Polytope of Type {16,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12518)
Rank : 3
Schlafli Type : {16,6}
Number of vertices, edges, etc : 144, 432, 54
Order of s₀s₁s₂ : 48
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,6}*864a
   3-fold quotients : {16,6}*576
   4-fold quotients : {4,6}*432a
   6-fold quotients : {8,6}*288
   8-fold quotients : {4,6}*216
   12-fold quotients : {4,6}*144
   24-fold quotients : {4,6}*72
   27-fold quotients : {16,2}*64
   54-fold quotients : {8,2}*32
   108-fold quotients : {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope