Polytope of Type {4,16,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,16,10}*1280a
Also Known As : {{4,16|2},{16,10|2}}. if this polytope has another name.
Group : SmallGroup(1280,323301)
Rank : 4
Schlafli Type : {4,16,10}
Number of vertices, edges, etc : 4, 32, 80, 10
Order of s0s1s2s3 : 80
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
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 : {4,8,10}*640a, {2,16,10}*640
   4-fold quotients : {4,4,10}*320, {2,8,10}*320
   5-fold quotients : {4,16,2}*256a
   8-fold quotients : {2,4,10}*160, {4,2,10}*160
   10-fold quotients : {4,8,2}*128a, {2,16,2}*128
   16-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {4,4,2}*64, {2,8,2}*64
   32-fold quotients : {2,2,5}*40
   40-fold quotients : {2,4,2}*32, {4,2,2}*32
   80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)(  8,168)
(  9,169)( 10,170)( 11,171)( 12,172)( 13,173)( 14,174)( 15,175)( 16,176)
( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,185)( 26,186)( 27,187)( 28,188)( 29,189)( 30,190)( 31,191)( 32,192)
( 33,193)( 34,194)( 35,195)( 36,196)( 37,197)( 38,198)( 39,199)( 40,200)
( 41,201)( 42,202)( 43,203)( 44,204)( 45,205)( 46,206)( 47,207)( 48,208)
( 49,209)( 50,210)( 51,211)( 52,212)( 53,213)( 54,214)( 55,215)( 56,216)
( 57,217)( 58,218)( 59,219)( 60,220)( 61,221)( 62,222)( 63,223)( 64,224)
( 65,225)( 66,226)( 67,227)( 68,228)( 69,229)( 70,230)( 71,231)( 72,232)
( 73,233)( 74,234)( 75,235)( 76,236)( 77,237)( 78,238)( 79,239)( 80,240)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)( 88,248)
( 89,249)( 90,250)( 91,251)( 92,252)( 93,253)( 94,254)( 95,255)( 96,256)
( 97,257)( 98,258)( 99,259)(100,260)(101,261)(102,262)(103,263)(104,264)
(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,271)(112,272)
(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)(120,280)
(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)(128,288)
(129,289)(130,290)(131,291)(132,292)(133,293)(134,294)(135,295)(136,296)
(137,297)(138,298)(139,299)(140,300)(141,301)(142,302)(143,303)(144,304)
(145,305)(146,306)(147,307)(148,308)(149,309)(150,310)(151,311)(152,312)
(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)(160,320)
(321,481)(322,482)(323,483)(324,484)(325,485)(326,486)(327,487)(328,488)
(329,489)(330,490)(331,491)(332,492)(333,493)(334,494)(335,495)(336,496)
(337,497)(338,498)(339,499)(340,500)(341,501)(342,502)(343,503)(344,504)
(345,505)(346,506)(347,507)(348,508)(349,509)(350,510)(351,511)(352,512)
(353,513)(354,514)(355,515)(356,516)(357,517)(358,518)(359,519)(360,520)
(361,521)(362,522)(363,523)(364,524)(365,525)(366,526)(367,527)(368,528)
(369,529)(370,530)(371,531)(372,532)(373,533)(374,534)(375,535)(376,536)
(377,537)(378,538)(379,539)(380,540)(381,541)(382,542)(383,543)(384,544)
(385,545)(386,546)(387,547)(388,548)(389,549)(390,550)(391,551)(392,552)
(393,553)(394,554)(395,555)(396,556)(397,557)(398,558)(399,559)(400,560)
(401,561)(402,562)(403,563)(404,564)(405,565)(406,566)(407,567)(408,568)
(409,569)(410,570)(411,571)(412,572)(413,573)(414,574)(415,575)(416,576)
(417,577)(418,578)(419,579)(420,580)(421,581)(422,582)(423,583)(424,584)
(425,585)(426,586)(427,587)(428,588)(429,589)(430,590)(431,591)(432,592)
(433,593)(434,594)(435,595)(436,596)(437,597)(438,598)(439,599)(440,600)
(441,601)(442,602)(443,603)(444,604)(445,605)(446,606)(447,607)(448,608)
(449,609)(450,610)(451,611)(452,612)(453,613)(454,614)(455,615)(456,616)
(457,617)(458,618)(459,619)(460,620)(461,621)(462,622)(463,623)(464,624)
(465,625)(466,626)(467,627)(468,628)(469,629)(470,630)(471,631)(472,632)
(473,633)(474,634)(475,635)(476,636)(477,637)(478,638)(479,639)(480,640);;
s1 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 41, 51)( 42, 52)( 43, 53)( 44, 54)( 45, 55)( 46, 56)
( 47, 57)( 48, 58)( 49, 59)( 50, 60)( 61, 71)( 62, 72)( 63, 73)( 64, 74)
( 65, 75)( 66, 76)( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 91, 96)( 92, 97)
( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)
(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)
(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)
(147,157)(148,158)(149,159)(150,160)(161,181)(162,182)(163,183)(164,184)
(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,196)(172,197)
(173,198)(174,199)(175,200)(176,191)(177,192)(178,193)(179,194)(180,195)
(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)
(209,239)(210,240)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)
(217,227)(218,228)(219,229)(220,230)(241,261)(242,262)(243,263)(244,264)
(245,265)(246,266)(247,267)(248,268)(249,269)(250,270)(251,276)(252,277)
(253,278)(254,279)(255,280)(256,271)(257,272)(258,273)(259,274)(260,275)
(281,311)(282,312)(283,313)(284,314)(285,315)(286,316)(287,317)(288,318)
(289,319)(290,320)(291,301)(292,302)(293,303)(294,304)(295,305)(296,306)
(297,307)(298,308)(299,309)(300,310)(321,361)(322,362)(323,363)(324,364)
(325,365)(326,366)(327,367)(328,368)(329,369)(330,370)(331,376)(332,377)
(333,378)(334,379)(335,380)(336,371)(337,372)(338,373)(339,374)(340,375)
(341,381)(342,382)(343,383)(344,384)(345,385)(346,386)(347,387)(348,388)
(349,389)(350,390)(351,396)(352,397)(353,398)(354,399)(355,400)(356,391)
(357,392)(358,393)(359,394)(360,395)(401,441)(402,442)(403,443)(404,444)
(405,445)(406,446)(407,447)(408,448)(409,449)(410,450)(411,456)(412,457)
(413,458)(414,459)(415,460)(416,451)(417,452)(418,453)(419,454)(420,455)
(421,461)(422,462)(423,463)(424,464)(425,465)(426,466)(427,467)(428,468)
(429,469)(430,470)(431,476)(432,477)(433,478)(434,479)(435,480)(436,471)
(437,472)(438,473)(439,474)(440,475)(481,541)(482,542)(483,543)(484,544)
(485,545)(486,546)(487,547)(488,548)(489,549)(490,550)(491,556)(492,557)
(493,558)(494,559)(495,560)(496,551)(497,552)(498,553)(499,554)(500,555)
(501,521)(502,522)(503,523)(504,524)(505,525)(506,526)(507,527)(508,528)
(509,529)(510,530)(511,536)(512,537)(513,538)(514,539)(515,540)(516,531)
(517,532)(518,533)(519,534)(520,535)(561,621)(562,622)(563,623)(564,624)
(565,625)(566,626)(567,627)(568,628)(569,629)(570,630)(571,636)(572,637)
(573,638)(574,639)(575,640)(576,631)(577,632)(578,633)(579,634)(580,635)
(581,601)(582,602)(583,603)(584,604)(585,605)(586,606)(587,607)(588,608)
(589,609)(590,610)(591,616)(592,617)(593,618)(594,619)(595,620)(596,611)
(597,612)(598,613)(599,614)(600,615);;
s2 := (  1,401)(  2,405)(  3,404)(  4,403)(  5,402)(  6,406)(  7,410)(  8,409)
(  9,408)( 10,407)( 11,416)( 12,420)( 13,419)( 14,418)( 15,417)( 16,411)
( 17,415)( 18,414)( 19,413)( 20,412)( 21,421)( 22,425)( 23,424)( 24,423)
( 25,422)( 26,426)( 27,430)( 28,429)( 29,428)( 30,427)( 31,436)( 32,440)
( 33,439)( 34,438)( 35,437)( 36,431)( 37,435)( 38,434)( 39,433)( 40,432)
( 41,451)( 42,455)( 43,454)( 44,453)( 45,452)( 46,456)( 47,460)( 48,459)
( 49,458)( 50,457)( 51,441)( 52,445)( 53,444)( 54,443)( 55,442)( 56,446)
( 57,450)( 58,449)( 59,448)( 60,447)( 61,471)( 62,475)( 63,474)( 64,473)
( 65,472)( 66,476)( 67,480)( 68,479)( 69,478)( 70,477)( 71,461)( 72,465)
( 73,464)( 74,463)( 75,462)( 76,466)( 77,470)( 78,469)( 79,468)( 80,467)
( 81,321)( 82,325)( 83,324)( 84,323)( 85,322)( 86,326)( 87,330)( 88,329)
( 89,328)( 90,327)( 91,336)( 92,340)( 93,339)( 94,338)( 95,337)( 96,331)
( 97,335)( 98,334)( 99,333)(100,332)(101,341)(102,345)(103,344)(104,343)
(105,342)(106,346)(107,350)(108,349)(109,348)(110,347)(111,356)(112,360)
(113,359)(114,358)(115,357)(116,351)(117,355)(118,354)(119,353)(120,352)
(121,371)(122,375)(123,374)(124,373)(125,372)(126,376)(127,380)(128,379)
(129,378)(130,377)(131,361)(132,365)(133,364)(134,363)(135,362)(136,366)
(137,370)(138,369)(139,368)(140,367)(141,391)(142,395)(143,394)(144,393)
(145,392)(146,396)(147,400)(148,399)(149,398)(150,397)(151,381)(152,385)
(153,384)(154,383)(155,382)(156,386)(157,390)(158,389)(159,388)(160,387)
(161,561)(162,565)(163,564)(164,563)(165,562)(166,566)(167,570)(168,569)
(169,568)(170,567)(171,576)(172,580)(173,579)(174,578)(175,577)(176,571)
(177,575)(178,574)(179,573)(180,572)(181,581)(182,585)(183,584)(184,583)
(185,582)(186,586)(187,590)(188,589)(189,588)(190,587)(191,596)(192,600)
(193,599)(194,598)(195,597)(196,591)(197,595)(198,594)(199,593)(200,592)
(201,611)(202,615)(203,614)(204,613)(205,612)(206,616)(207,620)(208,619)
(209,618)(210,617)(211,601)(212,605)(213,604)(214,603)(215,602)(216,606)
(217,610)(218,609)(219,608)(220,607)(221,631)(222,635)(223,634)(224,633)
(225,632)(226,636)(227,640)(228,639)(229,638)(230,637)(231,621)(232,625)
(233,624)(234,623)(235,622)(236,626)(237,630)(238,629)(239,628)(240,627)
(241,481)(242,485)(243,484)(244,483)(245,482)(246,486)(247,490)(248,489)
(249,488)(250,487)(251,496)(252,500)(253,499)(254,498)(255,497)(256,491)
(257,495)(258,494)(259,493)(260,492)(261,501)(262,505)(263,504)(264,503)
(265,502)(266,506)(267,510)(268,509)(269,508)(270,507)(271,516)(272,520)
(273,519)(274,518)(275,517)(276,511)(277,515)(278,514)(279,513)(280,512)
(281,531)(282,535)(283,534)(284,533)(285,532)(286,536)(287,540)(288,539)
(289,538)(290,537)(291,521)(292,525)(293,524)(294,523)(295,522)(296,526)
(297,530)(298,529)(299,528)(300,527)(301,551)(302,555)(303,554)(304,553)
(305,552)(306,556)(307,560)(308,559)(309,558)(310,557)(311,541)(312,545)
(313,544)(314,543)(315,542)(316,546)(317,550)(318,549)(319,548)(320,547);;
s3 := (  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)(  8, 86)
(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)( 16, 98)
( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,103)( 22,102)( 23,101)( 24,105)
( 25,104)( 26,108)( 27,107)( 28,106)( 29,110)( 30,109)( 31,113)( 32,112)
( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)
( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)( 48,126)
( 49,130)( 50,129)( 51,133)( 52,132)( 53,131)( 54,135)( 55,134)( 56,138)
( 57,137)( 58,136)( 59,140)( 60,139)( 61,143)( 62,142)( 63,141)( 64,145)
( 65,144)( 66,148)( 67,147)( 68,146)( 69,150)( 70,149)( 71,153)( 72,152)
( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)( 80,159)
(161,243)(162,242)(163,241)(164,245)(165,244)(166,248)(167,247)(168,246)
(169,250)(170,249)(171,253)(172,252)(173,251)(174,255)(175,254)(176,258)
(177,257)(178,256)(179,260)(180,259)(181,263)(182,262)(183,261)(184,265)
(185,264)(186,268)(187,267)(188,266)(189,270)(190,269)(191,273)(192,272)
(193,271)(194,275)(195,274)(196,278)(197,277)(198,276)(199,280)(200,279)
(201,283)(202,282)(203,281)(204,285)(205,284)(206,288)(207,287)(208,286)
(209,290)(210,289)(211,293)(212,292)(213,291)(214,295)(215,294)(216,298)
(217,297)(218,296)(219,300)(220,299)(221,303)(222,302)(223,301)(224,305)
(225,304)(226,308)(227,307)(228,306)(229,310)(230,309)(231,313)(232,312)
(233,311)(234,315)(235,314)(236,318)(237,317)(238,316)(239,320)(240,319)
(321,403)(322,402)(323,401)(324,405)(325,404)(326,408)(327,407)(328,406)
(329,410)(330,409)(331,413)(332,412)(333,411)(334,415)(335,414)(336,418)
(337,417)(338,416)(339,420)(340,419)(341,423)(342,422)(343,421)(344,425)
(345,424)(346,428)(347,427)(348,426)(349,430)(350,429)(351,433)(352,432)
(353,431)(354,435)(355,434)(356,438)(357,437)(358,436)(359,440)(360,439)
(361,443)(362,442)(363,441)(364,445)(365,444)(366,448)(367,447)(368,446)
(369,450)(370,449)(371,453)(372,452)(373,451)(374,455)(375,454)(376,458)
(377,457)(378,456)(379,460)(380,459)(381,463)(382,462)(383,461)(384,465)
(385,464)(386,468)(387,467)(388,466)(389,470)(390,469)(391,473)(392,472)
(393,471)(394,475)(395,474)(396,478)(397,477)(398,476)(399,480)(400,479)
(481,563)(482,562)(483,561)(484,565)(485,564)(486,568)(487,567)(488,566)
(489,570)(490,569)(491,573)(492,572)(493,571)(494,575)(495,574)(496,578)
(497,577)(498,576)(499,580)(500,579)(501,583)(502,582)(503,581)(504,585)
(505,584)(506,588)(507,587)(508,586)(509,590)(510,589)(511,593)(512,592)
(513,591)(514,595)(515,594)(516,598)(517,597)(518,596)(519,600)(520,599)
(521,603)(522,602)(523,601)(524,605)(525,604)(526,608)(527,607)(528,606)
(529,610)(530,609)(531,613)(532,612)(533,611)(534,615)(535,614)(536,618)
(537,617)(538,616)(539,620)(540,619)(541,623)(542,622)(543,621)(544,625)
(545,624)(546,628)(547,627)(548,626)(549,630)(550,629)(551,633)(552,632)
(553,631)(554,635)(555,634)(556,638)(557,637)(558,636)(559,640)(560,639);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*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(640)!(  1,161)(  2,162)(  3,163)(  4,164)(  5,165)(  6,166)(  7,167)
(  8,168)(  9,169)( 10,170)( 11,171)( 12,172)( 13,173)( 14,174)( 15,175)
( 16,176)( 17,177)( 18,178)( 19,179)( 20,180)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,185)( 26,186)( 27,187)( 28,188)( 29,189)( 30,190)( 31,191)
( 32,192)( 33,193)( 34,194)( 35,195)( 36,196)( 37,197)( 38,198)( 39,199)
( 40,200)( 41,201)( 42,202)( 43,203)( 44,204)( 45,205)( 46,206)( 47,207)
( 48,208)( 49,209)( 50,210)( 51,211)( 52,212)( 53,213)( 54,214)( 55,215)
( 56,216)( 57,217)( 58,218)( 59,219)( 60,220)( 61,221)( 62,222)( 63,223)
( 64,224)( 65,225)( 66,226)( 67,227)( 68,228)( 69,229)( 70,230)( 71,231)
( 72,232)( 73,233)( 74,234)( 75,235)( 76,236)( 77,237)( 78,238)( 79,239)
( 80,240)( 81,241)( 82,242)( 83,243)( 84,244)( 85,245)( 86,246)( 87,247)
( 88,248)( 89,249)( 90,250)( 91,251)( 92,252)( 93,253)( 94,254)( 95,255)
( 96,256)( 97,257)( 98,258)( 99,259)(100,260)(101,261)(102,262)(103,263)
(104,264)(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,271)
(112,272)(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)
(120,280)(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)
(128,288)(129,289)(130,290)(131,291)(132,292)(133,293)(134,294)(135,295)
(136,296)(137,297)(138,298)(139,299)(140,300)(141,301)(142,302)(143,303)
(144,304)(145,305)(146,306)(147,307)(148,308)(149,309)(150,310)(151,311)
(152,312)(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)
(160,320)(321,481)(322,482)(323,483)(324,484)(325,485)(326,486)(327,487)
(328,488)(329,489)(330,490)(331,491)(332,492)(333,493)(334,494)(335,495)
(336,496)(337,497)(338,498)(339,499)(340,500)(341,501)(342,502)(343,503)
(344,504)(345,505)(346,506)(347,507)(348,508)(349,509)(350,510)(351,511)
(352,512)(353,513)(354,514)(355,515)(356,516)(357,517)(358,518)(359,519)
(360,520)(361,521)(362,522)(363,523)(364,524)(365,525)(366,526)(367,527)
(368,528)(369,529)(370,530)(371,531)(372,532)(373,533)(374,534)(375,535)
(376,536)(377,537)(378,538)(379,539)(380,540)(381,541)(382,542)(383,543)
(384,544)(385,545)(386,546)(387,547)(388,548)(389,549)(390,550)(391,551)
(392,552)(393,553)(394,554)(395,555)(396,556)(397,557)(398,558)(399,559)
(400,560)(401,561)(402,562)(403,563)(404,564)(405,565)(406,566)(407,567)
(408,568)(409,569)(410,570)(411,571)(412,572)(413,573)(414,574)(415,575)
(416,576)(417,577)(418,578)(419,579)(420,580)(421,581)(422,582)(423,583)
(424,584)(425,585)(426,586)(427,587)(428,588)(429,589)(430,590)(431,591)
(432,592)(433,593)(434,594)(435,595)(436,596)(437,597)(438,598)(439,599)
(440,600)(441,601)(442,602)(443,603)(444,604)(445,605)(446,606)(447,607)
(448,608)(449,609)(450,610)(451,611)(452,612)(453,613)(454,614)(455,615)
(456,616)(457,617)(458,618)(459,619)(460,620)(461,621)(462,622)(463,623)
(464,624)(465,625)(466,626)(467,627)(468,628)(469,629)(470,630)(471,631)
(472,632)(473,633)(474,634)(475,635)(476,636)(477,637)(478,638)(479,639)
(480,640);
s1 := Sym(640)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 41, 51)( 42, 52)( 43, 53)( 44, 54)( 45, 55)
( 46, 56)( 47, 57)( 48, 58)( 49, 59)( 50, 60)( 61, 71)( 62, 72)( 63, 73)
( 64, 74)( 65, 75)( 66, 76)( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 91, 96)
( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)
(115,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)
(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)
(146,156)(147,157)(148,158)(149,159)(150,160)(161,181)(162,182)(163,183)
(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,196)
(172,197)(173,198)(174,199)(175,200)(176,191)(177,192)(178,193)(179,194)
(180,195)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)
(208,238)(209,239)(210,240)(211,221)(212,222)(213,223)(214,224)(215,225)
(216,226)(217,227)(218,228)(219,229)(220,230)(241,261)(242,262)(243,263)
(244,264)(245,265)(246,266)(247,267)(248,268)(249,269)(250,270)(251,276)
(252,277)(253,278)(254,279)(255,280)(256,271)(257,272)(258,273)(259,274)
(260,275)(281,311)(282,312)(283,313)(284,314)(285,315)(286,316)(287,317)
(288,318)(289,319)(290,320)(291,301)(292,302)(293,303)(294,304)(295,305)
(296,306)(297,307)(298,308)(299,309)(300,310)(321,361)(322,362)(323,363)
(324,364)(325,365)(326,366)(327,367)(328,368)(329,369)(330,370)(331,376)
(332,377)(333,378)(334,379)(335,380)(336,371)(337,372)(338,373)(339,374)
(340,375)(341,381)(342,382)(343,383)(344,384)(345,385)(346,386)(347,387)
(348,388)(349,389)(350,390)(351,396)(352,397)(353,398)(354,399)(355,400)
(356,391)(357,392)(358,393)(359,394)(360,395)(401,441)(402,442)(403,443)
(404,444)(405,445)(406,446)(407,447)(408,448)(409,449)(410,450)(411,456)
(412,457)(413,458)(414,459)(415,460)(416,451)(417,452)(418,453)(419,454)
(420,455)(421,461)(422,462)(423,463)(424,464)(425,465)(426,466)(427,467)
(428,468)(429,469)(430,470)(431,476)(432,477)(433,478)(434,479)(435,480)
(436,471)(437,472)(438,473)(439,474)(440,475)(481,541)(482,542)(483,543)
(484,544)(485,545)(486,546)(487,547)(488,548)(489,549)(490,550)(491,556)
(492,557)(493,558)(494,559)(495,560)(496,551)(497,552)(498,553)(499,554)
(500,555)(501,521)(502,522)(503,523)(504,524)(505,525)(506,526)(507,527)
(508,528)(509,529)(510,530)(511,536)(512,537)(513,538)(514,539)(515,540)
(516,531)(517,532)(518,533)(519,534)(520,535)(561,621)(562,622)(563,623)
(564,624)(565,625)(566,626)(567,627)(568,628)(569,629)(570,630)(571,636)
(572,637)(573,638)(574,639)(575,640)(576,631)(577,632)(578,633)(579,634)
(580,635)(581,601)(582,602)(583,603)(584,604)(585,605)(586,606)(587,607)
(588,608)(589,609)(590,610)(591,616)(592,617)(593,618)(594,619)(595,620)
(596,611)(597,612)(598,613)(599,614)(600,615);
s2 := Sym(640)!(  1,401)(  2,405)(  3,404)(  4,403)(  5,402)(  6,406)(  7,410)
(  8,409)(  9,408)( 10,407)( 11,416)( 12,420)( 13,419)( 14,418)( 15,417)
( 16,411)( 17,415)( 18,414)( 19,413)( 20,412)( 21,421)( 22,425)( 23,424)
( 24,423)( 25,422)( 26,426)( 27,430)( 28,429)( 29,428)( 30,427)( 31,436)
( 32,440)( 33,439)( 34,438)( 35,437)( 36,431)( 37,435)( 38,434)( 39,433)
( 40,432)( 41,451)( 42,455)( 43,454)( 44,453)( 45,452)( 46,456)( 47,460)
( 48,459)( 49,458)( 50,457)( 51,441)( 52,445)( 53,444)( 54,443)( 55,442)
( 56,446)( 57,450)( 58,449)( 59,448)( 60,447)( 61,471)( 62,475)( 63,474)
( 64,473)( 65,472)( 66,476)( 67,480)( 68,479)( 69,478)( 70,477)( 71,461)
( 72,465)( 73,464)( 74,463)( 75,462)( 76,466)( 77,470)( 78,469)( 79,468)
( 80,467)( 81,321)( 82,325)( 83,324)( 84,323)( 85,322)( 86,326)( 87,330)
( 88,329)( 89,328)( 90,327)( 91,336)( 92,340)( 93,339)( 94,338)( 95,337)
( 96,331)( 97,335)( 98,334)( 99,333)(100,332)(101,341)(102,345)(103,344)
(104,343)(105,342)(106,346)(107,350)(108,349)(109,348)(110,347)(111,356)
(112,360)(113,359)(114,358)(115,357)(116,351)(117,355)(118,354)(119,353)
(120,352)(121,371)(122,375)(123,374)(124,373)(125,372)(126,376)(127,380)
(128,379)(129,378)(130,377)(131,361)(132,365)(133,364)(134,363)(135,362)
(136,366)(137,370)(138,369)(139,368)(140,367)(141,391)(142,395)(143,394)
(144,393)(145,392)(146,396)(147,400)(148,399)(149,398)(150,397)(151,381)
(152,385)(153,384)(154,383)(155,382)(156,386)(157,390)(158,389)(159,388)
(160,387)(161,561)(162,565)(163,564)(164,563)(165,562)(166,566)(167,570)
(168,569)(169,568)(170,567)(171,576)(172,580)(173,579)(174,578)(175,577)
(176,571)(177,575)(178,574)(179,573)(180,572)(181,581)(182,585)(183,584)
(184,583)(185,582)(186,586)(187,590)(188,589)(189,588)(190,587)(191,596)
(192,600)(193,599)(194,598)(195,597)(196,591)(197,595)(198,594)(199,593)
(200,592)(201,611)(202,615)(203,614)(204,613)(205,612)(206,616)(207,620)
(208,619)(209,618)(210,617)(211,601)(212,605)(213,604)(214,603)(215,602)
(216,606)(217,610)(218,609)(219,608)(220,607)(221,631)(222,635)(223,634)
(224,633)(225,632)(226,636)(227,640)(228,639)(229,638)(230,637)(231,621)
(232,625)(233,624)(234,623)(235,622)(236,626)(237,630)(238,629)(239,628)
(240,627)(241,481)(242,485)(243,484)(244,483)(245,482)(246,486)(247,490)
(248,489)(249,488)(250,487)(251,496)(252,500)(253,499)(254,498)(255,497)
(256,491)(257,495)(258,494)(259,493)(260,492)(261,501)(262,505)(263,504)
(264,503)(265,502)(266,506)(267,510)(268,509)(269,508)(270,507)(271,516)
(272,520)(273,519)(274,518)(275,517)(276,511)(277,515)(278,514)(279,513)
(280,512)(281,531)(282,535)(283,534)(284,533)(285,532)(286,536)(287,540)
(288,539)(289,538)(290,537)(291,521)(292,525)(293,524)(294,523)(295,522)
(296,526)(297,530)(298,529)(299,528)(300,527)(301,551)(302,555)(303,554)
(304,553)(305,552)(306,556)(307,560)(308,559)(309,558)(310,557)(311,541)
(312,545)(313,544)(314,543)(315,542)(316,546)(317,550)(318,549)(319,548)
(320,547);
s3 := Sym(640)!(  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)
(  8, 86)(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)
( 16, 98)( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,103)( 22,102)( 23,101)
( 24,105)( 25,104)( 26,108)( 27,107)( 28,106)( 29,110)( 30,109)( 31,113)
( 32,112)( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)
( 40,119)( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)
( 48,126)( 49,130)( 50,129)( 51,133)( 52,132)( 53,131)( 54,135)( 55,134)
( 56,138)( 57,137)( 58,136)( 59,140)( 60,139)( 61,143)( 62,142)( 63,141)
( 64,145)( 65,144)( 66,148)( 67,147)( 68,146)( 69,150)( 70,149)( 71,153)
( 72,152)( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)
( 80,159)(161,243)(162,242)(163,241)(164,245)(165,244)(166,248)(167,247)
(168,246)(169,250)(170,249)(171,253)(172,252)(173,251)(174,255)(175,254)
(176,258)(177,257)(178,256)(179,260)(180,259)(181,263)(182,262)(183,261)
(184,265)(185,264)(186,268)(187,267)(188,266)(189,270)(190,269)(191,273)
(192,272)(193,271)(194,275)(195,274)(196,278)(197,277)(198,276)(199,280)
(200,279)(201,283)(202,282)(203,281)(204,285)(205,284)(206,288)(207,287)
(208,286)(209,290)(210,289)(211,293)(212,292)(213,291)(214,295)(215,294)
(216,298)(217,297)(218,296)(219,300)(220,299)(221,303)(222,302)(223,301)
(224,305)(225,304)(226,308)(227,307)(228,306)(229,310)(230,309)(231,313)
(232,312)(233,311)(234,315)(235,314)(236,318)(237,317)(238,316)(239,320)
(240,319)(321,403)(322,402)(323,401)(324,405)(325,404)(326,408)(327,407)
(328,406)(329,410)(330,409)(331,413)(332,412)(333,411)(334,415)(335,414)
(336,418)(337,417)(338,416)(339,420)(340,419)(341,423)(342,422)(343,421)
(344,425)(345,424)(346,428)(347,427)(348,426)(349,430)(350,429)(351,433)
(352,432)(353,431)(354,435)(355,434)(356,438)(357,437)(358,436)(359,440)
(360,439)(361,443)(362,442)(363,441)(364,445)(365,444)(366,448)(367,447)
(368,446)(369,450)(370,449)(371,453)(372,452)(373,451)(374,455)(375,454)
(376,458)(377,457)(378,456)(379,460)(380,459)(381,463)(382,462)(383,461)
(384,465)(385,464)(386,468)(387,467)(388,466)(389,470)(390,469)(391,473)
(392,472)(393,471)(394,475)(395,474)(396,478)(397,477)(398,476)(399,480)
(400,479)(481,563)(482,562)(483,561)(484,565)(485,564)(486,568)(487,567)
(488,566)(489,570)(490,569)(491,573)(492,572)(493,571)(494,575)(495,574)
(496,578)(497,577)(498,576)(499,580)(500,579)(501,583)(502,582)(503,581)
(504,585)(505,584)(506,588)(507,587)(508,586)(509,590)(510,589)(511,593)
(512,592)(513,591)(514,595)(515,594)(516,598)(517,597)(518,596)(519,600)
(520,599)(521,603)(522,602)(523,601)(524,605)(525,604)(526,608)(527,607)
(528,606)(529,610)(530,609)(531,613)(532,612)(533,611)(534,615)(535,614)
(536,618)(537,617)(538,616)(539,620)(540,619)(541,623)(542,622)(543,621)
(544,625)(545,624)(546,628)(547,627)(548,626)(549,630)(550,629)(551,633)
(552,632)(553,631)(554,635)(555,634)(556,638)(557,637)(558,636)(559,640)
(560,639);
poly := sub<Sym(640)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*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