(612,615)
(649,757)(650,758)(651,760)(652,759)(653,765)(654,766)(655,768)(656,767)
(657,761)(658,762)(659,764)(660,763)(661,789)(662,790)(663,792)(664,791)
(665,785)(666,786)(667,788)(668,787)(669,781)(670,782)(671,784)(672,783)
(673,777)(674,778)(675,780)(676,779)(677,773)(678,774)(679,776)(680,775)
(681,769)(682,770)(683,772)(684,771)(685,861)(686,862)(687,864)(688,863)
(689,857)(690,858)(691,860)(692,859)(693,853)(694,854)(695,856)(696,855)
(697,849)(698,850)(699,852)(700,851)(701,845)(702,846)(703,848)(704,847)
(705,841)(706,842)(707,844)(708,843)(709,837)(710,838)(711,840)(712,839)
(713,833)(714,834)(715,836)(716,835)(717,829)(718,830)(719,832)(720,831)
(721,825)(722,826)(723,828)(724,827)(725,821)(726,822)(727,824)(728,823)
(729,817)(730,818)(731,820)(732,819)(733,813)(734,814)(735,816)(736,815)
(737,809)(738,810)(739,812)(740,811)(741,805)(742,806)(743,808)(744,807)
(745,801)(746,802)(747,804)(748,803)(749,797)(750,798)(751,800)(752,799)
(753,793)(754,794)(755,796)(756,795);;
s2 := (  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)(  8,262)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,285)( 14,288)( 15,287)( 16,286)
( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)( 24,278)
( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)( 32,270)
( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)( 56,246)
( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)( 64,238)
( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,321)( 74,324)( 75,323)( 76,322)( 77,317)( 78,320)( 79,319)( 80,318)
( 81,313)( 82,316)( 83,315)( 84,314)( 85,309)( 86,312)( 87,311)( 88,310)
( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)( 96,302)
( 97,297)( 98,300)( 99,299)(100,298)(101,293)(102,296)(103,295)(104,294)
(105,289)(106,292)(107,291)(108,290)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)(128,390)
(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)(136,382)
(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)(144,374)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)(160,358)
(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)(168,350)
(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)(176,342)
(177,337)(178,340)(179,339)(180,338)(181,429)(182,432)(183,431)(184,430)
(185,425)(186,428)(187,427)(188,426)(189,421)(190,424)(191,423)(192,422)
(193,417)(194,420)(195,419)(196,418)(197,413)(198,416)(199,415)(200,414)
(201,409)(202,412)(203,411)(204,410)(205,405)(206,408)(207,407)(208,406)
(209,401)(210,404)(211,403)(212,402)(213,397)(214,400)(215,399)(216,398)
(433,685)(434,688)(435,687)(436,686)(437,693)(438,696)(439,695)(440,694)
(441,689)(442,692)(443,691)(444,690)(445,717)(446,720)(447,719)(448,718)
(449,713)(450,716)(451,715)(452,714)(453,709)(454,712)(455,711)(456,710)
(457,705)(458,708)(459,707)(460,706)(461,701)(462,704)(463,703)(464,702)
(465,697)(466,700)(467,699)(468,698)(469,649)(470,652)(471,651)(472,650)
(473,657)(474,660)(475,659)(476,658)(477,653)(478,656)(479,655)(480,654)
(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)(488,678)
(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)(496,670)
(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)(504,662)
(505,753)(506,756)(507,755)(508,754)(509,749)(510,752)(511,751)(512,750)
(513,745)(514,748)(515,747)(516,746)(517,741)(518,744)(519,743)(520,742)
(521,737)(522,740)(523,739)(524,738)(525,733)(526,736)(527,735)(528,734)
(529,729)(530,732)(531,731)(532,730)(533,725)(534,728)(535,727)(536,726)
(537,721)(538,724)(539,723)(540,722)(541,793)(542,796)(543,795)(544,794)
(545,801)(546,804)(547,803)(548,802)(549,797)(550,800)(551,799)(552,798)
(553,825)(554,828)(555,827)(556,826)(557,821)(558,824)(559,823)(560,822)
(561,817)(562,820)(563,819)(564,818)(565,813)(566,816)(567,815)(568,814)
(569,809)(570,812)(571,811)(572,810)(573,805)(574,808)(575,807)(576,806)
(577,757)(578,760)(579,759)(580,758)(581,765)(582,768)(583,767)(584,766)
(585,761)(586,764)(587,763)(588,762)(589,789)(590,792)(591,791)(592,790)
(593,785)(594,788)(595,787)(596,786)(597,781)(598,784)(599,783)(600,782)
(601,777)(602,780)(603,779)(604,778)(605,773)(606,776)(607,775)(608,774)
(609,769)(610,772)(611,771)(612,770)(613,861)(614,864)(615,863)(616,862)
(617,857)(618,860)(619,859)(620,858)(621,853)(622,856)(623,855)(624,854)
(625,849)(626,852)(627,851)(628,850)(629,845)(630,848)(631,847)(632,846)
(633,841)(634,844)(635,843)(636,842)(637,837)(638,840)(639,839)(640,838)
(641,833)(642,836)(643,835)(644,834)(645,829)(646,832)(647,831)(648,830);;
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*s0*s1*s2*s1*s2*s1, 
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*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)!(  1,435)(  2,436)(  3,433)(  4,434)(  5,439)(  6,440)(  7,437)
(  8,438)(  9,443)( 10,444)( 11,441)( 12,442)( 13,447)( 14,448)( 15,445)
( 16,446)( 17,451)( 18,452)( 19,449)( 20,450)( 21,455)( 22,456)( 23,453)
( 24,454)( 25,459)( 26,460)( 27,457)( 28,458)( 29,463)( 30,464)( 31,461)
( 32,462)( 33,467)( 34,468)( 35,465)( 36,466)( 37,471)( 38,472)( 39,469)
( 40,470)( 41,475)( 42,476)( 43,473)( 44,474)( 45,479)( 46,480)( 47,477)
( 48,478)( 49,483)( 50,484)( 51,481)( 52,482)( 53,487)( 54,488)( 55,485)
( 56,486)( 57,491)( 58,492)( 59,489)( 60,490)( 61,495)( 62,496)( 63,493)
( 64,494)( 65,499)( 66,500)( 67,497)( 68,498)( 69,503)( 70,504)( 71,501)
( 72,502)( 73,507)( 74,508)( 75,505)( 76,506)( 77,511)( 78,512)( 79,509)
( 80,510)( 81,515)( 82,516)( 83,513)( 84,514)( 85,519)( 86,520)( 87,517)
( 88,518)( 89,523)( 90,524)( 91,521)( 92,522)( 93,527)( 94,528)( 95,525)
( 96,526)( 97,531)( 98,532)( 99,529)(100,530)(101,535)(102,536)(103,533)
(104,534)(105,539)(106,540)(107,537)(108,538)(109,543)(110,544)(111,541)
(112,542)(113,547)(114,548)(115,545)(116,546)(117,551)(118,552)(119,549)
(120,550)(121,555)(122,556)(123,553)(124,554)(125,559)(126,560)(127,557)
(128,558)(129,563)(130,564)(131,561)(132,562)(133,567)(134,568)(135,565)
(136,566)(137,571)(138,572)(139,569)(140,570)(141,575)(142,576)(143,573)
(144,574)(145,579)(146,580)(147,577)(148,578)(149,583)(150,584)(151,581)
(152,582)(153,587)(154,588)(155,585)(156,586)(157,591)(158,592)(159,589)
(160,590)(161,595)(162,596)(163,593)(164,594)(165,599)(166,600)(167,597)
(168,598)(169,603)(170,604)(171,601)(172,602)(173,607)(174,608)(175,605)
(176,606)(177,611)(178,612)(179,609)(180,610)(181,615)(182,616)(183,613)
(184,614)(185,619)(186,620)(187,617)(188,618)(189,623)(190,624)(191,621)
(192,622)(193,627)(194,628)(195,625)(196,626)(197,631)(198,632)(199,629)
(200,630)(201,635)(202,636)(203,633)(204,634)(205,639)(206,640)(207,637)
(208,638)(209,643)(210,644)(211,641)(212,642)(213,647)(214,648)(215,645)
(216,646)(217,651)(218,652)(219,649)(220,650)(221,655)(222,656)(223,653)
(224,654)(225,659)(226,660)(227,657)(228,658)(229,663)(230,664)(231,661)
(232,662)(233,667)(234,668)(235,665)(236,666)(237,671)(238,672)(239,669)
(240,670)(241,675)(242,676)(243,673)(244,674)(245,679)(246,680)(247,677)
(248,678)(249,683)(250,684)(251,681)(252,682)(253,687)(254,688)(255,685)
(256,686)(257,691)(258,692)(259,689)(260,690)(261,695)(262,696)(263,693)
(264,694)(265,699)(266,700)(267,697)(268,698)(269,703)(270,704)(271,701)
(272,702)(273,707)(274,708)(275,705)(276,706)(277,711)(278,712)(279,709)
(280,710)(281,715)(282,716)(283,713)(284,714)(285,719)(286,720)(287,717)
(288,718)(289,723)(290,724)(291,721)(292,722)(293,727)(294,728)(295,725)
(296,726)(297,731)(298,732)(299,729)(300,730)(301,735)(302,736)(303,733)
(304,734)(305,739)(306,740)(307,737)(308,738)(309,743)(310,744)(311,741)
(312,742)(313,747)(314,748)(315,745)(316,746)(317,751)(318,752)(319,749)
(320,750)(321,755)(322,756)(323,753)(324,754)(325,759)(326,760)(327,757)
(328,758)(329,763)(330,764)(331,761)(332,762)(333,767)(334,768)(335,765)
(336,766)(337,771)(338,772)(339,769)(340,770)(341,775)(342,776)(343,773)
(344,774)(345,779)(346,780)(347,777)(348,778)(349,783)(350,784)(351,781)
(352,782)(353,787)(354,788)(355,785)(356,786)(357,791)(358,792)(359,789)
(360,790)(361,795)(362,796)(363,793)(364,794)(365,799)(366,800)(367,797)
(368,798)(369,803)(370,804)(371,801)(372,802)(373,807)(374,808)(375,805)
(376,806)(377,811)(378,812)(379,809)(380,810)(381,815)(382,816)(383,813)
(384,814)(385,819)(386,820)(387,817)(388,818)(389,823)(390,824)(391,821)
(392,822)(393,827)(394,828)(395,825)(396,826)(397,831)(398,832)(399,829)
(400,830)(401,835)(402,836)(403,833)(404,834)(405,839)(406,840)(407,837)
(408,838)(409,843)(410,844)(411,841)(412,842)(413,847)(414,848)(415,845)
(416,846)(417,851)(418,852)(419,849)(420,850)(421,855)(422,856)(423,853)
(424,854)(425,859)(426,860)(427,857)(428,858)(429,863)(430,864)(431,861)
(432,862);
s1 := Sym(864)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)
( 43,104)( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)
( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)
( 59, 88)( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)
( 67, 80)( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)
(114,118)(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)
(146,214)(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)
(154,206)(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)
(162,198)(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)
(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)
(178,182)(179,184)(180,183)(217,325)(218,326)(219,328)(220,327)(221,333)
(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)
(230,358)(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)
(238,350)(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)
(246,342)(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,429)
(254,430)(255,432)(256,431)(257,425)(258,426)(259,428)(260,427)(261,421)
(262,422)(263,424)(264,423)(265,417)(266,418)(267,420)(268,419)(269,413)
(270,414)(271,416)(272,415)(273,409)(274,410)(275,412)(276,411)(277,405)
(278,406)(279,408)(280,407)(281,401)(282,402)(283,404)(284,403)(285,397)
(286,398)(287,400)(288,399)(289,393)(290,394)(291,396)(292,395)(293,389)
(294,390)(295,392)(296,391)(297,385)(298,386)(299,388)(300,387)(301,381)
(302,382)(303,384)(304,383)(305,377)(306,378)(307,380)(308,379)(309,373)
(310,374)(311,376)(312,375)(313,369)(314,370)(315,372)(316,371)(317,365)
(318,366)(319,368)(320,367)(321,361)(322,362)(323,364)(324,363)(435,436)
(437,441)(438,442)(439,444)(440,443)(445,465)(446,466)(447,468)(448,467)
(449,461)(450,462)(451,464)(452,463)(453,457)(454,458)(455,460)(456,459)
(469,537)(470,538)(471,540)(472,539)(473,533)(474,534)(475,536)(476,535)
(477,529)(478,530)(479,532)(480,531)(481,525)(482,526)(483,528)(484,527)
(485,521)(486,522)(487,524)(488,523)(489,517)(490,518)(491,520)(492,519)
(493,513)(494,514)(495,516)(496,515)(497,509)(498,510)(499,512)(500,511)
(501,505)(502,506)(503,508)(504,507)(543,544)(545,549)(546,550)(547,552)
(548,551)(553,573)(554,574)(555,576)(556,575)(557,569)(558,570)(559,572)
(560,571)(561,565)(562,566)(563,568)(564,567)(577,645)(578,646)(579,648)
(580,647)(581,641)(582,642)(583,644)(584,643)(585,637)(586,638)(587,640)
(588,639)(589,633)(590,634)(591,636)(592,635)(593,629)(594,630)(595,632)
(596,631)(597,625)(598,626)(599,628)(600,627)(601,621)(602,622)(603,624)
(604,623)(605,617)(606,618)(607,620)(608,619)(609,613)(610,614)(611,616)
(612,615)(649,757)(650,758)(651,760)(652,759)(653,765)(654,766)(655,768)
(656,767)(657,761)(658,762)(659,764)(660,763)(661,789)(662,790)(663,792)
(664,791)(665,785)(666,786)(667,788)(668,787)(669,781)(670,782)(671,784)
(672,783)(673,777)(674,778)(675,780)(676,779)(677,773)(678,774)(679,776)
(680,775)(681,769)(682,770)(683,772)(684,771)(685,861)(686,862)(687,864)
(688,863)(689,857)(690,858)(691,860)(692,859)(693,853)(694,854)(695,856)
(696,855)(697,849)(698,850)(699,852)(700,851)(701,845)(702,846)(703,848)
(704,847)(705,841)(706,842)(707,844)(708,843)(709,837)(710,838)(711,840)
(712,839)(713,833)(714,834)(715,836)(716,835)(717,829)(718,830)(719,832)
(720,831)(721,825)(722,826)(723,828)(724,827)(725,821)(726,822)(727,824)
(728,823)(729,817)(730,818)(731,820)(732,819)(733,813)(734,814)(735,816)
(736,815)(737,809)(738,810)(739,812)(740,811)(741,805)(742,806)(743,808)
(744,807)(745,801)(746,802)(747,804)(748,803)(749,797)(750,798)(751,800)
(752,799)(753,793)(754,794)(755,796)(756,795);
s2 := Sym(864)!(  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)
(  8,262)(  9,257)( 10,260)( 11,259)( 12,258)( 13,285)( 14,288)( 15,287)
( 16,286)( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)
( 24,278)( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)
( 32,270)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)
( 56,246)( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)
( 64,238)( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,321)( 74,324)( 75,323)( 76,322)( 77,317)( 78,320)( 79,319)
( 80,318)( 81,313)( 82,316)( 83,315)( 84,314)( 85,309)( 86,312)( 87,311)
( 88,310)( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)
( 96,302)( 97,297)( 98,300)( 99,299)(100,298)(101,293)(102,296)(103,295)
(104,294)(105,289)(106,292)(107,291)(108,290)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)
(128,390)(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)
(136,382)(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)
(144,374)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)
(160,358)(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)
(168,350)(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)
(176,342)(177,337)(178,340)(179,339)(180,338)(181,429)(182,432)(183,431)
(184,430)(185,425)(186,428)(187,427)(188,426)(189,421)(190,424)(191,423)
(192,422)(193,417)(194,420)(195,419)(196,418)(197,413)(198,416)(199,415)
(200,414)(201,409)(202,412)(203,411)(204,410)(205,405)(206,408)(207,407)
(208,406)(209,401)(210,404)(211,403)(212,402)(213,397)(214,400)(215,399)
(216,398)(433,685)(434,688)(435,687)(436,686)(437,693)(438,696)(439,695)
(440,694)(441,689)(442,692)(443,691)(444,690)(445,717)(446,720)(447,719)
(448,718)(449,713)(450,716)(451,715)(452,714)(453,709)(454,712)(455,711)
(456,710)(457,705)(458,708)(459,707)(460,706)(461,701)(462,704)(463,703)
(464,702)(465,697)(466,700)(467,699)(468,698)(469,649)(470,652)(471,651)
(472,650)(473,657)(474,660)(475,659)(476,658)(477,653)(478,656)(479,655)
(480,654)(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)
(488,678)(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)
(496,670)(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)
(504,662)(505,753)(506,756)(507,755)(508,754)(509,749)(510,752)(511,751)
(512,750)(513,745)(514,748)(515,747)(516,746)(517,741)(518,744)(519,743)
(520,742)(521,737)(522,740)(523,739)(524,738)(525,733)(526,736)(527,735)
(528,734)(529,729)(530,732)(531,731)(532,730)(533,725)(534,728)(535,727)
(536,726)(537,721)(538,724)(539,723)(540,722)(541,793)(542,796)(543,795)
(544,794)(545,801)(546,804)(547,803)(548,802)(549,797)(550,800)(551,799)
(552,798)(553,825)(554,828)(555,827)(556,826)(557,821)(558,824)(559,823)
(560,822)(561,817)(562,820)(563,819)(564,818)(565,813)(566,816)(567,815)
(568,814)(569,809)(570,812)(571,811)(572,810)(573,805)(574,808)(575,807)
(576,806)(577,757)(578,760)(579,759)(580,758)(581,765)(582,768)(583,767)
(584,766)(585,761)(586,764)(587,763)(588,762)(589,789)(590,792)(591,791)
(592,790)(593,785)(594,788)(595,787)(596,786)(597,781)(598,784)(599,783)
(600,782)(601,777)(602,780)(603,779)(604,778)(605,773)(606,776)(607,775)
(608,774)(609,769)(610,772)(611,771)(612,770)(613,861)(614,864)(615,863)
(616,862)(617,857)(618,860)(619,859)(620,858)(621,853)(622,856)(623,855)
(624,854)(625,849)(626,852)(627,851)(628,850)(629,845)(630,848)(631,847)
(632,846)(633,841)(634,844)(635,843)(636,842)(637,837)(638,840)(639,839)
(640,838)(641,833)(642,836)(643,835)(644,834)(645,829)(646,832)(647,831)
(648,830);
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*s0*s1*s2*s1*s2*s1, 
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*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