Polytope of Type {52,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {52,8}*1664a
if this polytope has a name.
Group : SmallGroup(1664,3429)
Rank : 3
Schlafli Type : {52,8}
Number of vertices, edges, etc : 104, 416, 16
Order of s₀s₁s₂ : 104
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 : {52,4}*832, {52,8}*832a, {52,8}*832b
   4-fold quotients : {52,4}*416, {26,8}*416
   8-fold quotients : {52,2}*208, {26,4}*208
   13-fold quotients : {4,8}*128a
   16-fold quotients : {26,2}*104
   26-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
   32-fold quotients : {13,2}*52
   52-fold quotients : {4,4}*32, {2,8}*32
   104-fold quotients : {2,4}*16, {4,2}*16
   208-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope