Polytope of Type {6,140}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,140}*1680a
Also Known As : {6,140|2}. if this polytope has another name.
Group : SmallGroup(1680,798)
Rank : 3
Schlafli Type : {6,140}
Number of vertices, edges, etc : 6, 420, 140
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 : {2,140}*560
5-fold quotients : {6,28}*336a
6-fold quotients : {2,70}*280
7-fold quotients : {6,20}*240a
10-fold quotients : {6,14}*168
12-fold quotients : {2,35}*140
14-fold quotients : {6,10}*120
15-fold quotients : {2,28}*112
21-fold quotients : {2,20}*80
30-fold quotients : {2,14}*56
35-fold quotients : {6,4}*48a
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 : {2,4}*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)(246,281)(247,282)
(248,283)(249,284)(250,285)(251,286)(252,287)(253,288)(254,289)(255,290)
(256,291)(257,292)(258,293)(259,294)(260,295)(261,296)(262,297)(263,298)
(264,299)(265,300)(266,301)(267,302)(268,303)(269,304)(270,305)(271,306)
(272,307)(273,308)(274,309)(275,310)(276,311)(277,312)(278,313)(279,314)
(280,315)(351,386)(352,387)(353,388)(354,389)(355,390)(356,391)(357,392)
(358,393)(359,394)(360,395)(361,396)(362,397)(363,398)(364,399)(365,400)
(366,401)(367,402)(368,403)(369,404)(370,405)(371,406)(372,407)(373,408)
(374,409)(375,410)(376,411)(377,412)(378,413)(379,414)(380,415)(381,416)
(382,417)(383,418)(384,419)(385,420);;
s1 := ( 1, 36)( 2, 42)( 3, 41)( 4, 40)( 5, 39)( 6, 38)( 7, 37)( 8, 64)
( 9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 57)( 16, 63)
( 17, 62)( 18, 61)( 19, 60)( 20, 59)( 21, 58)( 22, 50)( 23, 56)( 24, 55)
( 25, 54)( 26, 53)( 27, 52)( 28, 51)( 29, 43)( 30, 49)( 31, 48)( 32, 47)
( 33, 46)( 34, 45)( 35, 44)( 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)(106,141)(107,147)(108,146)(109,145)
(110,144)(111,143)(112,142)(113,169)(114,175)(115,174)(116,173)(117,172)
(118,171)(119,170)(120,162)(121,168)(122,167)(123,166)(124,165)(125,164)
(126,163)(127,155)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)
(134,148)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(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)
(211,351)(212,357)(213,356)(214,355)(215,354)(216,353)(217,352)(218,379)
(219,385)(220,384)(221,383)(222,382)(223,381)(224,380)(225,372)(226,378)
(227,377)(228,376)(229,375)(230,374)(231,373)(232,365)(233,371)(234,370)
(235,369)(236,368)(237,367)(238,366)(239,358)(240,364)(241,363)(242,362)
(243,361)(244,360)(245,359)(246,316)(247,322)(248,321)(249,320)(250,319)
(251,318)(252,317)(253,344)(254,350)(255,349)(256,348)(257,347)(258,346)
(259,345)(260,337)(261,343)(262,342)(263,341)(264,340)(265,339)(266,338)
(267,330)(268,336)(269,335)(270,334)(271,333)(272,332)(273,331)(274,323)
(275,329)(276,328)(277,327)(278,326)(279,325)(280,324)(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);;
s2 := ( 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)( 36,254)( 37,253)( 38,259)( 39,258)( 40,257)
( 41,256)( 42,255)( 43,247)( 44,246)( 45,252)( 46,251)( 47,250)( 48,249)
( 49,248)( 50,275)( 51,274)( 52,280)( 53,279)( 54,278)( 55,277)( 56,276)
( 57,268)( 58,267)( 59,273)( 60,272)( 61,271)( 62,270)( 63,269)( 64,261)
( 65,260)( 66,266)( 67,265)( 68,264)( 69,263)( 70,262)( 71,289)( 72,288)
( 73,294)( 74,293)( 75,292)( 76,291)( 77,290)( 78,282)( 79,281)( 80,287)
( 81,286)( 82,285)( 83,284)( 84,283)( 85,310)( 86,309)( 87,315)( 88,314)
( 89,313)( 90,312)( 91,311)( 92,303)( 93,302)( 94,308)( 95,307)( 96,306)
( 97,305)( 98,304)( 99,296)(100,295)(101,301)(102,300)(103,299)(104,298)
(105,297)(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)(141,359)(142,358)(143,364)(144,363)
(145,362)(146,361)(147,360)(148,352)(149,351)(150,357)(151,356)(152,355)
(153,354)(154,353)(155,380)(156,379)(157,385)(158,384)(159,383)(160,382)
(161,381)(162,373)(163,372)(164,378)(165,377)(166,376)(167,375)(168,374)
(169,366)(170,365)(171,371)(172,370)(173,369)(174,368)(175,367)(176,394)
(177,393)(178,399)(179,398)(180,397)(181,396)(182,395)(183,387)(184,386)
(185,392)(186,391)(187,390)(188,389)(189,388)(190,415)(191,414)(192,420)
(193,419)(194,418)(195,417)(196,416)(197,408)(198,407)(199,413)(200,412)
(201,411)(202,410)(203,409)(204,401)(205,400)(206,406)(207,405)(208,404)
(209,403)(210,402);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)(246,281)
(247,282)(248,283)(249,284)(250,285)(251,286)(252,287)(253,288)(254,289)
(255,290)(256,291)(257,292)(258,293)(259,294)(260,295)(261,296)(262,297)
(263,298)(264,299)(265,300)(266,301)(267,302)(268,303)(269,304)(270,305)
(271,306)(272,307)(273,308)(274,309)(275,310)(276,311)(277,312)(278,313)
(279,314)(280,315)(351,386)(352,387)(353,388)(354,389)(355,390)(356,391)
(357,392)(358,393)(359,394)(360,395)(361,396)(362,397)(363,398)(364,399)
(365,400)(366,401)(367,402)(368,403)(369,404)(370,405)(371,406)(372,407)
(373,408)(374,409)(375,410)(376,411)(377,412)(378,413)(379,414)(380,415)
(381,416)(382,417)(383,418)(384,419)(385,420);
s1 := Sym(420)!( 1, 36)( 2, 42)( 3, 41)( 4, 40)( 5, 39)( 6, 38)( 7, 37)
( 8, 64)( 9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 57)
( 16, 63)( 17, 62)( 18, 61)( 19, 60)( 20, 59)( 21, 58)( 22, 50)( 23, 56)
( 24, 55)( 25, 54)( 26, 53)( 27, 52)( 28, 51)( 29, 43)( 30, 49)( 31, 48)
( 32, 47)( 33, 46)( 34, 45)( 35, 44)( 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)(106,141)(107,147)(108,146)
(109,145)(110,144)(111,143)(112,142)(113,169)(114,175)(115,174)(116,173)
(117,172)(118,171)(119,170)(120,162)(121,168)(122,167)(123,166)(124,165)
(125,164)(126,163)(127,155)(128,161)(129,160)(130,159)(131,158)(132,157)
(133,156)(134,148)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)
(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)(211,351)(212,357)(213,356)(214,355)(215,354)(216,353)(217,352)
(218,379)(219,385)(220,384)(221,383)(222,382)(223,381)(224,380)(225,372)
(226,378)(227,377)(228,376)(229,375)(230,374)(231,373)(232,365)(233,371)
(234,370)(235,369)(236,368)(237,367)(238,366)(239,358)(240,364)(241,363)
(242,362)(243,361)(244,360)(245,359)(246,316)(247,322)(248,321)(249,320)
(250,319)(251,318)(252,317)(253,344)(254,350)(255,349)(256,348)(257,347)
(258,346)(259,345)(260,337)(261,343)(262,342)(263,341)(264,340)(265,339)
(266,338)(267,330)(268,336)(269,335)(270,334)(271,333)(272,332)(273,331)
(274,323)(275,329)(276,328)(277,327)(278,326)(279,325)(280,324)(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);
s2 := 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)( 36,254)( 37,253)( 38,259)( 39,258)
( 40,257)( 41,256)( 42,255)( 43,247)( 44,246)( 45,252)( 46,251)( 47,250)
( 48,249)( 49,248)( 50,275)( 51,274)( 52,280)( 53,279)( 54,278)( 55,277)
( 56,276)( 57,268)( 58,267)( 59,273)( 60,272)( 61,271)( 62,270)( 63,269)
( 64,261)( 65,260)( 66,266)( 67,265)( 68,264)( 69,263)( 70,262)( 71,289)
( 72,288)( 73,294)( 74,293)( 75,292)( 76,291)( 77,290)( 78,282)( 79,281)
( 80,287)( 81,286)( 82,285)( 83,284)( 84,283)( 85,310)( 86,309)( 87,315)
( 88,314)( 89,313)( 90,312)( 91,311)( 92,303)( 93,302)( 94,308)( 95,307)
( 96,306)( 97,305)( 98,304)( 99,296)(100,295)(101,301)(102,300)(103,299)
(104,298)(105,297)(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)(141,359)(142,358)(143,364)
(144,363)(145,362)(146,361)(147,360)(148,352)(149,351)(150,357)(151,356)
(152,355)(153,354)(154,353)(155,380)(156,379)(157,385)(158,384)(159,383)
(160,382)(161,381)(162,373)(163,372)(164,378)(165,377)(166,376)(167,375)
(168,374)(169,366)(170,365)(171,371)(172,370)(173,369)(174,368)(175,367)
(176,394)(177,393)(178,399)(179,398)(180,397)(181,396)(182,395)(183,387)
(184,386)(185,392)(186,391)(187,390)(188,389)(189,388)(190,415)(191,414)
(192,420)(193,419)(194,418)(195,417)(196,416)(197,408)(198,407)(199,413)
(200,412)(201,411)(202,410)(203,409)(204,401)(205,400)(206,406)(207,405)
(208,404)(209,403)(210,402);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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