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Polytope of Type {4,16,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,16,10}*1280b
if this polytope has a name.
Group : SmallGroup(1280,323449)
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
   4-fold quotients : {4,4,10}*320, {2,8,10}*320
   5-fold quotients : {4,16,2}*256b
   8-fold quotients : {2,4,10}*160, {4,2,10}*160
   10-fold quotients : {4,8,2}*128a
   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)
( 41,206)( 42,207)( 43,208)( 44,209)( 45,210)( 46,201)( 47,202)( 48,203)
( 49,204)( 50,205)( 51,216)( 52,217)( 53,218)( 54,219)( 55,220)( 56,211)
( 57,212)( 58,213)( 59,214)( 60,215)( 61,226)( 62,227)( 63,228)( 64,229)
( 65,230)( 66,221)( 67,222)( 68,223)( 69,224)( 70,225)( 71,236)( 72,237)
( 73,238)( 74,239)( 75,240)( 76,231)( 77,232)( 78,233)( 79,234)( 80,235)
( 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)
(121,286)(122,287)(123,288)(124,289)(125,290)(126,281)(127,282)(128,283)
(129,284)(130,285)(131,296)(132,297)(133,298)(134,299)(135,300)(136,291)
(137,292)(138,293)(139,294)(140,295)(141,306)(142,307)(143,308)(144,309)
(145,310)(146,301)(147,302)(148,303)(149,304)(150,305)(151,316)(152,317)
(153,318)(154,319)(155,320)(156,311)(157,312)(158,313)(159,314)(160,315)
(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)
(361,526)(362,527)(363,528)(364,529)(365,530)(366,521)(367,522)(368,523)
(369,524)(370,525)(371,536)(372,537)(373,538)(374,539)(375,540)(376,531)
(377,532)(378,533)(379,534)(380,535)(381,546)(382,547)(383,548)(384,549)
(385,550)(386,541)(387,542)(388,543)(389,544)(390,545)(391,556)(392,557)
(393,558)(394,559)(395,560)(396,551)(397,552)(398,553)(399,554)(400,555)
(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)
(441,606)(442,607)(443,608)(444,609)(445,610)(446,601)(447,602)(448,603)
(449,604)(450,605)(451,616)(452,617)(453,618)(454,619)(455,620)(456,611)
(457,612)(458,613)(459,614)(460,615)(461,626)(462,627)(463,628)(464,629)
(465,630)(466,621)(467,622)(468,623)(469,624)(470,625)(471,636)(472,637)
(473,638)(474,639)(475,640)(476,631)(477,632)(478,633)(479,634)(480,635);;
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)(481,546)(482,547)(483,548)(484,549)
(485,550)(486,541)(487,542)(488,543)(489,544)(490,545)(491,551)(492,552)
(493,553)(494,554)(495,555)(496,556)(497,557)(498,558)(499,559)(500,560)
(501,526)(502,527)(503,528)(504,529)(505,530)(506,521)(507,522)(508,523)
(509,524)(510,525)(511,531)(512,532)(513,533)(514,534)(515,535)(516,536)
(517,537)(518,538)(519,539)(520,540)(561,626)(562,627)(563,628)(564,629)
(565,630)(566,621)(567,622)(568,623)(569,624)(570,625)(571,631)(572,632)
(573,633)(574,634)(575,635)(576,636)(577,637)(578,638)(579,639)(580,640)
(581,606)(582,607)(583,608)(584,609)(585,610)(586,601)(587,602)(588,603)
(589,604)(590,605)(591,611)(592,612)(593,613)(594,614)(595,615)(596,616)
(597,617)(598,618)(599,619)(600,620);;
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)( 21,426)( 22,430)( 23,429)( 24,428)
( 25,427)( 26,421)( 27,425)( 28,424)( 29,423)( 30,422)( 31,431)( 32,435)
( 33,434)( 34,433)( 35,432)( 36,436)( 37,440)( 38,439)( 39,438)( 40,437)
( 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)( 61,476)( 62,480)( 63,479)( 64,478)
( 65,477)( 66,471)( 67,475)( 68,474)( 69,473)( 70,472)( 71,466)( 72,470)
( 73,469)( 74,468)( 75,467)( 76,461)( 77,465)( 78,464)( 79,463)( 80,462)
( 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)(101,346)(102,350)(103,349)(104,348)
(105,347)(106,341)(107,345)(108,344)(109,343)(110,342)(111,351)(112,355)
(113,354)(114,353)(115,352)(116,356)(117,360)(118,359)(119,358)(120,357)
(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)(141,396)(142,400)(143,399)(144,398)
(145,397)(146,391)(147,395)(148,394)(149,393)(150,392)(151,386)(152,390)
(153,389)(154,388)(155,387)(156,381)(157,385)(158,384)(159,383)(160,382)
(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)(181,586)(182,590)(183,589)(184,588)
(185,587)(186,581)(187,585)(188,584)(189,583)(190,582)(191,591)(192,595)
(193,594)(194,593)(195,592)(196,596)(197,600)(198,599)(199,598)(200,597)
(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)(221,636)(222,640)(223,639)(224,638)
(225,637)(226,631)(227,635)(228,634)(229,633)(230,632)(231,626)(232,630)
(233,629)(234,628)(235,627)(236,621)(237,625)(238,624)(239,623)(240,622)
(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)(261,506)(262,510)(263,509)(264,508)
(265,507)(266,501)(267,505)(268,504)(269,503)(270,502)(271,511)(272,515)
(273,514)(274,513)(275,512)(276,516)(277,520)(278,519)(279,518)(280,517)
(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)(301,556)(302,560)(303,559)(304,558)
(305,557)(306,551)(307,555)(308,554)(309,553)(310,552)(311,546)(312,550)
(313,549)(314,548)(315,547)(316,541)(317,545)(318,544)(319,543)(320,542);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)( 41,206)( 42,207)( 43,208)( 44,209)( 45,210)( 46,201)( 47,202)
( 48,203)( 49,204)( 50,205)( 51,216)( 52,217)( 53,218)( 54,219)( 55,220)
( 56,211)( 57,212)( 58,213)( 59,214)( 60,215)( 61,226)( 62,227)( 63,228)
( 64,229)( 65,230)( 66,221)( 67,222)( 68,223)( 69,224)( 70,225)( 71,236)
( 72,237)( 73,238)( 74,239)( 75,240)( 76,231)( 77,232)( 78,233)( 79,234)
( 80,235)( 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)(121,286)(122,287)(123,288)(124,289)(125,290)(126,281)(127,282)
(128,283)(129,284)(130,285)(131,296)(132,297)(133,298)(134,299)(135,300)
(136,291)(137,292)(138,293)(139,294)(140,295)(141,306)(142,307)(143,308)
(144,309)(145,310)(146,301)(147,302)(148,303)(149,304)(150,305)(151,316)
(152,317)(153,318)(154,319)(155,320)(156,311)(157,312)(158,313)(159,314)
(160,315)(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)(361,526)(362,527)(363,528)(364,529)(365,530)(366,521)(367,522)
(368,523)(369,524)(370,525)(371,536)(372,537)(373,538)(374,539)(375,540)
(376,531)(377,532)(378,533)(379,534)(380,535)(381,546)(382,547)(383,548)
(384,549)(385,550)(386,541)(387,542)(388,543)(389,544)(390,545)(391,556)
(392,557)(393,558)(394,559)(395,560)(396,551)(397,552)(398,553)(399,554)
(400,555)(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)(441,606)(442,607)(443,608)(444,609)(445,610)(446,601)(447,602)
(448,603)(449,604)(450,605)(451,616)(452,617)(453,618)(454,619)(455,620)
(456,611)(457,612)(458,613)(459,614)(460,615)(461,626)(462,627)(463,628)
(464,629)(465,630)(466,621)(467,622)(468,623)(469,624)(470,625)(471,636)
(472,637)(473,638)(474,639)(475,640)(476,631)(477,632)(478,633)(479,634)
(480,635);
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)(481,546)(482,547)(483,548)
(484,549)(485,550)(486,541)(487,542)(488,543)(489,544)(490,545)(491,551)
(492,552)(493,553)(494,554)(495,555)(496,556)(497,557)(498,558)(499,559)
(500,560)(501,526)(502,527)(503,528)(504,529)(505,530)(506,521)(507,522)
(508,523)(509,524)(510,525)(511,531)(512,532)(513,533)(514,534)(515,535)
(516,536)(517,537)(518,538)(519,539)(520,540)(561,626)(562,627)(563,628)
(564,629)(565,630)(566,621)(567,622)(568,623)(569,624)(570,625)(571,631)
(572,632)(573,633)(574,634)(575,635)(576,636)(577,637)(578,638)(579,639)
(580,640)(581,606)(582,607)(583,608)(584,609)(585,610)(586,601)(587,602)
(588,603)(589,604)(590,605)(591,611)(592,612)(593,613)(594,614)(595,615)
(596,616)(597,617)(598,618)(599,619)(600,620);
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)( 21,426)( 22,430)( 23,429)
( 24,428)( 25,427)( 26,421)( 27,425)( 28,424)( 29,423)( 30,422)( 31,431)
( 32,435)( 33,434)( 34,433)( 35,432)( 36,436)( 37,440)( 38,439)( 39,438)
( 40,437)( 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)( 61,476)( 62,480)( 63,479)
( 64,478)( 65,477)( 66,471)( 67,475)( 68,474)( 69,473)( 70,472)( 71,466)
( 72,470)( 73,469)( 74,468)( 75,467)( 76,461)( 77,465)( 78,464)( 79,463)
( 80,462)( 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)(101,346)(102,350)(103,349)
(104,348)(105,347)(106,341)(107,345)(108,344)(109,343)(110,342)(111,351)
(112,355)(113,354)(114,353)(115,352)(116,356)(117,360)(118,359)(119,358)
(120,357)(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)(141,396)(142,400)(143,399)
(144,398)(145,397)(146,391)(147,395)(148,394)(149,393)(150,392)(151,386)
(152,390)(153,389)(154,388)(155,387)(156,381)(157,385)(158,384)(159,383)
(160,382)(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)(181,586)(182,590)(183,589)
(184,588)(185,587)(186,581)(187,585)(188,584)(189,583)(190,582)(191,591)
(192,595)(193,594)(194,593)(195,592)(196,596)(197,600)(198,599)(199,598)
(200,597)(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)(221,636)(222,640)(223,639)
(224,638)(225,637)(226,631)(227,635)(228,634)(229,633)(230,632)(231,626)
(232,630)(233,629)(234,628)(235,627)(236,621)(237,625)(238,624)(239,623)
(240,622)(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)(261,506)(262,510)(263,509)
(264,508)(265,507)(266,501)(267,505)(268,504)(269,503)(270,502)(271,511)
(272,515)(273,514)(274,513)(275,512)(276,516)(277,520)(278,519)(279,518)
(280,517)(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)(301,556)(302,560)(303,559)
(304,558)(305,557)(306,551)(307,555)(308,554)(309,553)(310,552)(311,546)
(312,550)(313,549)(314,548)(315,547)(316,541)(317,545)(318,544)(319,543)
(320,542);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope