Polytope of Type {4,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,78}*1248
if this polytope has a name.
Group : SmallGroup(1248,1441)
Rank : 3
Schlafli Type : {4,78}
Number of vertices, edges, etc : 8, 312, 156
Order of s₀s₁s₂ : 78
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 : {4,39}*624, {4,78}*624b, {4,78}*624c
   4-fold quotients : {4,39}*312, {2,78}*312
   8-fold quotients : {2,39}*156
   12-fold quotients : {2,26}*104
   13-fold quotients : {4,6}*96
   24-fold quotients : {2,13}*52
   26-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   52-fold quotients : {4,3}*24, {2,6}*24
   104-fold quotients : {2,3}*12
   156-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,315)(  2,316)(  3,313)(  4,314)(  5,319)(  6,320)(  7,317)(  8,318)
(  9,323)( 10,324)( 11,321)( 12,322)( 13,327)( 14,328)( 15,325)( 16,326)
( 17,331)( 18,332)( 19,329)( 20,330)( 21,335)( 22,336)( 23,333)( 24,334)
( 25,339)( 26,340)( 27,337)( 28,338)( 29,343)( 30,344)( 31,341)( 32,342)
( 33,347)( 34,348)( 35,345)( 36,346)( 37,351)( 38,352)( 39,349)( 40,350)
( 41,355)( 42,356)( 43,353)( 44,354)( 45,359)( 46,360)( 47,357)( 48,358)
( 49,363)( 50,364)( 51,361)( 52,362)( 53,367)( 54,368)( 55,365)( 56,366)
( 57,371)( 58,372)( 59,369)( 60,370)( 61,375)( 62,376)( 63,373)( 64,374)
( 65,379)( 66,380)( 67,377)( 68,378)( 69,383)( 70,384)( 71,381)( 72,382)
( 73,387)( 74,388)( 75,385)( 76,386)( 77,391)( 78,392)( 79,389)( 80,390)
( 81,395)( 82,396)( 83,393)( 84,394)( 85,399)( 86,400)( 87,397)( 88,398)
( 89,403)( 90,404)( 91,401)( 92,402)( 93,407)( 94,408)( 95,405)( 96,406)
( 97,411)( 98,412)( 99,409)(100,410)(101,415)(102,416)(103,413)(104,414)
(105,419)(106,420)(107,417)(108,418)(109,423)(110,424)(111,421)(112,422)
(113,427)(114,428)(115,425)(116,426)(117,431)(118,432)(119,429)(120,430)
(121,435)(122,436)(123,433)(124,434)(125,439)(126,440)(127,437)(128,438)
(129,443)(130,444)(131,441)(132,442)(133,447)(134,448)(135,445)(136,446)
(137,451)(138,452)(139,449)(140,450)(141,455)(142,456)(143,453)(144,454)
(145,459)(146,460)(147,457)(148,458)(149,463)(150,464)(151,461)(152,462)
(153,467)(154,468)(155,465)(156,466)(157,471)(158,472)(159,469)(160,470)
(161,475)(162,476)(163,473)(164,474)(165,479)(166,480)(167,477)(168,478)
(169,483)(170,484)(171,481)(172,482)(173,487)(174,488)(175,485)(176,486)
(177,491)(178,492)(179,489)(180,490)(181,495)(182,496)(183,493)(184,494)
(185,499)(186,500)(187,497)(188,498)(189,503)(190,504)(191,501)(192,502)
(193,507)(194,508)(195,505)(196,506)(197,511)(198,512)(199,509)(200,510)
(201,515)(202,516)(203,513)(204,514)(205,519)(206,520)(207,517)(208,518)
(209,523)(210,524)(211,521)(212,522)(213,527)(214,528)(215,525)(216,526)
(217,531)(218,532)(219,529)(220,530)(221,535)(222,536)(223,533)(224,534)
(225,539)(226,540)(227,537)(228,538)(229,543)(230,544)(231,541)(232,542)
(233,547)(234,548)(235,545)(236,546)(237,551)(238,552)(239,549)(240,550)
(241,555)(242,556)(243,553)(244,554)(245,559)(246,560)(247,557)(248,558)
(249,563)(250,564)(251,561)(252,562)(253,567)(254,568)(255,565)(256,566)
(257,571)(258,572)(259,569)(260,570)(261,575)(262,576)(263,573)(264,574)
(265,579)(266,580)(267,577)(268,578)(269,583)(270,584)(271,581)(272,582)
(273,587)(274,588)(275,585)(276,586)(277,591)(278,592)(279,589)(280,590)
(281,595)(282,596)(283,593)(284,594)(285,599)(286,600)(287,597)(288,598)
(289,603)(290,604)(291,601)(292,602)(293,607)(294,608)(295,605)(296,606)
(297,611)(298,612)(299,609)(300,610)(301,615)(302,616)(303,613)(304,614)
(305,619)(306,620)(307,617)(308,618)(309,623)(310,624)(311,621)(312,622);;
s1 := (  3,  4)(  5, 49)(  6, 50)(  7, 52)(  8, 51)(  9, 45)( 10, 46)( 11, 48)
( 12, 47)( 13, 41)( 14, 42)( 15, 44)( 16, 43)( 17, 37)( 18, 38)( 19, 40)
( 20, 39)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 25, 29)( 26, 30)( 27, 32)
( 28, 31)( 53,105)( 54,106)( 55,108)( 56,107)( 57,153)( 58,154)( 59,156)
( 60,155)( 61,149)( 62,150)( 63,152)( 64,151)( 65,145)( 66,146)( 67,148)
( 68,147)( 69,141)( 70,142)( 71,144)( 72,143)( 73,137)( 74,138)( 75,140)
( 76,139)( 77,133)( 78,134)( 79,136)( 80,135)( 81,129)( 82,130)( 83,132)
( 84,131)( 85,125)( 86,126)( 87,128)( 88,127)( 89,121)( 90,122)( 91,124)
( 92,123)( 93,117)( 94,118)( 95,120)( 96,119)( 97,113)( 98,114)( 99,116)
(100,115)(101,109)(102,110)(103,112)(104,111)(159,160)(161,205)(162,206)
(163,208)(164,207)(165,201)(166,202)(167,204)(168,203)(169,197)(170,198)
(171,200)(172,199)(173,193)(174,194)(175,196)(176,195)(177,189)(178,190)
(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)(209,261)(210,262)
(211,264)(212,263)(213,309)(214,310)(215,312)(216,311)(217,305)(218,306)
(219,308)(220,307)(221,301)(222,302)(223,304)(224,303)(225,297)(226,298)
(227,300)(228,299)(229,293)(230,294)(231,296)(232,295)(233,289)(234,290)
(235,292)(236,291)(237,285)(238,286)(239,288)(240,287)(241,281)(242,282)
(243,284)(244,283)(245,277)(246,278)(247,280)(248,279)(249,273)(250,274)
(251,276)(252,275)(253,269)(254,270)(255,272)(256,271)(257,265)(258,266)
(259,268)(260,267)(315,316)(317,361)(318,362)(319,364)(320,363)(321,357)
(322,358)(323,360)(324,359)(325,353)(326,354)(327,356)(328,355)(329,349)
(330,350)(331,352)(332,351)(333,345)(334,346)(335,348)(336,347)(337,341)
(338,342)(339,344)(340,343)(365,417)(366,418)(367,420)(368,419)(369,465)
(370,466)(371,468)(372,467)(373,461)(374,462)(375,464)(376,463)(377,457)
(378,458)(379,460)(380,459)(381,453)(382,454)(383,456)(384,455)(385,449)
(386,450)(387,452)(388,451)(389,445)(390,446)(391,448)(392,447)(393,441)
(394,442)(395,444)(396,443)(397,437)(398,438)(399,440)(400,439)(401,433)
(402,434)(403,436)(404,435)(405,429)(406,430)(407,432)(408,431)(409,425)
(410,426)(411,428)(412,427)(413,421)(414,422)(415,424)(416,423)(471,472)
(473,517)(474,518)(475,520)(476,519)(477,513)(478,514)(479,516)(480,515)
(481,509)(482,510)(483,512)(484,511)(485,505)(486,506)(487,508)(488,507)
(489,501)(490,502)(491,504)(492,503)(493,497)(494,498)(495,500)(496,499)
(521,573)(522,574)(523,576)(524,575)(525,621)(526,622)(527,624)(528,623)
(529,617)(530,618)(531,620)(532,619)(533,613)(534,614)(535,616)(536,615)
(537,609)(538,610)(539,612)(540,611)(541,605)(542,606)(543,608)(544,607)
(545,601)(546,602)(547,604)(548,603)(549,597)(550,598)(551,600)(552,599)
(553,593)(554,594)(555,596)(556,595)(557,589)(558,590)(559,592)(560,591)
(561,585)(562,586)(563,588)(564,587)(565,581)(566,582)(567,584)(568,583)
(569,577)(570,578)(571,580)(572,579);;
s2 := (  1,213)(  2,216)(  3,215)(  4,214)(  5,209)(  6,212)(  7,211)(  8,210)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,249)( 18,252)( 19,251)( 20,250)( 21,245)( 22,248)( 23,247)( 24,246)
( 25,241)( 26,244)( 27,243)( 28,242)( 29,237)( 30,240)( 31,239)( 32,238)
( 33,233)( 34,236)( 35,235)( 36,234)( 37,229)( 38,232)( 39,231)( 40,230)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,217)( 50,220)( 51,219)( 52,218)( 53,161)( 54,164)( 55,163)( 56,162)
( 57,157)( 58,160)( 59,159)( 60,158)( 61,205)( 62,208)( 63,207)( 64,206)
( 65,201)( 66,204)( 67,203)( 68,202)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,185)( 82,188)( 83,187)( 84,186)( 85,181)( 86,184)( 87,183)( 88,182)
( 89,177)( 90,180)( 91,179)( 92,178)( 93,173)( 94,176)( 95,175)( 96,174)
( 97,169)( 98,172)( 99,171)(100,170)(101,165)(102,168)(103,167)(104,166)
(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)(112,262)
(113,309)(114,312)(115,311)(116,310)(117,305)(118,308)(119,307)(120,306)
(121,301)(122,304)(123,303)(124,302)(125,297)(126,300)(127,299)(128,298)
(129,293)(130,296)(131,295)(132,294)(133,289)(134,292)(135,291)(136,290)
(137,285)(138,288)(139,287)(140,286)(141,281)(142,284)(143,283)(144,282)
(145,277)(146,280)(147,279)(148,278)(149,273)(150,276)(151,275)(152,274)
(153,269)(154,272)(155,271)(156,270)(313,525)(314,528)(315,527)(316,526)
(317,521)(318,524)(319,523)(320,522)(321,569)(322,572)(323,571)(324,570)
(325,565)(326,568)(327,567)(328,566)(329,561)(330,564)(331,563)(332,562)
(333,557)(334,560)(335,559)(336,558)(337,553)(338,556)(339,555)(340,554)
(341,549)(342,552)(343,551)(344,550)(345,545)(346,548)(347,547)(348,546)
(349,541)(350,544)(351,543)(352,542)(353,537)(354,540)(355,539)(356,538)
(357,533)(358,536)(359,535)(360,534)(361,529)(362,532)(363,531)(364,530)
(365,473)(366,476)(367,475)(368,474)(369,469)(370,472)(371,471)(372,470)
(373,517)(374,520)(375,519)(376,518)(377,513)(378,516)(379,515)(380,514)
(381,509)(382,512)(383,511)(384,510)(385,505)(386,508)(387,507)(388,506)
(389,501)(390,504)(391,503)(392,502)(393,497)(394,500)(395,499)(396,498)
(397,493)(398,496)(399,495)(400,494)(401,489)(402,492)(403,491)(404,490)
(405,485)(406,488)(407,487)(408,486)(409,481)(410,484)(411,483)(412,482)
(413,477)(414,480)(415,479)(416,478)(417,577)(418,580)(419,579)(420,578)
(421,573)(422,576)(423,575)(424,574)(425,621)(426,624)(427,623)(428,622)
(429,617)(430,620)(431,619)(432,618)(433,613)(434,616)(435,615)(436,614)
(437,609)(438,612)(439,611)(440,610)(441,605)(442,608)(443,607)(444,606)
(445,601)(446,604)(447,603)(448,602)(449,597)(450,600)(451,599)(452,598)
(453,593)(454,596)(455,595)(456,594)(457,589)(458,592)(459,591)(460,590)
(461,585)(462,588)(463,587)(464,586)(465,581)(466,584)(467,583)(468,582);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(624)!(  1,315)(  2,316)(  3,313)(  4,314)(  5,319)(  6,320)(  7,317)
(  8,318)(  9,323)( 10,324)( 11,321)( 12,322)( 13,327)( 14,328)( 15,325)
( 16,326)( 17,331)( 18,332)( 19,329)( 20,330)( 21,335)( 22,336)( 23,333)
( 24,334)( 25,339)( 26,340)( 27,337)( 28,338)( 29,343)( 30,344)( 31,341)
( 32,342)( 33,347)( 34,348)( 35,345)( 36,346)( 37,351)( 38,352)( 39,349)
( 40,350)( 41,355)( 42,356)( 43,353)( 44,354)( 45,359)( 46,360)( 47,357)
( 48,358)( 49,363)( 50,364)( 51,361)( 52,362)( 53,367)( 54,368)( 55,365)
( 56,366)( 57,371)( 58,372)( 59,369)( 60,370)( 61,375)( 62,376)( 63,373)
( 64,374)( 65,379)( 66,380)( 67,377)( 68,378)( 69,383)( 70,384)( 71,381)
( 72,382)( 73,387)( 74,388)( 75,385)( 76,386)( 77,391)( 78,392)( 79,389)
( 80,390)( 81,395)( 82,396)( 83,393)( 84,394)( 85,399)( 86,400)( 87,397)
( 88,398)( 89,403)( 90,404)( 91,401)( 92,402)( 93,407)( 94,408)( 95,405)
( 96,406)( 97,411)( 98,412)( 99,409)(100,410)(101,415)(102,416)(103,413)
(104,414)(105,419)(106,420)(107,417)(108,418)(109,423)(110,424)(111,421)
(112,422)(113,427)(114,428)(115,425)(116,426)(117,431)(118,432)(119,429)
(120,430)(121,435)(122,436)(123,433)(124,434)(125,439)(126,440)(127,437)
(128,438)(129,443)(130,444)(131,441)(132,442)(133,447)(134,448)(135,445)
(136,446)(137,451)(138,452)(139,449)(140,450)(141,455)(142,456)(143,453)
(144,454)(145,459)(146,460)(147,457)(148,458)(149,463)(150,464)(151,461)
(152,462)(153,467)(154,468)(155,465)(156,466)(157,471)(158,472)(159,469)
(160,470)(161,475)(162,476)(163,473)(164,474)(165,479)(166,480)(167,477)
(168,478)(169,483)(170,484)(171,481)(172,482)(173,487)(174,488)(175,485)
(176,486)(177,491)(178,492)(179,489)(180,490)(181,495)(182,496)(183,493)
(184,494)(185,499)(186,500)(187,497)(188,498)(189,503)(190,504)(191,501)
(192,502)(193,507)(194,508)(195,505)(196,506)(197,511)(198,512)(199,509)
(200,510)(201,515)(202,516)(203,513)(204,514)(205,519)(206,520)(207,517)
(208,518)(209,523)(210,524)(211,521)(212,522)(213,527)(214,528)(215,525)
(216,526)(217,531)(218,532)(219,529)(220,530)(221,535)(222,536)(223,533)
(224,534)(225,539)(226,540)(227,537)(228,538)(229,543)(230,544)(231,541)
(232,542)(233,547)(234,548)(235,545)(236,546)(237,551)(238,552)(239,549)
(240,550)(241,555)(242,556)(243,553)(244,554)(245,559)(246,560)(247,557)
(248,558)(249,563)(250,564)(251,561)(252,562)(253,567)(254,568)(255,565)
(256,566)(257,571)(258,572)(259,569)(260,570)(261,575)(262,576)(263,573)
(264,574)(265,579)(266,580)(267,577)(268,578)(269,583)(270,584)(271,581)
(272,582)(273,587)(274,588)(275,585)(276,586)(277,591)(278,592)(279,589)
(280,590)(281,595)(282,596)(283,593)(284,594)(285,599)(286,600)(287,597)
(288,598)(289,603)(290,604)(291,601)(292,602)(293,607)(294,608)(295,605)
(296,606)(297,611)(298,612)(299,609)(300,610)(301,615)(302,616)(303,613)
(304,614)(305,619)(306,620)(307,617)(308,618)(309,623)(310,624)(311,621)
(312,622);
s1 := Sym(624)!(  3,  4)(  5, 49)(  6, 50)(  7, 52)(  8, 51)(  9, 45)( 10, 46)
( 11, 48)( 12, 47)( 13, 41)( 14, 42)( 15, 44)( 16, 43)( 17, 37)( 18, 38)
( 19, 40)( 20, 39)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 25, 29)( 26, 30)
( 27, 32)( 28, 31)( 53,105)( 54,106)( 55,108)( 56,107)( 57,153)( 58,154)
( 59,156)( 60,155)( 61,149)( 62,150)( 63,152)( 64,151)( 65,145)( 66,146)
( 67,148)( 68,147)( 69,141)( 70,142)( 71,144)( 72,143)( 73,137)( 74,138)
( 75,140)( 76,139)( 77,133)( 78,134)( 79,136)( 80,135)( 81,129)( 82,130)
( 83,132)( 84,131)( 85,125)( 86,126)( 87,128)( 88,127)( 89,121)( 90,122)
( 91,124)( 92,123)( 93,117)( 94,118)( 95,120)( 96,119)( 97,113)( 98,114)
( 99,116)(100,115)(101,109)(102,110)(103,112)(104,111)(159,160)(161,205)
(162,206)(163,208)(164,207)(165,201)(166,202)(167,204)(168,203)(169,197)
(170,198)(171,200)(172,199)(173,193)(174,194)(175,196)(176,195)(177,189)
(178,190)(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)(209,261)
(210,262)(211,264)(212,263)(213,309)(214,310)(215,312)(216,311)(217,305)
(218,306)(219,308)(220,307)(221,301)(222,302)(223,304)(224,303)(225,297)
(226,298)(227,300)(228,299)(229,293)(230,294)(231,296)(232,295)(233,289)
(234,290)(235,292)(236,291)(237,285)(238,286)(239,288)(240,287)(241,281)
(242,282)(243,284)(244,283)(245,277)(246,278)(247,280)(248,279)(249,273)
(250,274)(251,276)(252,275)(253,269)(254,270)(255,272)(256,271)(257,265)
(258,266)(259,268)(260,267)(315,316)(317,361)(318,362)(319,364)(320,363)
(321,357)(322,358)(323,360)(324,359)(325,353)(326,354)(327,356)(328,355)
(329,349)(330,350)(331,352)(332,351)(333,345)(334,346)(335,348)(336,347)
(337,341)(338,342)(339,344)(340,343)(365,417)(366,418)(367,420)(368,419)
(369,465)(370,466)(371,468)(372,467)(373,461)(374,462)(375,464)(376,463)
(377,457)(378,458)(379,460)(380,459)(381,453)(382,454)(383,456)(384,455)
(385,449)(386,450)(387,452)(388,451)(389,445)(390,446)(391,448)(392,447)
(393,441)(394,442)(395,444)(396,443)(397,437)(398,438)(399,440)(400,439)
(401,433)(402,434)(403,436)(404,435)(405,429)(406,430)(407,432)(408,431)
(409,425)(410,426)(411,428)(412,427)(413,421)(414,422)(415,424)(416,423)
(471,472)(473,517)(474,518)(475,520)(476,519)(477,513)(478,514)(479,516)
(480,515)(481,509)(482,510)(483,512)(484,511)(485,505)(486,506)(487,508)
(488,507)(489,501)(490,502)(491,504)(492,503)(493,497)(494,498)(495,500)
(496,499)(521,573)(522,574)(523,576)(524,575)(525,621)(526,622)(527,624)
(528,623)(529,617)(530,618)(531,620)(532,619)(533,613)(534,614)(535,616)
(536,615)(537,609)(538,610)(539,612)(540,611)(541,605)(542,606)(543,608)
(544,607)(545,601)(546,602)(547,604)(548,603)(549,597)(550,598)(551,600)
(552,599)(553,593)(554,594)(555,596)(556,595)(557,589)(558,590)(559,592)
(560,591)(561,585)(562,586)(563,588)(564,587)(565,581)(566,582)(567,584)
(568,583)(569,577)(570,578)(571,580)(572,579);
s2 := Sym(624)!(  1,213)(  2,216)(  3,215)(  4,214)(  5,209)(  6,212)(  7,211)
(  8,210)(  9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,249)( 18,252)( 19,251)( 20,250)( 21,245)( 22,248)( 23,247)
( 24,246)( 25,241)( 26,244)( 27,243)( 28,242)( 29,237)( 30,240)( 31,239)
( 32,238)( 33,233)( 34,236)( 35,235)( 36,234)( 37,229)( 38,232)( 39,231)
( 40,230)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,217)( 50,220)( 51,219)( 52,218)( 53,161)( 54,164)( 55,163)
( 56,162)( 57,157)( 58,160)( 59,159)( 60,158)( 61,205)( 62,208)( 63,207)
( 64,206)( 65,201)( 66,204)( 67,203)( 68,202)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)
( 80,190)( 81,185)( 82,188)( 83,187)( 84,186)( 85,181)( 86,184)( 87,183)
( 88,182)( 89,177)( 90,180)( 91,179)( 92,178)( 93,173)( 94,176)( 95,175)
( 96,174)( 97,169)( 98,172)( 99,171)(100,170)(101,165)(102,168)(103,167)
(104,166)(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)
(112,262)(113,309)(114,312)(115,311)(116,310)(117,305)(118,308)(119,307)
(120,306)(121,301)(122,304)(123,303)(124,302)(125,297)(126,300)(127,299)
(128,298)(129,293)(130,296)(131,295)(132,294)(133,289)(134,292)(135,291)
(136,290)(137,285)(138,288)(139,287)(140,286)(141,281)(142,284)(143,283)
(144,282)(145,277)(146,280)(147,279)(148,278)(149,273)(150,276)(151,275)
(152,274)(153,269)(154,272)(155,271)(156,270)(313,525)(314,528)(315,527)
(316,526)(317,521)(318,524)(319,523)(320,522)(321,569)(322,572)(323,571)
(324,570)(325,565)(326,568)(327,567)(328,566)(329,561)(330,564)(331,563)
(332,562)(333,557)(334,560)(335,559)(336,558)(337,553)(338,556)(339,555)
(340,554)(341,549)(342,552)(343,551)(344,550)(345,545)(346,548)(347,547)
(348,546)(349,541)(350,544)(351,543)(352,542)(353,537)(354,540)(355,539)
(356,538)(357,533)(358,536)(359,535)(360,534)(361,529)(362,532)(363,531)
(364,530)(365,473)(366,476)(367,475)(368,474)(369,469)(370,472)(371,471)
(372,470)(373,517)(374,520)(375,519)(376,518)(377,513)(378,516)(379,515)
(380,514)(381,509)(382,512)(383,511)(384,510)(385,505)(386,508)(387,507)
(388,506)(389,501)(390,504)(391,503)(392,502)(393,497)(394,500)(395,499)
(396,498)(397,493)(398,496)(399,495)(400,494)(401,489)(402,492)(403,491)
(404,490)(405,485)(406,488)(407,487)(408,486)(409,481)(410,484)(411,483)
(412,482)(413,477)(414,480)(415,479)(416,478)(417,577)(418,580)(419,579)
(420,578)(421,573)(422,576)(423,575)(424,574)(425,621)(426,624)(427,623)
(428,622)(429,617)(430,620)(431,619)(432,618)(433,613)(434,616)(435,615)
(436,614)(437,609)(438,612)(439,611)(440,610)(441,605)(442,608)(443,607)
(444,606)(445,601)(446,604)(447,603)(448,602)(449,597)(450,600)(451,599)
(452,598)(453,593)(454,596)(455,595)(456,594)(457,589)(458,592)(459,591)
(460,590)(461,585)(462,588)(463,587)(464,586)(465,581)(466,584)(467,583)
(468,582);
poly := sub<Sym(624)|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 >;

References : None.
to this polytope