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