Polytope of Type {140,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {140,6}*1680a
Also Known As : {140,6|2}. if this polytope has another name.
Group : SmallGroup(1680,798)
Rank : 3
Schlafli Type : {140,6}
Number of vertices, edges, etc : 140, 420, 6
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 : {70,6}*840
   3-fold quotients : {140,2}*560
   5-fold quotients : {28,6}*336a
   6-fold quotients : {70,2}*280
   7-fold quotients : {20,6}*240a
   10-fold quotients : {14,6}*168
   12-fold quotients : {35,2}*140
   14-fold quotients : {10,6}*120
   15-fold quotients : {28,2}*112
   21-fold quotients : {20,2}*80
   30-fold quotients : {14,2}*56
   35-fold quotients : {4,6}*48a
   42-fold quotients : {10,2}*40
   60-fold quotients : {7,2}*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 := (  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)( 12, 32)
( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)
( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)
( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)(114,140)
(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)(122,132)
(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)(148,169)
(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)(156,168)
(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)(179,180)
(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,197)
(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(211,316)(212,322)
(213,321)(214,320)(215,319)(216,318)(217,317)(218,344)(219,350)(220,349)
(221,348)(222,347)(223,346)(224,345)(225,337)(226,343)(227,342)(228,341)
(229,340)(230,339)(231,338)(232,330)(233,336)(234,335)(235,334)(236,333)
(237,332)(238,331)(239,323)(240,329)(241,328)(242,327)(243,326)(244,325)
(245,324)(246,351)(247,357)(248,356)(249,355)(250,354)(251,353)(252,352)
(253,379)(254,385)(255,384)(256,383)(257,382)(258,381)(259,380)(260,372)
(261,378)(262,377)(263,376)(264,375)(265,374)(266,373)(267,365)(268,371)
(269,370)(270,369)(271,368)(272,367)(273,366)(274,358)(275,364)(276,363)
(277,362)(278,361)(279,360)(280,359)(281,386)(282,392)(283,391)(284,390)
(285,389)(286,388)(287,387)(288,414)(289,420)(290,419)(291,418)(292,417)
(293,416)(294,415)(295,407)(296,413)(297,412)(298,411)(299,410)(300,409)
(301,408)(302,400)(303,406)(304,405)(305,404)(306,403)(307,402)(308,401)
(309,393)(310,399)(311,398)(312,397)(313,396)(314,395)(315,394);;
s1 := (  1,219)(  2,218)(  3,224)(  4,223)(  5,222)(  6,221)(  7,220)(  8,212)
(  9,211)( 10,217)( 11,216)( 12,215)( 13,214)( 14,213)( 15,240)( 16,239)
( 17,245)( 18,244)( 19,243)( 20,242)( 21,241)( 22,233)( 23,232)( 24,238)
( 25,237)( 26,236)( 27,235)( 28,234)( 29,226)( 30,225)( 31,231)( 32,230)
( 33,229)( 34,228)( 35,227)( 36,289)( 37,288)( 38,294)( 39,293)( 40,292)
( 41,291)( 42,290)( 43,282)( 44,281)( 45,287)( 46,286)( 47,285)( 48,284)
( 49,283)( 50,310)( 51,309)( 52,315)( 53,314)( 54,313)( 55,312)( 56,311)
( 57,303)( 58,302)( 59,308)( 60,307)( 61,306)( 62,305)( 63,304)( 64,296)
( 65,295)( 66,301)( 67,300)( 68,299)( 69,298)( 70,297)( 71,254)( 72,253)
( 73,259)( 74,258)( 75,257)( 76,256)( 77,255)( 78,247)( 79,246)( 80,252)
( 81,251)( 82,250)( 83,249)( 84,248)( 85,275)( 86,274)( 87,280)( 88,279)
( 89,278)( 90,277)( 91,276)( 92,268)( 93,267)( 94,273)( 95,272)( 96,271)
( 97,270)( 98,269)( 99,261)(100,260)(101,266)(102,265)(103,264)(104,263)
(105,262)(106,324)(107,323)(108,329)(109,328)(110,327)(111,326)(112,325)
(113,317)(114,316)(115,322)(116,321)(117,320)(118,319)(119,318)(120,345)
(121,344)(122,350)(123,349)(124,348)(125,347)(126,346)(127,338)(128,337)
(129,343)(130,342)(131,341)(132,340)(133,339)(134,331)(135,330)(136,336)
(137,335)(138,334)(139,333)(140,332)(141,394)(142,393)(143,399)(144,398)
(145,397)(146,396)(147,395)(148,387)(149,386)(150,392)(151,391)(152,390)
(153,389)(154,388)(155,415)(156,414)(157,420)(158,419)(159,418)(160,417)
(161,416)(162,408)(163,407)(164,413)(165,412)(166,411)(167,410)(168,409)
(169,401)(170,400)(171,406)(172,405)(173,404)(174,403)(175,402)(176,359)
(177,358)(178,364)(179,363)(180,362)(181,361)(182,360)(183,352)(184,351)
(185,357)(186,356)(187,355)(188,354)(189,353)(190,380)(191,379)(192,385)
(193,384)(194,383)(195,382)(196,381)(197,373)(198,372)(199,378)(200,377)
(201,376)(202,375)(203,374)(204,366)(205,365)(206,371)(207,370)(208,369)
(209,368)(210,367);;
s2 := (  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 41)(  7, 42)(  8, 43)
(  9, 44)( 10, 45)( 11, 46)( 12, 47)( 13, 48)( 14, 49)( 15, 50)( 16, 51)
( 17, 52)( 18, 53)( 19, 54)( 20, 55)( 21, 56)( 22, 57)( 23, 58)( 24, 59)
( 25, 60)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 66)( 32, 67)
( 33, 68)( 34, 69)( 35, 70)(106,141)(107,142)(108,143)(109,144)(110,145)
(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)
(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)
(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)
(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(211,246)(212,247)
(213,248)(214,249)(215,250)(216,251)(217,252)(218,253)(219,254)(220,255)
(221,256)(222,257)(223,258)(224,259)(225,260)(226,261)(227,262)(228,263)
(229,264)(230,265)(231,266)(232,267)(233,268)(234,269)(235,270)(236,271)
(237,272)(238,273)(239,274)(240,275)(241,276)(242,277)(243,278)(244,279)
(245,280)(316,351)(317,352)(318,353)(319,354)(320,355)(321,356)(322,357)
(323,358)(324,359)(325,360)(326,361)(327,362)(328,363)(329,364)(330,365)
(331,366)(332,367)(333,368)(334,369)(335,370)(336,371)(337,372)(338,373)
(339,374)(340,375)(341,376)(342,377)(343,378)(344,379)(345,380)(346,381)
(347,382)(348,383)(349,384)(350,385);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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*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*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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(420)!(  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)
( 12, 32)( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)
( 20, 24)( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)
( 54, 60)( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)
( 80,104)( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)
( 88, 96)( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)
(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)
(122,132)(123,131)(124,130)(125,129)(126,128)(142,147)(143,146)(144,145)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,162)
(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(177,182)(178,181)
(179,180)(183,204)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)
(190,197)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)(211,316)
(212,322)(213,321)(214,320)(215,319)(216,318)(217,317)(218,344)(219,350)
(220,349)(221,348)(222,347)(223,346)(224,345)(225,337)(226,343)(227,342)
(228,341)(229,340)(230,339)(231,338)(232,330)(233,336)(234,335)(235,334)
(236,333)(237,332)(238,331)(239,323)(240,329)(241,328)(242,327)(243,326)
(244,325)(245,324)(246,351)(247,357)(248,356)(249,355)(250,354)(251,353)
(252,352)(253,379)(254,385)(255,384)(256,383)(257,382)(258,381)(259,380)
(260,372)(261,378)(262,377)(263,376)(264,375)(265,374)(266,373)(267,365)
(268,371)(269,370)(270,369)(271,368)(272,367)(273,366)(274,358)(275,364)
(276,363)(277,362)(278,361)(279,360)(280,359)(281,386)(282,392)(283,391)
(284,390)(285,389)(286,388)(287,387)(288,414)(289,420)(290,419)(291,418)
(292,417)(293,416)(294,415)(295,407)(296,413)(297,412)(298,411)(299,410)
(300,409)(301,408)(302,400)(303,406)(304,405)(305,404)(306,403)(307,402)
(308,401)(309,393)(310,399)(311,398)(312,397)(313,396)(314,395)(315,394);
s1 := Sym(420)!(  1,219)(  2,218)(  3,224)(  4,223)(  5,222)(  6,221)(  7,220)
(  8,212)(  9,211)( 10,217)( 11,216)( 12,215)( 13,214)( 14,213)( 15,240)
( 16,239)( 17,245)( 18,244)( 19,243)( 20,242)( 21,241)( 22,233)( 23,232)
( 24,238)( 25,237)( 26,236)( 27,235)( 28,234)( 29,226)( 30,225)( 31,231)
( 32,230)( 33,229)( 34,228)( 35,227)( 36,289)( 37,288)( 38,294)( 39,293)
( 40,292)( 41,291)( 42,290)( 43,282)( 44,281)( 45,287)( 46,286)( 47,285)
( 48,284)( 49,283)( 50,310)( 51,309)( 52,315)( 53,314)( 54,313)( 55,312)
( 56,311)( 57,303)( 58,302)( 59,308)( 60,307)( 61,306)( 62,305)( 63,304)
( 64,296)( 65,295)( 66,301)( 67,300)( 68,299)( 69,298)( 70,297)( 71,254)
( 72,253)( 73,259)( 74,258)( 75,257)( 76,256)( 77,255)( 78,247)( 79,246)
( 80,252)( 81,251)( 82,250)( 83,249)( 84,248)( 85,275)( 86,274)( 87,280)
( 88,279)( 89,278)( 90,277)( 91,276)( 92,268)( 93,267)( 94,273)( 95,272)
( 96,271)( 97,270)( 98,269)( 99,261)(100,260)(101,266)(102,265)(103,264)
(104,263)(105,262)(106,324)(107,323)(108,329)(109,328)(110,327)(111,326)
(112,325)(113,317)(114,316)(115,322)(116,321)(117,320)(118,319)(119,318)
(120,345)(121,344)(122,350)(123,349)(124,348)(125,347)(126,346)(127,338)
(128,337)(129,343)(130,342)(131,341)(132,340)(133,339)(134,331)(135,330)
(136,336)(137,335)(138,334)(139,333)(140,332)(141,394)(142,393)(143,399)
(144,398)(145,397)(146,396)(147,395)(148,387)(149,386)(150,392)(151,391)
(152,390)(153,389)(154,388)(155,415)(156,414)(157,420)(158,419)(159,418)
(160,417)(161,416)(162,408)(163,407)(164,413)(165,412)(166,411)(167,410)
(168,409)(169,401)(170,400)(171,406)(172,405)(173,404)(174,403)(175,402)
(176,359)(177,358)(178,364)(179,363)(180,362)(181,361)(182,360)(183,352)
(184,351)(185,357)(186,356)(187,355)(188,354)(189,353)(190,380)(191,379)
(192,385)(193,384)(194,383)(195,382)(196,381)(197,373)(198,372)(199,378)
(200,377)(201,376)(202,375)(203,374)(204,366)(205,365)(206,371)(207,370)
(208,369)(209,368)(210,367);
s2 := Sym(420)!(  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 41)(  7, 42)
(  8, 43)(  9, 44)( 10, 45)( 11, 46)( 12, 47)( 13, 48)( 14, 49)( 15, 50)
( 16, 51)( 17, 52)( 18, 53)( 19, 54)( 20, 55)( 21, 56)( 22, 57)( 23, 58)
( 24, 59)( 25, 60)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 66)
( 32, 67)( 33, 68)( 34, 69)( 35, 70)(106,141)(107,142)(108,143)(109,144)
(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)
(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)
(126,161)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)
(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(211,246)
(212,247)(213,248)(214,249)(215,250)(216,251)(217,252)(218,253)(219,254)
(220,255)(221,256)(222,257)(223,258)(224,259)(225,260)(226,261)(227,262)
(228,263)(229,264)(230,265)(231,266)(232,267)(233,268)(234,269)(235,270)
(236,271)(237,272)(238,273)(239,274)(240,275)(241,276)(242,277)(243,278)
(244,279)(245,280)(316,351)(317,352)(318,353)(319,354)(320,355)(321,356)
(322,357)(323,358)(324,359)(325,360)(326,361)(327,362)(328,363)(329,364)
(330,365)(331,366)(332,367)(333,368)(334,369)(335,370)(336,371)(337,372)
(338,373)(339,374)(340,375)(341,376)(342,377)(343,378)(344,379)(345,380)
(346,381)(347,382)(348,383)(349,384)(350,385);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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*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*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*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 >;
References : None.
to this polytope