Polytope of Type {8,4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,20}*1280b
if this polytope has a name.
Group : SmallGroup(1280,200904)
Rank : 4
Schlafli Type : {8,4,20}
Number of vertices, edges, etc : 8, 16, 40, 20
Order of s₀s₁s₂s₃ : 40
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 : {4,4,20}*640, {8,4,10}*640b
   4-fold quotients : {2,4,20}*320, {4,2,20}*320, {4,4,10}*320
   5-fold quotients : {8,4,4}*256b
   8-fold quotients : {2,2,20}*160, {2,4,10}*160, {4,2,10}*160
   10-fold quotients : {4,4,4}*128, {8,4,2}*128b
   16-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {2,2,5}*40
   40-fold quotients : {2,2,4}*32, {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, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,121)( 47,122)( 48,123)
( 49,124)( 50,125)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,131)
( 57,132)( 58,133)( 59,134)( 60,135)( 61,146)( 62,147)( 63,148)( 64,149)
( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)( 72,157)
( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)
(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)
(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)
(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)
(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)
(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)
(201,286)(202,287)(203,288)(204,289)(205,290)(206,281)(207,282)(208,283)
(209,284)(210,285)(211,296)(212,297)(213,298)(214,299)(215,300)(216,291)
(217,292)(218,293)(219,294)(220,295)(221,306)(222,307)(223,308)(224,309)
(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)(232,317)
(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)(240,315)
(321,401)(322,402)(323,403)(324,404)(325,405)(326,406)(327,407)(328,408)
(329,409)(330,410)(331,411)(332,412)(333,413)(334,414)(335,415)(336,416)
(337,417)(338,418)(339,419)(340,420)(341,421)(342,422)(343,423)(344,424)
(345,425)(346,426)(347,427)(348,428)(349,429)(350,430)(351,431)(352,432)
(353,433)(354,434)(355,435)(356,436)(357,437)(358,438)(359,439)(360,440)
(361,446)(362,447)(363,448)(364,449)(365,450)(366,441)(367,442)(368,443)
(369,444)(370,445)(371,456)(372,457)(373,458)(374,459)(375,460)(376,451)
(377,452)(378,453)(379,454)(380,455)(381,466)(382,467)(383,468)(384,469)
(385,470)(386,461)(387,462)(388,463)(389,464)(390,465)(391,476)(392,477)
(393,478)(394,479)(395,480)(396,471)(397,472)(398,473)(399,474)(400,475)
(481,561)(482,562)(483,563)(484,564)(485,565)(486,566)(487,567)(488,568)
(489,569)(490,570)(491,571)(492,572)(493,573)(494,574)(495,575)(496,576)
(497,577)(498,578)(499,579)(500,580)(501,581)(502,582)(503,583)(504,584)
(505,585)(506,586)(507,587)(508,588)(509,589)(510,590)(511,591)(512,592)
(513,593)(514,594)(515,595)(516,596)(517,597)(518,598)(519,599)(520,600)
(521,606)(522,607)(523,608)(524,609)(525,610)(526,601)(527,602)(528,603)
(529,604)(530,605)(531,616)(532,617)(533,618)(534,619)(535,620)(536,611)
(537,612)(538,613)(539,614)(540,615)(541,626)(542,627)(543,628)(544,629)
(545,630)(546,621)(547,622)(548,623)(549,624)(550,625)(551,636)(552,637)
(553,638)(554,639)(555,640)(556,631)(557,632)(558,633)(559,634)(560,635);;
s1 := ( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 41, 46)( 42, 47)( 43, 48)( 44, 49)( 45, 50)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 81,121)( 82,122)( 83,123)( 84,124)
( 85,125)( 86,126)( 87,127)( 88,128)( 89,129)( 90,130)( 91,131)( 92,132)
( 93,133)( 94,134)( 95,135)( 96,136)( 97,137)( 98,138)( 99,139)(100,140)
(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)(108,143)
(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)(116,151)
(117,152)(118,153)(119,154)(120,155)(161,181)(162,182)(163,183)(164,184)
(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)
(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)
(201,226)(202,227)(203,228)(204,229)(205,230)(206,221)(207,222)(208,223)
(209,224)(210,225)(211,236)(212,237)(213,238)(214,239)(215,240)(216,231)
(217,232)(218,233)(219,234)(220,235)(241,301)(242,302)(243,303)(244,304)
(245,305)(246,306)(247,307)(248,308)(249,309)(250,310)(251,311)(252,312)
(253,313)(254,314)(255,315)(256,316)(257,317)(258,318)(259,319)(260,320)
(261,281)(262,282)(263,283)(264,284)(265,285)(266,286)(267,287)(268,288)
(269,289)(270,290)(271,291)(272,292)(273,293)(274,294)(275,295)(276,296)
(277,297)(278,298)(279,299)(280,300)(321,361)(322,362)(323,363)(324,364)
(325,365)(326,366)(327,367)(328,368)(329,369)(330,370)(331,371)(332,372)
(333,373)(334,374)(335,375)(336,376)(337,377)(338,378)(339,379)(340,380)
(341,386)(342,387)(343,388)(344,389)(345,390)(346,381)(347,382)(348,383)
(349,384)(350,385)(351,396)(352,397)(353,398)(354,399)(355,400)(356,391)
(357,392)(358,393)(359,394)(360,395)(421,426)(422,427)(423,428)(424,429)
(425,430)(431,436)(432,437)(433,438)(434,439)(435,440)(441,446)(442,447)
(443,448)(444,449)(445,450)(451,456)(452,457)(453,458)(454,459)(455,460)
(481,541)(482,542)(483,543)(484,544)(485,545)(486,546)(487,547)(488,548)
(489,549)(490,550)(491,551)(492,552)(493,553)(494,554)(495,555)(496,556)
(497,557)(498,558)(499,559)(500,560)(501,521)(502,522)(503,523)(504,524)
(505,525)(506,526)(507,527)(508,528)(509,529)(510,530)(511,531)(512,532)
(513,533)(514,534)(515,535)(516,536)(517,537)(518,538)(519,539)(520,540)
(561,581)(562,582)(563,583)(564,584)(565,585)(566,586)(567,587)(568,588)
(569,589)(570,590)(571,591)(572,592)(573,593)(574,594)(575,595)(576,596)
(577,597)(578,598)(579,599)(580,600)(601,626)(602,627)(603,628)(604,629)
(605,630)(606,621)(607,622)(608,623)(609,624)(610,625)(611,636)(612,637)
(613,638)(614,639)(615,640)(616,631)(617,632)(618,633)(619,634)(620,635);;
s2 := (  1,241)(  2,245)(  3,244)(  4,243)(  5,242)(  6,246)(  7,250)(  8,249)
(  9,248)( 10,247)( 11,251)( 12,255)( 13,254)( 14,253)( 15,252)( 16,256)
( 17,260)( 18,259)( 19,258)( 20,257)( 21,266)( 22,270)( 23,269)( 24,268)
( 25,267)( 26,261)( 27,265)( 28,264)( 29,263)( 30,262)( 31,276)( 32,280)
( 33,279)( 34,278)( 35,277)( 36,271)( 37,275)( 38,274)( 39,273)( 40,272)
( 41,286)( 42,290)( 43,289)( 44,288)( 45,287)( 46,281)( 47,285)( 48,284)
( 49,283)( 50,282)( 51,296)( 52,300)( 53,299)( 54,298)( 55,297)( 56,291)
( 57,295)( 58,294)( 59,293)( 60,292)( 61,301)( 62,305)( 63,304)( 64,303)
( 65,302)( 66,306)( 67,310)( 68,309)( 69,308)( 70,307)( 71,311)( 72,315)
( 73,314)( 74,313)( 75,312)( 76,316)( 77,320)( 78,319)( 79,318)( 80,317)
( 81,161)( 82,165)( 83,164)( 84,163)( 85,162)( 86,166)( 87,170)( 88,169)
( 89,168)( 90,167)( 91,171)( 92,175)( 93,174)( 94,173)( 95,172)( 96,176)
( 97,180)( 98,179)( 99,178)(100,177)(101,186)(102,190)(103,189)(104,188)
(105,187)(106,181)(107,185)(108,184)(109,183)(110,182)(111,196)(112,200)
(113,199)(114,198)(115,197)(116,191)(117,195)(118,194)(119,193)(120,192)
(121,206)(122,210)(123,209)(124,208)(125,207)(126,201)(127,205)(128,204)
(129,203)(130,202)(131,216)(132,220)(133,219)(134,218)(135,217)(136,211)
(137,215)(138,214)(139,213)(140,212)(141,221)(142,225)(143,224)(144,223)
(145,222)(146,226)(147,230)(148,229)(149,228)(150,227)(151,231)(152,235)
(153,234)(154,233)(155,232)(156,236)(157,240)(158,239)(159,238)(160,237)
(321,561)(322,565)(323,564)(324,563)(325,562)(326,566)(327,570)(328,569)
(329,568)(330,567)(331,571)(332,575)(333,574)(334,573)(335,572)(336,576)
(337,580)(338,579)(339,578)(340,577)(341,586)(342,590)(343,589)(344,588)
(345,587)(346,581)(347,585)(348,584)(349,583)(350,582)(351,596)(352,600)
(353,599)(354,598)(355,597)(356,591)(357,595)(358,594)(359,593)(360,592)
(361,606)(362,610)(363,609)(364,608)(365,607)(366,601)(367,605)(368,604)
(369,603)(370,602)(371,616)(372,620)(373,619)(374,618)(375,617)(376,611)
(377,615)(378,614)(379,613)(380,612)(381,621)(382,625)(383,624)(384,623)
(385,622)(386,626)(387,630)(388,629)(389,628)(390,627)(391,631)(392,635)
(393,634)(394,633)(395,632)(396,636)(397,640)(398,639)(399,638)(400,637)
(401,481)(402,485)(403,484)(404,483)(405,482)(406,486)(407,490)(408,489)
(409,488)(410,487)(411,491)(412,495)(413,494)(414,493)(415,492)(416,496)
(417,500)(418,499)(419,498)(420,497)(421,506)(422,510)(423,509)(424,508)
(425,507)(426,501)(427,505)(428,504)(429,503)(430,502)(431,516)(432,520)
(433,519)(434,518)(435,517)(436,511)(437,515)(438,514)(439,513)(440,512)
(441,526)(442,530)(443,529)(444,528)(445,527)(446,521)(447,525)(448,524)
(449,523)(450,522)(451,536)(452,540)(453,539)(454,538)(455,537)(456,531)
(457,535)(458,534)(459,533)(460,532)(461,541)(462,545)(463,544)(464,543)
(465,542)(466,546)(467,550)(468,549)(469,548)(470,547)(471,551)(472,555)
(473,554)(474,553)(475,552)(476,556)(477,560)(478,559)(479,558)(480,557);;
s3 := (  1,403)(  2,402)(  3,401)(  4,405)(  5,404)(  6,408)(  7,407)(  8,406)
(  9,410)( 10,409)( 11,413)( 12,412)( 13,411)( 14,415)( 15,414)( 16,418)
( 17,417)( 18,416)( 19,420)( 20,419)( 21,423)( 22,422)( 23,421)( 24,425)
( 25,424)( 26,428)( 27,427)( 28,426)( 29,430)( 30,429)( 31,433)( 32,432)
( 33,431)( 34,435)( 35,434)( 36,438)( 37,437)( 38,436)( 39,440)( 40,439)
( 41,443)( 42,442)( 43,441)( 44,445)( 45,444)( 46,448)( 47,447)( 48,446)
( 49,450)( 50,449)( 51,453)( 52,452)( 53,451)( 54,455)( 55,454)( 56,458)
( 57,457)( 58,456)( 59,460)( 60,459)( 61,463)( 62,462)( 63,461)( 64,465)
( 65,464)( 66,468)( 67,467)( 68,466)( 69,470)( 70,469)( 71,473)( 72,472)
( 73,471)( 74,475)( 75,474)( 76,478)( 77,477)( 78,476)( 79,480)( 80,479)
( 81,323)( 82,322)( 83,321)( 84,325)( 85,324)( 86,328)( 87,327)( 88,326)
( 89,330)( 90,329)( 91,333)( 92,332)( 93,331)( 94,335)( 95,334)( 96,338)
( 97,337)( 98,336)( 99,340)(100,339)(101,343)(102,342)(103,341)(104,345)
(105,344)(106,348)(107,347)(108,346)(109,350)(110,349)(111,353)(112,352)
(113,351)(114,355)(115,354)(116,358)(117,357)(118,356)(119,360)(120,359)
(121,363)(122,362)(123,361)(124,365)(125,364)(126,368)(127,367)(128,366)
(129,370)(130,369)(131,373)(132,372)(133,371)(134,375)(135,374)(136,378)
(137,377)(138,376)(139,380)(140,379)(141,383)(142,382)(143,381)(144,385)
(145,384)(146,388)(147,387)(148,386)(149,390)(150,389)(151,393)(152,392)
(153,391)(154,395)(155,394)(156,398)(157,397)(158,396)(159,400)(160,399)
(161,573)(162,572)(163,571)(164,575)(165,574)(166,578)(167,577)(168,576)
(169,580)(170,579)(171,563)(172,562)(173,561)(174,565)(175,564)(176,568)
(177,567)(178,566)(179,570)(180,569)(181,593)(182,592)(183,591)(184,595)
(185,594)(186,598)(187,597)(188,596)(189,600)(190,599)(191,583)(192,582)
(193,581)(194,585)(195,584)(196,588)(197,587)(198,586)(199,590)(200,589)
(201,613)(202,612)(203,611)(204,615)(205,614)(206,618)(207,617)(208,616)
(209,620)(210,619)(211,603)(212,602)(213,601)(214,605)(215,604)(216,608)
(217,607)(218,606)(219,610)(220,609)(221,633)(222,632)(223,631)(224,635)
(225,634)(226,638)(227,637)(228,636)(229,640)(230,639)(231,623)(232,622)
(233,621)(234,625)(235,624)(236,628)(237,627)(238,626)(239,630)(240,629)
(241,493)(242,492)(243,491)(244,495)(245,494)(246,498)(247,497)(248,496)
(249,500)(250,499)(251,483)(252,482)(253,481)(254,485)(255,484)(256,488)
(257,487)(258,486)(259,490)(260,489)(261,513)(262,512)(263,511)(264,515)
(265,514)(266,518)(267,517)(268,516)(269,520)(270,519)(271,503)(272,502)
(273,501)(274,505)(275,504)(276,508)(277,507)(278,506)(279,510)(280,509)
(281,533)(282,532)(283,531)(284,535)(285,534)(286,538)(287,537)(288,536)
(289,540)(290,539)(291,523)(292,522)(293,521)(294,525)(295,524)(296,528)
(297,527)(298,526)(299,530)(300,529)(301,553)(302,552)(303,551)(304,555)
(305,554)(306,558)(307,557)(308,556)(309,560)(310,559)(311,543)(312,542)
(313,541)(314,545)(315,544)(316,548)(317,547)(318,546)(319,550)(320,549);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,101)( 22,102)( 23,103)
( 24,104)( 25,105)( 26,106)( 27,107)( 28,108)( 29,109)( 30,110)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,121)( 47,122)
( 48,123)( 49,124)( 50,125)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)
( 56,131)( 57,132)( 58,133)( 59,134)( 60,135)( 61,146)( 62,147)( 63,148)
( 64,149)( 65,150)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,156)
( 72,157)( 73,158)( 74,159)( 75,160)( 76,151)( 77,152)( 78,153)( 79,154)
( 80,155)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)
(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)
(176,256)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)
(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)
(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)
(200,280)(201,286)(202,287)(203,288)(204,289)(205,290)(206,281)(207,282)
(208,283)(209,284)(210,285)(211,296)(212,297)(213,298)(214,299)(215,300)
(216,291)(217,292)(218,293)(219,294)(220,295)(221,306)(222,307)(223,308)
(224,309)(225,310)(226,301)(227,302)(228,303)(229,304)(230,305)(231,316)
(232,317)(233,318)(234,319)(235,320)(236,311)(237,312)(238,313)(239,314)
(240,315)(321,401)(322,402)(323,403)(324,404)(325,405)(326,406)(327,407)
(328,408)(329,409)(330,410)(331,411)(332,412)(333,413)(334,414)(335,415)
(336,416)(337,417)(338,418)(339,419)(340,420)(341,421)(342,422)(343,423)
(344,424)(345,425)(346,426)(347,427)(348,428)(349,429)(350,430)(351,431)
(352,432)(353,433)(354,434)(355,435)(356,436)(357,437)(358,438)(359,439)
(360,440)(361,446)(362,447)(363,448)(364,449)(365,450)(366,441)(367,442)
(368,443)(369,444)(370,445)(371,456)(372,457)(373,458)(374,459)(375,460)
(376,451)(377,452)(378,453)(379,454)(380,455)(381,466)(382,467)(383,468)
(384,469)(385,470)(386,461)(387,462)(388,463)(389,464)(390,465)(391,476)
(392,477)(393,478)(394,479)(395,480)(396,471)(397,472)(398,473)(399,474)
(400,475)(481,561)(482,562)(483,563)(484,564)(485,565)(486,566)(487,567)
(488,568)(489,569)(490,570)(491,571)(492,572)(493,573)(494,574)(495,575)
(496,576)(497,577)(498,578)(499,579)(500,580)(501,581)(502,582)(503,583)
(504,584)(505,585)(506,586)(507,587)(508,588)(509,589)(510,590)(511,591)
(512,592)(513,593)(514,594)(515,595)(516,596)(517,597)(518,598)(519,599)
(520,600)(521,606)(522,607)(523,608)(524,609)(525,610)(526,601)(527,602)
(528,603)(529,604)(530,605)(531,616)(532,617)(533,618)(534,619)(535,620)
(536,611)(537,612)(538,613)(539,614)(540,615)(541,626)(542,627)(543,628)
(544,629)(545,630)(546,621)(547,622)(548,623)(549,624)(550,625)(551,636)
(552,637)(553,638)(554,639)(555,640)(556,631)(557,632)(558,633)(559,634)
(560,635);
s1 := Sym(640)!( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 41, 46)( 42, 47)( 43, 48)( 44, 49)( 45, 50)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 81,121)( 82,122)( 83,123)
( 84,124)( 85,125)( 86,126)( 87,127)( 88,128)( 89,129)( 90,130)( 91,131)
( 92,132)( 93,133)( 94,134)( 95,135)( 96,136)( 97,137)( 98,138)( 99,139)
(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)(106,141)(107,142)
(108,143)(109,144)(110,145)(111,156)(112,157)(113,158)(114,159)(115,160)
(116,151)(117,152)(118,153)(119,154)(120,155)(161,181)(162,182)(163,183)
(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)
(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)
(180,200)(201,226)(202,227)(203,228)(204,229)(205,230)(206,221)(207,222)
(208,223)(209,224)(210,225)(211,236)(212,237)(213,238)(214,239)(215,240)
(216,231)(217,232)(218,233)(219,234)(220,235)(241,301)(242,302)(243,303)
(244,304)(245,305)(246,306)(247,307)(248,308)(249,309)(250,310)(251,311)
(252,312)(253,313)(254,314)(255,315)(256,316)(257,317)(258,318)(259,319)
(260,320)(261,281)(262,282)(263,283)(264,284)(265,285)(266,286)(267,287)
(268,288)(269,289)(270,290)(271,291)(272,292)(273,293)(274,294)(275,295)
(276,296)(277,297)(278,298)(279,299)(280,300)(321,361)(322,362)(323,363)
(324,364)(325,365)(326,366)(327,367)(328,368)(329,369)(330,370)(331,371)
(332,372)(333,373)(334,374)(335,375)(336,376)(337,377)(338,378)(339,379)
(340,380)(341,386)(342,387)(343,388)(344,389)(345,390)(346,381)(347,382)
(348,383)(349,384)(350,385)(351,396)(352,397)(353,398)(354,399)(355,400)
(356,391)(357,392)(358,393)(359,394)(360,395)(421,426)(422,427)(423,428)
(424,429)(425,430)(431,436)(432,437)(433,438)(434,439)(435,440)(441,446)
(442,447)(443,448)(444,449)(445,450)(451,456)(452,457)(453,458)(454,459)
(455,460)(481,541)(482,542)(483,543)(484,544)(485,545)(486,546)(487,547)
(488,548)(489,549)(490,550)(491,551)(492,552)(493,553)(494,554)(495,555)
(496,556)(497,557)(498,558)(499,559)(500,560)(501,521)(502,522)(503,523)
(504,524)(505,525)(506,526)(507,527)(508,528)(509,529)(510,530)(511,531)
(512,532)(513,533)(514,534)(515,535)(516,536)(517,537)(518,538)(519,539)
(520,540)(561,581)(562,582)(563,583)(564,584)(565,585)(566,586)(567,587)
(568,588)(569,589)(570,590)(571,591)(572,592)(573,593)(574,594)(575,595)
(576,596)(577,597)(578,598)(579,599)(580,600)(601,626)(602,627)(603,628)
(604,629)(605,630)(606,621)(607,622)(608,623)(609,624)(610,625)(611,636)
(612,637)(613,638)(614,639)(615,640)(616,631)(617,632)(618,633)(619,634)
(620,635);
s2 := Sym(640)!(  1,241)(  2,245)(  3,244)(  4,243)(  5,242)(  6,246)(  7,250)
(  8,249)(  9,248)( 10,247)( 11,251)( 12,255)( 13,254)( 14,253)( 15,252)
( 16,256)( 17,260)( 18,259)( 19,258)( 20,257)( 21,266)( 22,270)( 23,269)
( 24,268)( 25,267)( 26,261)( 27,265)( 28,264)( 29,263)( 30,262)( 31,276)
( 32,280)( 33,279)( 34,278)( 35,277)( 36,271)( 37,275)( 38,274)( 39,273)
( 40,272)( 41,286)( 42,290)( 43,289)( 44,288)( 45,287)( 46,281)( 47,285)
( 48,284)( 49,283)( 50,282)( 51,296)( 52,300)( 53,299)( 54,298)( 55,297)
( 56,291)( 57,295)( 58,294)( 59,293)( 60,292)( 61,301)( 62,305)( 63,304)
( 64,303)( 65,302)( 66,306)( 67,310)( 68,309)( 69,308)( 70,307)( 71,311)
( 72,315)( 73,314)( 74,313)( 75,312)( 76,316)( 77,320)( 78,319)( 79,318)
( 80,317)( 81,161)( 82,165)( 83,164)( 84,163)( 85,162)( 86,166)( 87,170)
( 88,169)( 89,168)( 90,167)( 91,171)( 92,175)( 93,174)( 94,173)( 95,172)
( 96,176)( 97,180)( 98,179)( 99,178)(100,177)(101,186)(102,190)(103,189)
(104,188)(105,187)(106,181)(107,185)(108,184)(109,183)(110,182)(111,196)
(112,200)(113,199)(114,198)(115,197)(116,191)(117,195)(118,194)(119,193)
(120,192)(121,206)(122,210)(123,209)(124,208)(125,207)(126,201)(127,205)
(128,204)(129,203)(130,202)(131,216)(132,220)(133,219)(134,218)(135,217)
(136,211)(137,215)(138,214)(139,213)(140,212)(141,221)(142,225)(143,224)
(144,223)(145,222)(146,226)(147,230)(148,229)(149,228)(150,227)(151,231)
(152,235)(153,234)(154,233)(155,232)(156,236)(157,240)(158,239)(159,238)
(160,237)(321,561)(322,565)(323,564)(324,563)(325,562)(326,566)(327,570)
(328,569)(329,568)(330,567)(331,571)(332,575)(333,574)(334,573)(335,572)
(336,576)(337,580)(338,579)(339,578)(340,577)(341,586)(342,590)(343,589)
(344,588)(345,587)(346,581)(347,585)(348,584)(349,583)(350,582)(351,596)
(352,600)(353,599)(354,598)(355,597)(356,591)(357,595)(358,594)(359,593)
(360,592)(361,606)(362,610)(363,609)(364,608)(365,607)(366,601)(367,605)
(368,604)(369,603)(370,602)(371,616)(372,620)(373,619)(374,618)(375,617)
(376,611)(377,615)(378,614)(379,613)(380,612)(381,621)(382,625)(383,624)
(384,623)(385,622)(386,626)(387,630)(388,629)(389,628)(390,627)(391,631)
(392,635)(393,634)(394,633)(395,632)(396,636)(397,640)(398,639)(399,638)
(400,637)(401,481)(402,485)(403,484)(404,483)(405,482)(406,486)(407,490)
(408,489)(409,488)(410,487)(411,491)(412,495)(413,494)(414,493)(415,492)
(416,496)(417,500)(418,499)(419,498)(420,497)(421,506)(422,510)(423,509)
(424,508)(425,507)(426,501)(427,505)(428,504)(429,503)(430,502)(431,516)
(432,520)(433,519)(434,518)(435,517)(436,511)(437,515)(438,514)(439,513)
(440,512)(441,526)(442,530)(443,529)(444,528)(445,527)(446,521)(447,525)
(448,524)(449,523)(450,522)(451,536)(452,540)(453,539)(454,538)(455,537)
(456,531)(457,535)(458,534)(459,533)(460,532)(461,541)(462,545)(463,544)
(464,543)(465,542)(466,546)(467,550)(468,549)(469,548)(470,547)(471,551)
(472,555)(473,554)(474,553)(475,552)(476,556)(477,560)(478,559)(479,558)
(480,557);
s3 := Sym(640)!(  1,403)(  2,402)(  3,401)(  4,405)(  5,404)(  6,408)(  7,407)
(  8,406)(  9,410)( 10,409)( 11,413)( 12,412)( 13,411)( 14,415)( 15,414)
( 16,418)( 17,417)( 18,416)( 19,420)( 20,419)( 21,423)( 22,422)( 23,421)
( 24,425)( 25,424)( 26,428)( 27,427)( 28,426)( 29,430)( 30,429)( 31,433)
( 32,432)( 33,431)( 34,435)( 35,434)( 36,438)( 37,437)( 38,436)( 39,440)
( 40,439)( 41,443)( 42,442)( 43,441)( 44,445)( 45,444)( 46,448)( 47,447)
( 48,446)( 49,450)( 50,449)( 51,453)( 52,452)( 53,451)( 54,455)( 55,454)
( 56,458)( 57,457)( 58,456)( 59,460)( 60,459)( 61,463)( 62,462)( 63,461)
( 64,465)( 65,464)( 66,468)( 67,467)( 68,466)( 69,470)( 70,469)( 71,473)
( 72,472)( 73,471)( 74,475)( 75,474)( 76,478)( 77,477)( 78,476)( 79,480)
( 80,479)( 81,323)( 82,322)( 83,321)( 84,325)( 85,324)( 86,328)( 87,327)
( 88,326)( 89,330)( 90,329)( 91,333)( 92,332)( 93,331)( 94,335)( 95,334)
( 96,338)( 97,337)( 98,336)( 99,340)(100,339)(101,343)(102,342)(103,341)
(104,345)(105,344)(106,348)(107,347)(108,346)(109,350)(110,349)(111,353)
(112,352)(113,351)(114,355)(115,354)(116,358)(117,357)(118,356)(119,360)
(120,359)(121,363)(122,362)(123,361)(124,365)(125,364)(126,368)(127,367)
(128,366)(129,370)(130,369)(131,373)(132,372)(133,371)(134,375)(135,374)
(136,378)(137,377)(138,376)(139,380)(140,379)(141,383)(142,382)(143,381)
(144,385)(145,384)(146,388)(147,387)(148,386)(149,390)(150,389)(151,393)
(152,392)(153,391)(154,395)(155,394)(156,398)(157,397)(158,396)(159,400)
(160,399)(161,573)(162,572)(163,571)(164,575)(165,574)(166,578)(167,577)
(168,576)(169,580)(170,579)(171,563)(172,562)(173,561)(174,565)(175,564)
(176,568)(177,567)(178,566)(179,570)(180,569)(181,593)(182,592)(183,591)
(184,595)(185,594)(186,598)(187,597)(188,596)(189,600)(190,599)(191,583)
(192,582)(193,581)(194,585)(195,584)(196,588)(197,587)(198,586)(199,590)
(200,589)(201,613)(202,612)(203,611)(204,615)(205,614)(206,618)(207,617)
(208,616)(209,620)(210,619)(211,603)(212,602)(213,601)(214,605)(215,604)
(216,608)(217,607)(218,606)(219,610)(220,609)(221,633)(222,632)(223,631)
(224,635)(225,634)(226,638)(227,637)(228,636)(229,640)(230,639)(231,623)
(232,622)(233,621)(234,625)(235,624)(236,628)(237,627)(238,626)(239,630)
(240,629)(241,493)(242,492)(243,491)(244,495)(245,494)(246,498)(247,497)
(248,496)(249,500)(250,499)(251,483)(252,482)(253,481)(254,485)(255,484)
(256,488)(257,487)(258,486)(259,490)(260,489)(261,513)(262,512)(263,511)
(264,515)(265,514)(266,518)(267,517)(268,516)(269,520)(270,519)(271,503)
(272,502)(273,501)(274,505)(275,504)(276,508)(277,507)(278,506)(279,510)
(280,509)(281,533)(282,532)(283,531)(284,535)(285,534)(286,538)(287,537)
(288,536)(289,540)(290,539)(291,523)(292,522)(293,521)(294,525)(295,524)
(296,528)(297,527)(298,526)(299,530)(300,529)(301,553)(302,552)(303,551)
(304,555)(305,554)(306,558)(307,557)(308,556)(309,560)(310,559)(311,543)
(312,542)(313,541)(314,545)(315,544)(316,548)(317,547)(318,546)(319,550)
(320,549);
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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