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Polytope of Type {4,210}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,210}*1680a
Also Known As : {4,210|2}. if this polytope has another name.
Group : SmallGroup(1680,914)
Rank : 3
Schlafli Type : {4,210}
Number of vertices, edges, etc : 4, 420, 210
Order of s0s1s2 : 420
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {2,210}*840
   3-fold quotients : {4,70}*560
   4-fold quotients : {2,105}*420
   5-fold quotients : {4,42}*336a
   6-fold quotients : {2,70}*280
   7-fold quotients : {4,30}*240a
   10-fold quotients : {2,42}*168
   12-fold quotients : {2,35}*140
   14-fold quotients : {2,30}*120
   15-fold quotients : {4,14}*112
   20-fold quotients : {2,21}*84
   21-fold quotients : {4,10}*80
   28-fold quotients : {2,15}*60
   30-fold quotients : {2,14}*56
   35-fold quotients : {4,6}*48a
   42-fold quotients : {2,10}*40
   60-fold quotients : {2,7}*28
   70-fold quotients : {2,6}*24
   84-fold quotients : {2,5}*20
   105-fold quotients : {4,2}*16
   140-fold quotients : {2,3}*12
   210-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,323)
(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,330)(226,331)
(227,332)(228,333)(229,334)(230,335)(231,336)(232,337)(233,338)(234,339)
(235,340)(236,341)(237,342)(238,343)(239,344)(240,345)(241,346)(242,347)
(243,348)(244,349)(245,350)(246,351)(247,352)(248,353)(249,354)(250,355)
(251,356)(252,357)(253,358)(254,359)(255,360)(256,361)(257,362)(258,363)
(259,364)(260,365)(261,366)(262,367)(263,368)(264,369)(265,370)(266,371)
(267,372)(268,373)(269,374)(270,375)(271,376)(272,377)(273,378)(274,379)
(275,380)(276,381)(277,382)(278,383)(279,384)(280,385)(281,386)(282,387)
(283,388)(284,389)(285,390)(286,391)(287,392)(288,393)(289,394)(290,395)
(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,402)(298,403)
(299,404)(300,405)(301,406)(302,407)(303,408)(304,409)(305,410)(306,411)
(307,412)(308,413)(309,414)(310,415)(311,416)(312,417)(313,418)(314,419)
(315,420);;
s1 := (  1,211)(  2,217)(  3,216)(  4,215)(  5,214)(  6,213)(  7,212)(  8,239)
(  9,245)( 10,244)( 11,243)( 12,242)( 13,241)( 14,240)( 15,232)( 16,238)
( 17,237)( 18,236)( 19,235)( 20,234)( 21,233)( 22,225)( 23,231)( 24,230)
( 25,229)( 26,228)( 27,227)( 28,226)( 29,218)( 30,224)( 31,223)( 32,222)
( 33,221)( 34,220)( 35,219)( 36,281)( 37,287)( 38,286)( 39,285)( 40,284)
( 41,283)( 42,282)( 43,309)( 44,315)( 45,314)( 46,313)( 47,312)( 48,311)
( 49,310)( 50,302)( 51,308)( 52,307)( 53,306)( 54,305)( 55,304)( 56,303)
( 57,295)( 58,301)( 59,300)( 60,299)( 61,298)( 62,297)( 63,296)( 64,288)
( 65,294)( 66,293)( 67,292)( 68,291)( 69,290)( 70,289)( 71,246)( 72,252)
( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,274)( 79,280)( 80,279)
( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)( 88,271)
( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)( 96,263)
( 97,262)( 98,261)( 99,253)(100,259)(101,258)(102,257)(103,256)(104,255)
(105,254)(106,316)(107,322)(108,321)(109,320)(110,319)(111,318)(112,317)
(113,344)(114,350)(115,349)(116,348)(117,347)(118,346)(119,345)(120,337)
(121,343)(122,342)(123,341)(124,340)(125,339)(126,338)(127,330)(128,336)
(129,335)(130,334)(131,333)(132,332)(133,331)(134,323)(135,329)(136,328)
(137,327)(138,326)(139,325)(140,324)(141,386)(142,392)(143,391)(144,390)
(145,389)(146,388)(147,387)(148,414)(149,420)(150,419)(151,418)(152,417)
(153,416)(154,415)(155,407)(156,413)(157,412)(158,411)(159,410)(160,409)
(161,408)(162,400)(163,406)(164,405)(165,404)(166,403)(167,402)(168,401)
(169,393)(170,399)(171,398)(172,397)(173,396)(174,395)(175,394)(176,351)
(177,357)(178,356)(179,355)(180,354)(181,353)(182,352)(183,379)(184,385)
(185,384)(186,383)(187,382)(188,381)(189,380)(190,372)(191,378)(192,377)
(193,376)(194,375)(195,374)(196,373)(197,365)(198,371)(199,370)(200,369)
(201,368)(202,367)(203,366)(204,358)(205,364)(206,363)(207,362)(208,361)
(209,360)(210,359);;
s2 := (  1, 44)(  2, 43)(  3, 49)(  4, 48)(  5, 47)(  6, 46)(  7, 45)(  8, 37)
(  9, 36)( 10, 42)( 11, 41)( 12, 40)( 13, 39)( 14, 38)( 15, 65)( 16, 64)
( 17, 70)( 18, 69)( 19, 68)( 20, 67)( 21, 66)( 22, 58)( 23, 57)( 24, 63)
( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 51)( 30, 50)( 31, 56)( 32, 55)
( 33, 54)( 34, 53)( 35, 52)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)
( 76, 81)( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)
( 91,101)( 92, 93)( 94, 98)( 95, 97)(106,149)(107,148)(108,154)(109,153)
(110,152)(111,151)(112,150)(113,142)(114,141)(115,147)(116,146)(117,145)
(118,144)(119,143)(120,170)(121,169)(122,175)(123,174)(124,173)(125,172)
(126,171)(127,163)(128,162)(129,168)(130,167)(131,166)(132,165)(133,164)
(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)(176,184)
(177,183)(178,189)(179,188)(180,187)(181,186)(182,185)(190,205)(191,204)
(192,210)(193,209)(194,208)(195,207)(196,206)(197,198)(199,203)(200,202)
(211,254)(212,253)(213,259)(214,258)(215,257)(216,256)(217,255)(218,247)
(219,246)(220,252)(221,251)(222,250)(223,249)(224,248)(225,275)(226,274)
(227,280)(228,279)(229,278)(230,277)(231,276)(232,268)(233,267)(234,273)
(235,272)(236,271)(237,270)(238,269)(239,261)(240,260)(241,266)(242,265)
(243,264)(244,263)(245,262)(281,289)(282,288)(283,294)(284,293)(285,292)
(286,291)(287,290)(295,310)(296,309)(297,315)(298,314)(299,313)(300,312)
(301,311)(302,303)(304,308)(305,307)(316,359)(317,358)(318,364)(319,363)
(320,362)(321,361)(322,360)(323,352)(324,351)(325,357)(326,356)(327,355)
(328,354)(329,353)(330,380)(331,379)(332,385)(333,384)(334,383)(335,382)
(336,381)(337,373)(338,372)(339,378)(340,377)(341,376)(342,375)(343,374)
(344,366)(345,365)(346,371)(347,370)(348,369)(349,368)(350,367)(386,394)
(387,393)(388,399)(389,398)(390,397)(391,396)(392,395)(400,415)(401,414)
(402,420)(403,419)(404,418)(405,417)(406,416)(407,408)(409,413)(410,412);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(420)!(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)
(218,323)(219,324)(220,325)(221,326)(222,327)(223,328)(224,329)(225,330)
(226,331)(227,332)(228,333)(229,334)(230,335)(231,336)(232,337)(233,338)
(234,339)(235,340)(236,341)(237,342)(238,343)(239,344)(240,345)(241,346)
(242,347)(243,348)(244,349)(245,350)(246,351)(247,352)(248,353)(249,354)
(250,355)(251,356)(252,357)(253,358)(254,359)(255,360)(256,361)(257,362)
(258,363)(259,364)(260,365)(261,366)(262,367)(263,368)(264,369)(265,370)
(266,371)(267,372)(268,373)(269,374)(270,375)(271,376)(272,377)(273,378)
(274,379)(275,380)(276,381)(277,382)(278,383)(279,384)(280,385)(281,386)
(282,387)(283,388)(284,389)(285,390)(286,391)(287,392)(288,393)(289,394)
(290,395)(291,396)(292,397)(293,398)(294,399)(295,400)(296,401)(297,402)
(298,403)(299,404)(300,405)(301,406)(302,407)(303,408)(304,409)(305,410)
(306,411)(307,412)(308,413)(309,414)(310,415)(311,416)(312,417)(313,418)
(314,419)(315,420);
s1 := Sym(420)!(  1,211)(  2,217)(  3,216)(  4,215)(  5,214)(  6,213)(  7,212)
(  8,239)(  9,245)( 10,244)( 11,243)( 12,242)( 13,241)( 14,240)( 15,232)
( 16,238)( 17,237)( 18,236)( 19,235)( 20,234)( 21,233)( 22,225)( 23,231)
( 24,230)( 25,229)( 26,228)( 27,227)( 28,226)( 29,218)( 30,224)( 31,223)
( 32,222)( 33,221)( 34,220)( 35,219)( 36,281)( 37,287)( 38,286)( 39,285)
( 40,284)( 41,283)( 42,282)( 43,309)( 44,315)( 45,314)( 46,313)( 47,312)
( 48,311)( 49,310)( 50,302)( 51,308)( 52,307)( 53,306)( 54,305)( 55,304)
( 56,303)( 57,295)( 58,301)( 59,300)( 60,299)( 61,298)( 62,297)( 63,296)
( 64,288)( 65,294)( 66,293)( 67,292)( 68,291)( 69,290)( 70,289)( 71,246)
( 72,252)( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,274)( 79,280)
( 80,279)( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)
( 88,271)( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)
( 96,263)( 97,262)( 98,261)( 99,253)(100,259)(101,258)(102,257)(103,256)
(104,255)(105,254)(106,316)(107,322)(108,321)(109,320)(110,319)(111,318)
(112,317)(113,344)(114,350)(115,349)(116,348)(117,347)(118,346)(119,345)
(120,337)(121,343)(122,342)(123,341)(124,340)(125,339)(126,338)(127,330)
(128,336)(129,335)(130,334)(131,333)(132,332)(133,331)(134,323)(135,329)
(136,328)(137,327)(138,326)(139,325)(140,324)(141,386)(142,392)(143,391)
(144,390)(145,389)(146,388)(147,387)(148,414)(149,420)(150,419)(151,418)
(152,417)(153,416)(154,415)(155,407)(156,413)(157,412)(158,411)(159,410)
(160,409)(161,408)(162,400)(163,406)(164,405)(165,404)(166,403)(167,402)
(168,401)(169,393)(170,399)(171,398)(172,397)(173,396)(174,395)(175,394)
(176,351)(177,357)(178,356)(179,355)(180,354)(181,353)(182,352)(183,379)
(184,385)(185,384)(186,383)(187,382)(188,381)(189,380)(190,372)(191,378)
(192,377)(193,376)(194,375)(195,374)(196,373)(197,365)(198,371)(199,370)
(200,369)(201,368)(202,367)(203,366)(204,358)(205,364)(206,363)(207,362)
(208,361)(209,360)(210,359);
s2 := Sym(420)!(  1, 44)(  2, 43)(  3, 49)(  4, 48)(  5, 47)(  6, 46)(  7, 45)
(  8, 37)(  9, 36)( 10, 42)( 11, 41)( 12, 40)( 13, 39)( 14, 38)( 15, 65)
( 16, 64)( 17, 70)( 18, 69)( 19, 68)( 20, 67)( 21, 66)( 22, 58)( 23, 57)
( 24, 63)( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 51)( 30, 50)( 31, 56)
( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 71, 79)( 72, 78)( 73, 84)( 74, 83)
( 75, 82)( 76, 81)( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)
( 90,102)( 91,101)( 92, 93)( 94, 98)( 95, 97)(106,149)(107,148)(108,154)
(109,153)(110,152)(111,151)(112,150)(113,142)(114,141)(115,147)(116,146)
(117,145)(118,144)(119,143)(120,170)(121,169)(122,175)(123,174)(124,173)
(125,172)(126,171)(127,163)(128,162)(129,168)(130,167)(131,166)(132,165)
(133,164)(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)
(176,184)(177,183)(178,189)(179,188)(180,187)(181,186)(182,185)(190,205)
(191,204)(192,210)(193,209)(194,208)(195,207)(196,206)(197,198)(199,203)
(200,202)(211,254)(212,253)(213,259)(214,258)(215,257)(216,256)(217,255)
(218,247)(219,246)(220,252)(221,251)(222,250)(223,249)(224,248)(225,275)
(226,274)(227,280)(228,279)(229,278)(230,277)(231,276)(232,268)(233,267)
(234,273)(235,272)(236,271)(237,270)(238,269)(239,261)(240,260)(241,266)
(242,265)(243,264)(244,263)(245,262)(281,289)(282,288)(283,294)(284,293)
(285,292)(286,291)(287,290)(295,310)(296,309)(297,315)(298,314)(299,313)
(300,312)(301,311)(302,303)(304,308)(305,307)(316,359)(317,358)(318,364)
(319,363)(320,362)(321,361)(322,360)(323,352)(324,351)(325,357)(326,356)
(327,355)(328,354)(329,353)(330,380)(331,379)(332,385)(333,384)(334,383)
(335,382)(336,381)(337,373)(338,372)(339,378)(340,377)(341,376)(342,375)
(343,374)(344,366)(345,365)(346,371)(347,370)(348,369)(349,368)(350,367)
(386,394)(387,393)(388,399)(389,398)(390,397)(391,396)(392,395)(400,415)
(401,414)(402,420)(403,419)(404,418)(405,417)(406,416)(407,408)(409,413)
(410,412);
poly := sub<Sym(420)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope