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Polytope of Type {12,70}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,70}*1680
Also Known As : {12,70|2}. if this polytope has another name.
Group : SmallGroup(1680,797)
Rank : 3
Schlafli Type : {12,70}
Number of vertices, edges, etc : 12, 420, 70
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 : {6,70}*840
   3-fold quotients : {4,70}*560
   5-fold quotients : {12,14}*336
   6-fold quotients : {2,70}*280
   7-fold quotients : {12,10}*240
   10-fold quotients : {6,14}*168
   12-fold quotients : {2,35}*140
   14-fold quotients : {6,10}*120
   15-fold quotients : {4,14}*112
   21-fold quotients : {4,10}*80
   30-fold quotients : {2,14}*56
   35-fold quotients : {12,2}*48
   42-fold quotients : {2,10}*40
   60-fold quotients : {2,7}*28
   70-fold quotients : {6,2}*24
   84-fold quotients : {2,5}*20
   105-fold quotients : {4,2}*16
   140-fold quotients : {3,2}*12
   210-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)( 41, 76)( 42, 77)( 43, 78)
( 44, 79)( 45, 80)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)( 51, 86)
( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91)( 57, 92)( 58, 93)( 59, 94)
( 60, 95)( 61, 96)( 62, 97)( 63, 98)( 64, 99)( 65,100)( 66,101)( 67,102)
( 68,103)( 69,104)( 70,105)(141,176)(142,177)(143,178)(144,179)(145,180)
(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)
(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)
(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)(169,204)
(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(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)(246,386)(247,387)(248,388)(249,389)(250,390)(251,391)(252,392)
(253,393)(254,394)(255,395)(256,396)(257,397)(258,398)(259,399)(260,400)
(261,401)(262,402)(263,403)(264,404)(265,405)(266,406)(267,407)(268,408)
(269,409)(270,410)(271,411)(272,412)(273,413)(274,414)(275,415)(276,416)
(277,417)(278,418)(279,419)(280,420)(281,351)(282,352)(283,353)(284,354)
(285,355)(286,356)(287,357)(288,358)(289,359)(290,360)(291,361)(292,362)
(293,363)(294,364)(295,365)(296,366)(297,367)(298,368)(299,369)(300,370)
(301,371)(302,372)(303,373)(304,374)(305,375)(306,376)(307,377)(308,378)
(309,379)(310,380)(311,381)(312,382)(313,383)(314,384)(315,385);;
s1 := (  1,246)(  2,252)(  3,251)(  4,250)(  5,249)(  6,248)(  7,247)(  8,274)
(  9,280)( 10,279)( 11,278)( 12,277)( 13,276)( 14,275)( 15,267)( 16,273)
( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,260)( 23,266)( 24,265)
( 25,264)( 26,263)( 27,262)( 28,261)( 29,253)( 30,259)( 31,258)( 32,257)
( 33,256)( 34,255)( 35,254)( 36,211)( 37,217)( 38,216)( 39,215)( 40,214)
( 41,213)( 42,212)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)( 56,233)
( 57,225)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)( 64,218)
( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,281)( 72,287)
( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,309)( 79,315)( 80,314)
( 81,313)( 82,312)( 83,311)( 84,310)( 85,302)( 86,308)( 87,307)( 88,306)
( 89,305)( 90,304)( 91,303)( 92,295)( 93,301)( 94,300)( 95,299)( 96,298)
( 97,297)( 98,296)( 99,288)(100,294)(101,293)(102,292)(103,291)(104,290)
(105,289)(106,351)(107,357)(108,356)(109,355)(110,354)(111,353)(112,352)
(113,379)(114,385)(115,384)(116,383)(117,382)(118,381)(119,380)(120,372)
(121,378)(122,377)(123,376)(124,375)(125,374)(126,373)(127,365)(128,371)
(129,370)(130,369)(131,368)(132,367)(133,366)(134,358)(135,364)(136,363)
(137,362)(138,361)(139,360)(140,359)(141,316)(142,322)(143,321)(144,320)
(145,319)(146,318)(147,317)(148,344)(149,350)(150,349)(151,348)(152,347)
(153,346)(154,345)(155,337)(156,343)(157,342)(158,341)(159,340)(160,339)
(161,338)(162,330)(163,336)(164,335)(165,334)(166,333)(167,332)(168,331)
(169,323)(170,329)(171,328)(172,327)(173,326)(174,325)(175,324)(176,386)
(177,392)(178,391)(179,390)(180,389)(181,388)(182,387)(183,414)(184,420)
(185,419)(186,418)(187,417)(188,416)(189,415)(190,407)(191,413)(192,412)
(193,411)(194,410)(195,409)(196,408)(197,400)(198,406)(199,405)(200,404)
(201,403)(202,402)(203,401)(204,393)(205,399)(206,398)(207,397)(208,396)
(209,395)(210,394);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)( 57, 58)
( 59, 63)( 60, 62)( 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)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)(125,137)
(126,136)(127,128)(129,133)(130,132)(141,149)(142,148)(143,154)(144,153)
(145,152)(146,151)(147,150)(155,170)(156,169)(157,175)(158,174)(159,173)
(160,172)(161,171)(162,163)(164,168)(165,167)(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)(211,219)(212,218)
(213,224)(214,223)(215,222)(216,221)(217,220)(225,240)(226,239)(227,245)
(228,244)(229,243)(230,242)(231,241)(232,233)(234,238)(235,237)(246,254)
(247,253)(248,259)(249,258)(250,257)(251,256)(252,255)(260,275)(261,274)
(262,280)(263,279)(264,278)(265,277)(266,276)(267,268)(269,273)(270,272)
(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)(316,324)(317,323)(318,329)(319,328)(320,327)(321,326)(322,325)
(330,345)(331,344)(332,350)(333,349)(334,348)(335,347)(336,346)(337,338)
(339,343)(340,342)(351,359)(352,358)(353,364)(354,363)(355,362)(356,361)
(357,360)(365,380)(366,379)(367,385)(368,384)(369,383)(370,382)(371,381)
(372,373)(374,378)(375,377)(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*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)( 41, 76)( 42, 77)
( 43, 78)( 44, 79)( 45, 80)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)
( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91)( 57, 92)( 58, 93)
( 59, 94)( 60, 95)( 61, 96)( 62, 97)( 63, 98)( 64, 99)( 65,100)( 66,101)
( 67,102)( 68,103)( 69,104)( 70,105)(141,176)(142,177)(143,178)(144,179)
(145,180)(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)
(153,188)(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)
(161,196)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)
(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(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)(246,386)(247,387)(248,388)(249,389)(250,390)(251,391)
(252,392)(253,393)(254,394)(255,395)(256,396)(257,397)(258,398)(259,399)
(260,400)(261,401)(262,402)(263,403)(264,404)(265,405)(266,406)(267,407)
(268,408)(269,409)(270,410)(271,411)(272,412)(273,413)(274,414)(275,415)
(276,416)(277,417)(278,418)(279,419)(280,420)(281,351)(282,352)(283,353)
(284,354)(285,355)(286,356)(287,357)(288,358)(289,359)(290,360)(291,361)
(292,362)(293,363)(294,364)(295,365)(296,366)(297,367)(298,368)(299,369)
(300,370)(301,371)(302,372)(303,373)(304,374)(305,375)(306,376)(307,377)
(308,378)(309,379)(310,380)(311,381)(312,382)(313,383)(314,384)(315,385);
s1 := Sym(420)!(  1,246)(  2,252)(  3,251)(  4,250)(  5,249)(  6,248)(  7,247)
(  8,274)(  9,280)( 10,279)( 11,278)( 12,277)( 13,276)( 14,275)( 15,267)
( 16,273)( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,260)( 23,266)
( 24,265)( 25,264)( 26,263)( 27,262)( 28,261)( 29,253)( 30,259)( 31,258)
( 32,257)( 33,256)( 34,255)( 35,254)( 36,211)( 37,217)( 38,216)( 39,215)
( 40,214)( 41,213)( 42,212)( 43,239)( 44,245)( 45,244)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,232)( 51,238)( 52,237)( 53,236)( 54,235)( 55,234)
( 56,233)( 57,225)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)
( 64,218)( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,281)
( 72,287)( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,309)( 79,315)
( 80,314)( 81,313)( 82,312)( 83,311)( 84,310)( 85,302)( 86,308)( 87,307)
( 88,306)( 89,305)( 90,304)( 91,303)( 92,295)( 93,301)( 94,300)( 95,299)
( 96,298)( 97,297)( 98,296)( 99,288)(100,294)(101,293)(102,292)(103,291)
(104,290)(105,289)(106,351)(107,357)(108,356)(109,355)(110,354)(111,353)
(112,352)(113,379)(114,385)(115,384)(116,383)(117,382)(118,381)(119,380)
(120,372)(121,378)(122,377)(123,376)(124,375)(125,374)(126,373)(127,365)
(128,371)(129,370)(130,369)(131,368)(132,367)(133,366)(134,358)(135,364)
(136,363)(137,362)(138,361)(139,360)(140,359)(141,316)(142,322)(143,321)
(144,320)(145,319)(146,318)(147,317)(148,344)(149,350)(150,349)(151,348)
(152,347)(153,346)(154,345)(155,337)(156,343)(157,342)(158,341)(159,340)
(160,339)(161,338)(162,330)(163,336)(164,335)(165,334)(166,333)(167,332)
(168,331)(169,323)(170,329)(171,328)(172,327)(173,326)(174,325)(175,324)
(176,386)(177,392)(178,391)(179,390)(180,389)(181,388)(182,387)(183,414)
(184,420)(185,419)(186,418)(187,417)(188,416)(189,415)(190,407)(191,413)
(192,412)(193,411)(194,410)(195,409)(196,408)(197,400)(198,406)(199,405)
(200,404)(201,403)(202,402)(203,401)(204,393)(205,399)(206,398)(207,397)
(208,396)(209,395)(210,394);
s2 := Sym(420)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)
( 42, 45)( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 57, 58)( 59, 63)( 60, 62)( 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)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,128)(129,133)(130,132)(141,149)(142,148)(143,154)
(144,153)(145,152)(146,151)(147,150)(155,170)(156,169)(157,175)(158,174)
(159,173)(160,172)(161,171)(162,163)(164,168)(165,167)(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)(211,219)
(212,218)(213,224)(214,223)(215,222)(216,221)(217,220)(225,240)(226,239)
(227,245)(228,244)(229,243)(230,242)(231,241)(232,233)(234,238)(235,237)
(246,254)(247,253)(248,259)(249,258)(250,257)(251,256)(252,255)(260,275)
(261,274)(262,280)(263,279)(264,278)(265,277)(266,276)(267,268)(269,273)
(270,272)(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)(316,324)(317,323)(318,329)(319,328)(320,327)(321,326)
(322,325)(330,345)(331,344)(332,350)(333,349)(334,348)(335,347)(336,346)
(337,338)(339,343)(340,342)(351,359)(352,358)(353,364)(354,363)(355,362)
(356,361)(357,360)(365,380)(366,379)(367,385)(368,384)(369,383)(370,382)
(371,381)(372,373)(374,378)(375,377)(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*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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