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Polytope of Type {4,8,20}

Atlas Canonical Name : {4,8,20}*1280d
Also Known As : {{4,8|2},{8,20|2}}. if this polytope has another name.
Group : SmallGroup(1280,201309)
Rank : 4
Schlafli Type : {4,8,20}
Number of vertices, edges, etc : 4, 16, 80, 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, {2,8,20}*640a, {4,8,10}*640a
4-fold quotients : {2,4,20}*320, {4,2,20}*320, {4,4,10}*320, {2,8,10}*320
5-fold quotients : {4,8,4}*256d
8-fold quotients : {2,2,20}*160, {2,4,10}*160, {4,2,10}*160
10-fold quotients : {4,4,4}*128, {2,8,4}*128a, {4,8,2}*128a
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, {2,8,2}*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)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)( 72,152)
( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
(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)
(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288)
(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)(216,296)
(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)
(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)(232,312)
(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)(240,320)
(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)
(361,441)(362,442)(363,443)(364,444)(365,445)(366,446)(367,447)(368,448)
(369,449)(370,450)(371,451)(372,452)(373,453)(374,454)(375,455)(376,456)
(377,457)(378,458)(379,459)(380,460)(381,461)(382,462)(383,463)(384,464)
(385,465)(386,466)(387,467)(388,468)(389,469)(390,470)(391,471)(392,472)
(393,473)(394,474)(395,475)(396,476)(397,477)(398,478)(399,479)(400,480)
(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)
(521,601)(522,602)(523,603)(524,604)(525,605)(526,606)(527,607)(528,608)
(529,609)(530,610)(531,611)(532,612)(533,613)(534,614)(535,615)(536,616)
(537,617)(538,618)(539,619)(540,620)(541,621)(542,622)(543,623)(544,624)
(545,625)(546,626)(547,627)(548,628)(549,629)(550,630)(551,631)(552,632)
(553,633)(554,634)(555,635)(556,636)(557,637)(558,638)(559,639)(560,640);;
s1 := ( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 71, 76)
( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 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)
(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)(208,228)
(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)(216,236)
(217,237)(218,238)(219,239)(220,240)(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)(461,466)(462,467)
(463,468)(464,469)(465,470)(471,476)(472,477)(473,478)(474,479)(475,480)
(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)(601,621)(602,622)(603,623)(604,624)
(605,625)(606,626)(607,627)(608,628)(609,629)(610,630)(611,631)(612,632)
(613,633)(614,634)(615,635)(616,636)(617,637)(618,638)(619,639)(620,640);;
s2 := (  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)(  8,169)
(  9,168)( 10,167)( 11,171)( 12,175)( 13,174)( 14,173)( 15,172)( 16,176)
( 17,180)( 18,179)( 19,178)( 20,177)( 21,186)( 22,190)( 23,189)( 24,188)
( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,196)( 32,200)
( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)( 40,192)
( 41,201)( 42,205)( 43,204)( 44,203)( 45,202)( 46,206)( 47,210)( 48,209)
( 49,208)( 50,207)( 51,211)( 52,215)( 53,214)( 54,213)( 55,212)( 56,216)
( 57,220)( 58,219)( 59,218)( 60,217)( 61,226)( 62,230)( 63,229)( 64,228)
( 65,227)( 66,221)( 67,225)( 68,224)( 69,223)( 70,222)( 71,236)( 72,240)
( 73,239)( 74,238)( 75,237)( 76,231)( 77,235)( 78,234)( 79,233)( 80,232)
( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,251)( 92,255)( 93,254)( 94,253)( 95,252)( 96,256)
( 97,260)( 98,259)( 99,258)(100,257)(101,266)(102,270)(103,269)(104,268)
(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,276)(112,280)
(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)(120,272)
(121,281)(122,285)(123,284)(124,283)(125,282)(126,286)(127,290)(128,289)
(129,288)(130,287)(131,291)(132,295)(133,294)(134,293)(135,292)(136,296)
(137,300)(138,299)(139,298)(140,297)(141,306)(142,310)(143,309)(144,308)
(145,307)(146,301)(147,305)(148,304)(149,303)(150,302)(151,316)(152,320)
(153,319)(154,318)(155,317)(156,311)(157,315)(158,314)(159,313)(160,312)
(321,481)(322,485)(323,484)(324,483)(325,482)(326,486)(327,490)(328,489)
(329,488)(330,487)(331,491)(332,495)(333,494)(334,493)(335,492)(336,496)
(337,500)(338,499)(339,498)(340,497)(341,506)(342,510)(343,509)(344,508)
(345,507)(346,501)(347,505)(348,504)(349,503)(350,502)(351,516)(352,520)
(353,519)(354,518)(355,517)(356,511)(357,515)(358,514)(359,513)(360,512)
(361,521)(362,525)(363,524)(364,523)(365,522)(366,526)(367,530)(368,529)
(369,528)(370,527)(371,531)(372,535)(373,534)(374,533)(375,532)(376,536)
(377,540)(378,539)(379,538)(380,537)(381,546)(382,550)(383,549)(384,548)
(385,547)(386,541)(387,545)(388,544)(389,543)(390,542)(391,556)(392,560)
(393,559)(394,558)(395,557)(396,551)(397,555)(398,554)(399,553)(400,552)
(401,561)(402,565)(403,564)(404,563)(405,562)(406,566)(407,570)(408,569)
(409,568)(410,567)(411,571)(412,575)(413,574)(414,573)(415,572)(416,576)
(417,580)(418,579)(419,578)(420,577)(421,586)(422,590)(423,589)(424,588)
(425,587)(426,581)(427,585)(428,584)(429,583)(430,582)(431,596)(432,600)
(433,599)(434,598)(435,597)(436,591)(437,595)(438,594)(439,593)(440,592)
(441,601)(442,605)(443,604)(444,603)(445,602)(446,606)(447,610)(448,609)
(449,608)(450,607)(451,611)(452,615)(453,614)(454,613)(455,612)(456,616)
(457,620)(458,619)(459,618)(460,617)(461,626)(462,630)(463,629)(464,628)
(465,627)(466,621)(467,625)(468,624)(469,623)(470,622)(471,636)(472,640)
(473,639)(474,638)(475,637)(476,631)(477,635)(478,634)(479,633)(480,632);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)
( 48,128)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,151)
( 72,152)( 73,153)( 74,154)( 75,155)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)(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)(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)
(208,288)(209,289)(210,290)(211,291)(212,292)(213,293)(214,294)(215,295)
(216,296)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)
(224,304)(225,305)(226,306)(227,307)(228,308)(229,309)(230,310)(231,311)
(232,312)(233,313)(234,314)(235,315)(236,316)(237,317)(238,318)(239,319)
(240,320)(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)(361,441)(362,442)(363,443)(364,444)(365,445)(366,446)(367,447)
(368,448)(369,449)(370,450)(371,451)(372,452)(373,453)(374,454)(375,455)
(376,456)(377,457)(378,458)(379,459)(380,460)(381,461)(382,462)(383,463)
(384,464)(385,465)(386,466)(387,467)(388,468)(389,469)(390,470)(391,471)
(392,472)(393,473)(394,474)(395,475)(396,476)(397,477)(398,478)(399,479)
(400,480)(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)(521,601)(522,602)(523,603)(524,604)(525,605)(526,606)(527,607)
(528,608)(529,609)(530,610)(531,611)(532,612)(533,613)(534,614)(535,615)
(536,616)(537,617)(538,618)(539,619)(540,620)(541,621)(542,622)(543,623)
(544,624)(545,625)(546,626)(547,627)(548,628)(549,629)(550,630)(551,631)
(552,632)(553,633)(554,634)(555,635)(556,636)(557,637)(558,638)(559,639)
(560,640);
s1 := Sym(640)!( 21, 26)( 22, 27)( 23, 28)( 24, 29)( 25, 30)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)
( 71, 76)( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 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)(201,221)(202,222)(203,223)(204,224)(205,225)(206,226)(207,227)
(208,228)(209,229)(210,230)(211,231)(212,232)(213,233)(214,234)(215,235)
(216,236)(217,237)(218,238)(219,239)(220,240)(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)(461,466)
(462,467)(463,468)(464,469)(465,470)(471,476)(472,477)(473,478)(474,479)
(475,480)(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)(601,621)(602,622)(603,623)
(604,624)(605,625)(606,626)(607,627)(608,628)(609,629)(610,630)(611,631)
(612,632)(613,633)(614,634)(615,635)(616,636)(617,637)(618,638)(619,639)
(620,640);
s2 := Sym(640)!(  1,161)(  2,165)(  3,164)(  4,163)(  5,162)(  6,166)(  7,170)
(  8,169)(  9,168)( 10,167)( 11,171)( 12,175)( 13,174)( 14,173)( 15,172)
( 16,176)( 17,180)( 18,179)( 19,178)( 20,177)( 21,186)( 22,190)( 23,189)
( 24,188)( 25,187)( 26,181)( 27,185)( 28,184)( 29,183)( 30,182)( 31,196)
( 32,200)( 33,199)( 34,198)( 35,197)( 36,191)( 37,195)( 38,194)( 39,193)
( 40,192)( 41,201)( 42,205)( 43,204)( 44,203)( 45,202)( 46,206)( 47,210)
( 48,209)( 49,208)( 50,207)( 51,211)( 52,215)( 53,214)( 54,213)( 55,212)
( 56,216)( 57,220)( 58,219)( 59,218)( 60,217)( 61,226)( 62,230)( 63,229)
( 64,228)( 65,227)( 66,221)( 67,225)( 68,224)( 69,223)( 70,222)( 71,236)
( 72,240)( 73,239)( 74,238)( 75,237)( 76,231)( 77,235)( 78,234)( 79,233)
( 80,232)( 81,241)( 82,245)( 83,244)( 84,243)( 85,242)( 86,246)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,251)( 92,255)( 93,254)( 94,253)( 95,252)
( 96,256)( 97,260)( 98,259)( 99,258)(100,257)(101,266)(102,270)(103,269)
(104,268)(105,267)(106,261)(107,265)(108,264)(109,263)(110,262)(111,276)
(112,280)(113,279)(114,278)(115,277)(116,271)(117,275)(118,274)(119,273)
(120,272)(121,281)(122,285)(123,284)(124,283)(125,282)(126,286)(127,290)
(128,289)(129,288)(130,287)(131,291)(132,295)(133,294)(134,293)(135,292)
(136,296)(137,300)(138,299)(139,298)(140,297)(141,306)(142,310)(143,309)
(144,308)(145,307)(146,301)(147,305)(148,304)(149,303)(150,302)(151,316)
(152,320)(153,319)(154,318)(155,317)(156,311)(157,315)(158,314)(159,313)
(160,312)(321,481)(322,485)(323,484)(324,483)(325,482)(326,486)(327,490)
(328,489)(329,488)(330,487)(331,491)(332,495)(333,494)(334,493)(335,492)
(336,496)(337,500)(338,499)(339,498)(340,497)(341,506)(342,510)(343,509)
(344,508)(345,507)(346,501)(347,505)(348,504)(349,503)(350,502)(351,516)
(352,520)(353,519)(354,518)(355,517)(356,511)(357,515)(358,514)(359,513)
(360,512)(361,521)(362,525)(363,524)(364,523)(365,522)(366,526)(367,530)
(368,529)(369,528)(370,527)(371,531)(372,535)(373,534)(374,533)(375,532)
(376,536)(377,540)(378,539)(379,538)(380,537)(381,546)(382,550)(383,549)
(384,548)(385,547)(386,541)(387,545)(388,544)(389,543)(390,542)(391,556)
(392,560)(393,559)(394,558)(395,557)(396,551)(397,555)(398,554)(399,553)
(400,552)(401,561)(402,565)(403,564)(404,563)(405,562)(406,566)(407,570)
(408,569)(409,568)(410,567)(411,571)(412,575)(413,574)(414,573)(415,572)
(416,576)(417,580)(418,579)(419,578)(420,577)(421,586)(422,590)(423,589)
(424,588)(425,587)(426,581)(427,585)(428,584)(429,583)(430,582)(431,596)
(432,600)(433,599)(434,598)(435,597)(436,591)(437,595)(438,594)(439,593)
(440,592)(441,601)(442,605)(443,604)(444,603)(445,602)(446,606)(447,610)
(448,609)(449,608)(450,607)(451,611)(452,615)(453,614)(454,613)(455,612)
(456,616)(457,620)(458,619)(459,618)(460,617)(461,626)(462,630)(463,629)
(464,628)(465,627)(466,621)(467,625)(468,624)(469,623)(470,622)(471,636)
(472,640)(473,639)(474,638)(475,637)(476,631)(477,635)(478,634)(479,633)
(480,632);
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,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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