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Polytope of Type {20,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,42}*1680a
Also Known As : {20,42|2}. if this polytope has another name.
Group : SmallGroup(1680,770)
Rank : 3
Schlafli Type : {20,42}
Number of vertices, edges, etc : 20, 420, 42
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 : {10,42}*840
   3-fold quotients : {20,14}*560
   5-fold quotients : {4,42}*336a
   6-fold quotients : {10,14}*280
   7-fold quotients : {20,6}*240a
   10-fold quotients : {2,42}*168
   14-fold quotients : {10,6}*120
   15-fold quotients : {4,14}*112
   20-fold quotients : {2,21}*84
   21-fold quotients : {20,2}*80
   30-fold quotients : {2,14}*56
   35-fold quotients : {4,6}*48a
   42-fold quotients : {10,2}*40
   60-fold quotients : {2,7}*28
   70-fold quotients : {2,6}*24
   84-fold quotients : {5,2}*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 := (  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 22)
( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 64)( 44, 65)
( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 57)( 51, 58)( 52, 59)
( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 78, 99)( 79,100)( 80,101)( 81,102)
( 82,103)( 83,104)( 84,105)( 85, 92)( 86, 93)( 87, 94)( 88, 95)( 89, 96)
( 90, 97)( 91, 98)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)
(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)
(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,162)
(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,204)(184,205)
(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)
(193,200)(194,201)(195,202)(196,203)(211,316)(212,317)(213,318)(214,319)
(215,320)(216,321)(217,322)(218,344)(219,345)(220,346)(221,347)(222,348)
(223,349)(224,350)(225,337)(226,338)(227,339)(228,340)(229,341)(230,342)
(231,343)(232,330)(233,331)(234,332)(235,333)(236,334)(237,335)(238,336)
(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,351)
(247,352)(248,353)(249,354)(250,355)(251,356)(252,357)(253,379)(254,380)
(255,381)(256,382)(257,383)(258,384)(259,385)(260,372)(261,373)(262,374)
(263,375)(264,376)(265,377)(266,378)(267,365)(268,366)(269,367)(270,368)
(271,369)(272,370)(273,371)(274,358)(275,359)(276,360)(277,361)(278,362)
(279,363)(280,364)(281,386)(282,387)(283,388)(284,389)(285,390)(286,391)
(287,392)(288,414)(289,415)(290,416)(291,417)(292,418)(293,419)(294,420)
(295,407)(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)(302,400)
(303,401)(304,402)(305,403)(306,404)(307,405)(308,406)(309,393)(310,394)
(311,395)(312,396)(313,397)(314,398)(315,399);;
s1 := (  1,218)(  2,224)(  3,223)(  4,222)(  5,221)(  6,220)(  7,219)(  8,211)
(  9,217)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,239)( 16,245)
( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,232)( 23,238)( 24,237)
( 25,236)( 26,235)( 27,234)( 28,233)( 29,225)( 30,231)( 31,230)( 32,229)
( 33,228)( 34,227)( 35,226)( 36,288)( 37,294)( 38,293)( 39,292)( 40,291)
( 41,290)( 42,289)( 43,281)( 44,287)( 45,286)( 46,285)( 47,284)( 48,283)
( 49,282)( 50,309)( 51,315)( 52,314)( 53,313)( 54,312)( 55,311)( 56,310)
( 57,302)( 58,308)( 59,307)( 60,306)( 61,305)( 62,304)( 63,303)( 64,295)
( 65,301)( 66,300)( 67,299)( 68,298)( 69,297)( 70,296)( 71,253)( 72,259)
( 73,258)( 74,257)( 75,256)( 76,255)( 77,254)( 78,246)( 79,252)( 80,251)
( 81,250)( 82,249)( 83,248)( 84,247)( 85,274)( 86,280)( 87,279)( 88,278)
( 89,277)( 90,276)( 91,275)( 92,267)( 93,273)( 94,272)( 95,271)( 96,270)
( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)(104,262)
(105,261)(106,323)(107,329)(108,328)(109,327)(110,326)(111,325)(112,324)
(113,316)(114,322)(115,321)(116,320)(117,319)(118,318)(119,317)(120,344)
(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,337)(128,343)
(129,342)(130,341)(131,340)(132,339)(133,338)(134,330)(135,336)(136,335)
(137,334)(138,333)(139,332)(140,331)(141,393)(142,399)(143,398)(144,397)
(145,396)(146,395)(147,394)(148,386)(149,392)(150,391)(151,390)(152,389)
(153,388)(154,387)(155,414)(156,420)(157,419)(158,418)(159,417)(160,416)
(161,415)(162,407)(163,413)(164,412)(165,411)(166,410)(167,409)(168,408)
(169,400)(170,406)(171,405)(172,404)(173,403)(174,402)(175,401)(176,358)
(177,364)(178,363)(179,362)(180,361)(181,360)(182,359)(183,351)(184,357)
(185,356)(186,355)(187,354)(188,353)(189,352)(190,379)(191,385)(192,384)
(193,383)(194,382)(195,381)(196,380)(197,372)(198,378)(199,377)(200,376)
(201,375)(202,374)(203,373)(204,365)(205,371)(206,370)(207,369)(208,368)
(209,367)(210,366);;
s2 := (  1, 37)(  2, 36)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)(  8, 44)
(  9, 43)( 10, 49)( 11, 48)( 12, 47)( 13, 46)( 14, 45)( 15, 51)( 16, 50)
( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 58)( 23, 57)( 24, 63)
( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 65)( 30, 64)( 31, 70)( 32, 69)
( 33, 68)( 34, 67)( 35, 66)( 71, 72)( 73, 77)( 74, 76)( 78, 79)( 80, 84)
( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)( 99,100)
(101,105)(102,104)(106,142)(107,141)(108,147)(109,146)(110,145)(111,144)
(112,143)(113,149)(114,148)(115,154)(116,153)(117,152)(118,151)(119,150)
(120,156)(121,155)(122,161)(123,160)(124,159)(125,158)(126,157)(127,163)
(128,162)(129,168)(130,167)(131,166)(132,165)(133,164)(134,170)(135,169)
(136,175)(137,174)(138,173)(139,172)(140,171)(176,177)(178,182)(179,181)
(183,184)(185,189)(186,188)(190,191)(192,196)(193,195)(197,198)(199,203)
(200,202)(204,205)(206,210)(207,209)(211,247)(212,246)(213,252)(214,251)
(215,250)(216,249)(217,248)(218,254)(219,253)(220,259)(221,258)(222,257)
(223,256)(224,255)(225,261)(226,260)(227,266)(228,265)(229,264)(230,263)
(231,262)(232,268)(233,267)(234,273)(235,272)(236,271)(237,270)(238,269)
(239,275)(240,274)(241,280)(242,279)(243,278)(244,277)(245,276)(281,282)
(283,287)(284,286)(288,289)(290,294)(291,293)(295,296)(297,301)(298,300)
(302,303)(304,308)(305,307)(309,310)(311,315)(312,314)(316,352)(317,351)
(318,357)(319,356)(320,355)(321,354)(322,353)(323,359)(324,358)(325,364)
(326,363)(327,362)(328,361)(329,360)(330,366)(331,365)(332,371)(333,370)
(334,369)(335,368)(336,367)(337,373)(338,372)(339,378)(340,377)(341,376)
(342,375)(343,374)(344,380)(345,379)(346,385)(347,384)(348,383)(349,382)
(350,381)(386,387)(388,392)(389,391)(393,394)(395,399)(396,398)(400,401)
(402,406)(403,405)(407,408)(409,413)(410,412)(414,415)(416,420)(417,419);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(420)!(  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)
( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 64)
( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 78, 99)( 79,100)( 80,101)
( 81,102)( 82,103)( 83,104)( 84,105)( 85, 92)( 86, 93)( 87, 94)( 88, 95)
( 89, 96)( 90, 97)( 91, 98)(113,134)(114,135)(115,136)(116,137)(117,138)
(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)
(126,133)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)
(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,204)
(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)
(192,199)(193,200)(194,201)(195,202)(196,203)(211,316)(212,317)(213,318)
(214,319)(215,320)(216,321)(217,322)(218,344)(219,345)(220,346)(221,347)
(222,348)(223,349)(224,350)(225,337)(226,338)(227,339)(228,340)(229,341)
(230,342)(231,343)(232,330)(233,331)(234,332)(235,333)(236,334)(237,335)
(238,336)(239,323)(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)
(246,351)(247,352)(248,353)(249,354)(250,355)(251,356)(252,357)(253,379)
(254,380)(255,381)(256,382)(257,383)(258,384)(259,385)(260,372)(261,373)
(262,374)(263,375)(264,376)(265,377)(266,378)(267,365)(268,366)(269,367)
(270,368)(271,369)(272,370)(273,371)(274,358)(275,359)(276,360)(277,361)
(278,362)(279,363)(280,364)(281,386)(282,387)(283,388)(284,389)(285,390)
(286,391)(287,392)(288,414)(289,415)(290,416)(291,417)(292,418)(293,419)
(294,420)(295,407)(296,408)(297,409)(298,410)(299,411)(300,412)(301,413)
(302,400)(303,401)(304,402)(305,403)(306,404)(307,405)(308,406)(309,393)
(310,394)(311,395)(312,396)(313,397)(314,398)(315,399);
s1 := Sym(420)!(  1,218)(  2,224)(  3,223)(  4,222)(  5,221)(  6,220)(  7,219)
(  8,211)(  9,217)( 10,216)( 11,215)( 12,214)( 13,213)( 14,212)( 15,239)
( 16,245)( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,232)( 23,238)
( 24,237)( 25,236)( 26,235)( 27,234)( 28,233)( 29,225)( 30,231)( 31,230)
( 32,229)( 33,228)( 34,227)( 35,226)( 36,288)( 37,294)( 38,293)( 39,292)
( 40,291)( 41,290)( 42,289)( 43,281)( 44,287)( 45,286)( 46,285)( 47,284)
( 48,283)( 49,282)( 50,309)( 51,315)( 52,314)( 53,313)( 54,312)( 55,311)
( 56,310)( 57,302)( 58,308)( 59,307)( 60,306)( 61,305)( 62,304)( 63,303)
( 64,295)( 65,301)( 66,300)( 67,299)( 68,298)( 69,297)( 70,296)( 71,253)
( 72,259)( 73,258)( 74,257)( 75,256)( 76,255)( 77,254)( 78,246)( 79,252)
( 80,251)( 81,250)( 82,249)( 83,248)( 84,247)( 85,274)( 86,280)( 87,279)
( 88,278)( 89,277)( 90,276)( 91,275)( 92,267)( 93,273)( 94,272)( 95,271)
( 96,270)( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)
(104,262)(105,261)(106,323)(107,329)(108,328)(109,327)(110,326)(111,325)
(112,324)(113,316)(114,322)(115,321)(116,320)(117,319)(118,318)(119,317)
(120,344)(121,350)(122,349)(123,348)(124,347)(125,346)(126,345)(127,337)
(128,343)(129,342)(130,341)(131,340)(132,339)(133,338)(134,330)(135,336)
(136,335)(137,334)(138,333)(139,332)(140,331)(141,393)(142,399)(143,398)
(144,397)(145,396)(146,395)(147,394)(148,386)(149,392)(150,391)(151,390)
(152,389)(153,388)(154,387)(155,414)(156,420)(157,419)(158,418)(159,417)
(160,416)(161,415)(162,407)(163,413)(164,412)(165,411)(166,410)(167,409)
(168,408)(169,400)(170,406)(171,405)(172,404)(173,403)(174,402)(175,401)
(176,358)(177,364)(178,363)(179,362)(180,361)(181,360)(182,359)(183,351)
(184,357)(185,356)(186,355)(187,354)(188,353)(189,352)(190,379)(191,385)
(192,384)(193,383)(194,382)(195,381)(196,380)(197,372)(198,378)(199,377)
(200,376)(201,375)(202,374)(203,373)(204,365)(205,371)(206,370)(207,369)
(208,368)(209,367)(210,366);
s2 := Sym(420)!(  1, 37)(  2, 36)(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 38)
(  8, 44)(  9, 43)( 10, 49)( 11, 48)( 12, 47)( 13, 46)( 14, 45)( 15, 51)
( 16, 50)( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 52)( 22, 58)( 23, 57)
( 24, 63)( 25, 62)( 26, 61)( 27, 60)( 28, 59)( 29, 65)( 30, 64)( 31, 70)
( 32, 69)( 33, 68)( 34, 67)( 35, 66)( 71, 72)( 73, 77)( 74, 76)( 78, 79)
( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)( 94, 98)( 95, 97)
( 99,100)(101,105)(102,104)(106,142)(107,141)(108,147)(109,146)(110,145)
(111,144)(112,143)(113,149)(114,148)(115,154)(116,153)(117,152)(118,151)
(119,150)(120,156)(121,155)(122,161)(123,160)(124,159)(125,158)(126,157)
(127,163)(128,162)(129,168)(130,167)(131,166)(132,165)(133,164)(134,170)
(135,169)(136,175)(137,174)(138,173)(139,172)(140,171)(176,177)(178,182)
(179,181)(183,184)(185,189)(186,188)(190,191)(192,196)(193,195)(197,198)
(199,203)(200,202)(204,205)(206,210)(207,209)(211,247)(212,246)(213,252)
(214,251)(215,250)(216,249)(217,248)(218,254)(219,253)(220,259)(221,258)
(222,257)(223,256)(224,255)(225,261)(226,260)(227,266)(228,265)(229,264)
(230,263)(231,262)(232,268)(233,267)(234,273)(235,272)(236,271)(237,270)
(238,269)(239,275)(240,274)(241,280)(242,279)(243,278)(244,277)(245,276)
(281,282)(283,287)(284,286)(288,289)(290,294)(291,293)(295,296)(297,301)
(298,300)(302,303)(304,308)(305,307)(309,310)(311,315)(312,314)(316,352)
(317,351)(318,357)(319,356)(320,355)(321,354)(322,353)(323,359)(324,358)
(325,364)(326,363)(327,362)(328,361)(329,360)(330,366)(331,365)(332,371)
(333,370)(334,369)(335,368)(336,367)(337,373)(338,372)(339,378)(340,377)
(341,376)(342,375)(343,374)(344,380)(345,379)(346,385)(347,384)(348,383)
(349,382)(350,381)(386,387)(388,392)(389,391)(393,394)(395,399)(396,398)
(400,401)(402,406)(403,405)(407,408)(409,413)(410,412)(414,415)(416,420)
(417,419);
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >; 
 
References : None.
to this polytope