Polytope of Type {4,220}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,220}*1760
Also Known As : {4,220|2}. if this polytope has another name.
Group : SmallGroup(1760,991)
Rank : 3
Schlafli Type : {4,220}
Number of vertices, edges, etc : 4, 440, 220
Order of s0s1s2 : 220
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 : {2,220}*880, {4,110}*880
4-fold quotients : {2,110}*440
5-fold quotients : {4,44}*352
8-fold quotients : {2,55}*220
10-fold quotients : {2,44}*176, {4,22}*176
11-fold quotients : {4,20}*160
20-fold quotients : {2,22}*88
22-fold quotients : {2,20}*80, {4,10}*80
40-fold quotients : {2,11}*44
44-fold quotients : {2,10}*40
55-fold quotients : {4,4}*32
88-fold quotients : {2,5}*20
110-fold quotients : {2,4}*16, {4,2}*16
220-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (221,331)(222,332)(223,333)(224,334)(225,335)(226,336)(227,337)(228,338)
(229,339)(230,340)(231,341)(232,342)(233,343)(234,344)(235,345)(236,346)
(237,347)(238,348)(239,349)(240,350)(241,351)(242,352)(243,353)(244,354)
(245,355)(246,356)(247,357)(248,358)(249,359)(250,360)(251,361)(252,362)
(253,363)(254,364)(255,365)(256,366)(257,367)(258,368)(259,369)(260,370)
(261,371)(262,372)(263,373)(264,374)(265,375)(266,376)(267,377)(268,378)
(269,379)(270,380)(271,381)(272,382)(273,383)(274,384)(275,385)(276,386)
(277,387)(278,388)(279,389)(280,390)(281,391)(282,392)(283,393)(284,394)
(285,395)(286,396)(287,397)(288,398)(289,399)(290,400)(291,401)(292,402)
(293,403)(294,404)(295,405)(296,406)(297,407)(298,408)(299,409)(300,410)
(301,411)(302,412)(303,413)(304,414)(305,415)(306,416)(307,417)(308,418)
(309,419)(310,420)(311,421)(312,422)(313,423)(314,424)(315,425)(316,426)
(317,427)(318,428)(319,429)(320,430)(321,431)(322,432)(323,433)(324,434)
(325,435)(326,436)(327,437)(328,438)(329,439)(330,440);;
s1 := ( 1,221)( 2,231)( 3,230)( 4,229)( 5,228)( 6,227)( 7,226)( 8,225)
( 9,224)( 10,223)( 11,222)( 12,265)( 13,275)( 14,274)( 15,273)( 16,272)
( 17,271)( 18,270)( 19,269)( 20,268)( 21,267)( 22,266)( 23,254)( 24,264)
( 25,263)( 26,262)( 27,261)( 28,260)( 29,259)( 30,258)( 31,257)( 32,256)
( 33,255)( 34,243)( 35,253)( 36,252)( 37,251)( 38,250)( 39,249)( 40,248)
( 41,247)( 42,246)( 43,245)( 44,244)( 45,232)( 46,242)( 47,241)( 48,240)
( 49,239)( 50,238)( 51,237)( 52,236)( 53,235)( 54,234)( 55,233)( 56,276)
( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)( 64,279)
( 65,278)( 66,277)( 67,320)( 68,330)( 69,329)( 70,328)( 71,327)( 72,326)
( 73,325)( 74,324)( 75,323)( 76,322)( 77,321)( 78,309)( 79,319)( 80,318)
( 81,317)( 82,316)( 83,315)( 84,314)( 85,313)( 86,312)( 87,311)( 88,310)
( 89,298)( 90,308)( 91,307)( 92,306)( 93,305)( 94,304)( 95,303)( 96,302)
( 97,301)( 98,300)( 99,299)(100,287)(101,297)(102,296)(103,295)(104,294)
(105,293)(106,292)(107,291)(108,290)(109,289)(110,288)(111,331)(112,341)
(113,340)(114,339)(115,338)(116,337)(117,336)(118,335)(119,334)(120,333)
(121,332)(122,375)(123,385)(124,384)(125,383)(126,382)(127,381)(128,380)
(129,379)(130,378)(131,377)(132,376)(133,364)(134,374)(135,373)(136,372)
(137,371)(138,370)(139,369)(140,368)(141,367)(142,366)(143,365)(144,353)
(145,363)(146,362)(147,361)(148,360)(149,359)(150,358)(151,357)(152,356)
(153,355)(154,354)(155,342)(156,352)(157,351)(158,350)(159,349)(160,348)
(161,347)(162,346)(163,345)(164,344)(165,343)(166,386)(167,396)(168,395)
(169,394)(170,393)(171,392)(172,391)(173,390)(174,389)(175,388)(176,387)
(177,430)(178,440)(179,439)(180,438)(181,437)(182,436)(183,435)(184,434)
(185,433)(186,432)(187,431)(188,419)(189,429)(190,428)(191,427)(192,426)
(193,425)(194,424)(195,423)(196,422)(197,421)(198,420)(199,408)(200,418)
(201,417)(202,416)(203,415)(204,414)(205,413)(206,412)(207,411)(208,410)
(209,409)(210,397)(211,407)(212,406)(213,405)(214,404)(215,403)(216,402)
(217,401)(218,400)(219,399)(220,398);;
s2 := ( 1, 13)( 2, 12)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 18)( 8, 17)
( 9, 16)( 10, 15)( 11, 14)( 23, 46)( 24, 45)( 25, 55)( 26, 54)( 27, 53)
( 28, 52)( 29, 51)( 30, 50)( 31, 49)( 32, 48)( 33, 47)( 34, 35)( 36, 44)
( 37, 43)( 38, 42)( 39, 41)( 56, 68)( 57, 67)( 58, 77)( 59, 76)( 60, 75)
( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 78,101)( 79,100)
( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)
( 88,102)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(111,123)(112,122)
(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)
(121,124)(133,156)(134,155)(135,165)(136,164)(137,163)(138,162)(139,161)
(140,160)(141,159)(142,158)(143,157)(144,145)(146,154)(147,153)(148,152)
(149,151)(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)(172,183)
(173,182)(174,181)(175,180)(176,179)(188,211)(189,210)(190,220)(191,219)
(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)(199,200)
(201,209)(202,208)(203,207)(204,206)(221,288)(222,287)(223,297)(224,296)
(225,295)(226,294)(227,293)(228,292)(229,291)(230,290)(231,289)(232,277)
(233,276)(234,286)(235,285)(236,284)(237,283)(238,282)(239,281)(240,280)
(241,279)(242,278)(243,321)(244,320)(245,330)(246,329)(247,328)(248,327)
(249,326)(250,325)(251,324)(252,323)(253,322)(254,310)(255,309)(256,319)
(257,318)(258,317)(259,316)(260,315)(261,314)(262,313)(263,312)(264,311)
(265,299)(266,298)(267,308)(268,307)(269,306)(270,305)(271,304)(272,303)
(273,302)(274,301)(275,300)(331,398)(332,397)(333,407)(334,406)(335,405)
(336,404)(337,403)(338,402)(339,401)(340,400)(341,399)(342,387)(343,386)
(344,396)(345,395)(346,394)(347,393)(348,392)(349,391)(350,390)(351,389)
(352,388)(353,431)(354,430)(355,440)(356,439)(357,438)(358,437)(359,436)
(360,435)(361,434)(362,433)(363,432)(364,420)(365,419)(366,429)(367,428)
(368,427)(369,426)(370,425)(371,424)(372,423)(373,422)(374,421)(375,409)
(376,408)(377,418)(378,417)(379,416)(380,415)(381,414)(382,413)(383,412)
(384,411)(385,410);;
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*s0*s1*s0*s1*s0*s1,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(440)!(221,331)(222,332)(223,333)(224,334)(225,335)(226,336)(227,337)
(228,338)(229,339)(230,340)(231,341)(232,342)(233,343)(234,344)(235,345)
(236,346)(237,347)(238,348)(239,349)(240,350)(241,351)(242,352)(243,353)
(244,354)(245,355)(246,356)(247,357)(248,358)(249,359)(250,360)(251,361)
(252,362)(253,363)(254,364)(255,365)(256,366)(257,367)(258,368)(259,369)
(260,370)(261,371)(262,372)(263,373)(264,374)(265,375)(266,376)(267,377)
(268,378)(269,379)(270,380)(271,381)(272,382)(273,383)(274,384)(275,385)
(276,386)(277,387)(278,388)(279,389)(280,390)(281,391)(282,392)(283,393)
(284,394)(285,395)(286,396)(287,397)(288,398)(289,399)(290,400)(291,401)
(292,402)(293,403)(294,404)(295,405)(296,406)(297,407)(298,408)(299,409)
(300,410)(301,411)(302,412)(303,413)(304,414)(305,415)(306,416)(307,417)
(308,418)(309,419)(310,420)(311,421)(312,422)(313,423)(314,424)(315,425)
(316,426)(317,427)(318,428)(319,429)(320,430)(321,431)(322,432)(323,433)
(324,434)(325,435)(326,436)(327,437)(328,438)(329,439)(330,440);
s1 := Sym(440)!( 1,221)( 2,231)( 3,230)( 4,229)( 5,228)( 6,227)( 7,226)
( 8,225)( 9,224)( 10,223)( 11,222)( 12,265)( 13,275)( 14,274)( 15,273)
( 16,272)( 17,271)( 18,270)( 19,269)( 20,268)( 21,267)( 22,266)( 23,254)
( 24,264)( 25,263)( 26,262)( 27,261)( 28,260)( 29,259)( 30,258)( 31,257)
( 32,256)( 33,255)( 34,243)( 35,253)( 36,252)( 37,251)( 38,250)( 39,249)
( 40,248)( 41,247)( 42,246)( 43,245)( 44,244)( 45,232)( 46,242)( 47,241)
( 48,240)( 49,239)( 50,238)( 51,237)( 52,236)( 53,235)( 54,234)( 55,233)
( 56,276)( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)
( 64,279)( 65,278)( 66,277)( 67,320)( 68,330)( 69,329)( 70,328)( 71,327)
( 72,326)( 73,325)( 74,324)( 75,323)( 76,322)( 77,321)( 78,309)( 79,319)
( 80,318)( 81,317)( 82,316)( 83,315)( 84,314)( 85,313)( 86,312)( 87,311)
( 88,310)( 89,298)( 90,308)( 91,307)( 92,306)( 93,305)( 94,304)( 95,303)
( 96,302)( 97,301)( 98,300)( 99,299)(100,287)(101,297)(102,296)(103,295)
(104,294)(105,293)(106,292)(107,291)(108,290)(109,289)(110,288)(111,331)
(112,341)(113,340)(114,339)(115,338)(116,337)(117,336)(118,335)(119,334)
(120,333)(121,332)(122,375)(123,385)(124,384)(125,383)(126,382)(127,381)
(128,380)(129,379)(130,378)(131,377)(132,376)(133,364)(134,374)(135,373)
(136,372)(137,371)(138,370)(139,369)(140,368)(141,367)(142,366)(143,365)
(144,353)(145,363)(146,362)(147,361)(148,360)(149,359)(150,358)(151,357)
(152,356)(153,355)(154,354)(155,342)(156,352)(157,351)(158,350)(159,349)
(160,348)(161,347)(162,346)(163,345)(164,344)(165,343)(166,386)(167,396)
(168,395)(169,394)(170,393)(171,392)(172,391)(173,390)(174,389)(175,388)
(176,387)(177,430)(178,440)(179,439)(180,438)(181,437)(182,436)(183,435)
(184,434)(185,433)(186,432)(187,431)(188,419)(189,429)(190,428)(191,427)
(192,426)(193,425)(194,424)(195,423)(196,422)(197,421)(198,420)(199,408)
(200,418)(201,417)(202,416)(203,415)(204,414)(205,413)(206,412)(207,411)
(208,410)(209,409)(210,397)(211,407)(212,406)(213,405)(214,404)(215,403)
(216,402)(217,401)(218,400)(219,399)(220,398);
s2 := Sym(440)!( 1, 13)( 2, 12)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 18)
( 8, 17)( 9, 16)( 10, 15)( 11, 14)( 23, 46)( 24, 45)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 51)( 30, 50)( 31, 49)( 32, 48)( 33, 47)( 34, 35)
( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 56, 68)( 57, 67)( 58, 77)( 59, 76)
( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 78,101)
( 79,100)( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)
( 87,103)( 88,102)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(111,123)
(112,122)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)
(120,125)(121,124)(133,156)(134,155)(135,165)(136,164)(137,163)(138,162)
(139,161)(140,160)(141,159)(142,158)(143,157)(144,145)(146,154)(147,153)
(148,152)(149,151)(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)
(172,183)(173,182)(174,181)(175,180)(176,179)(188,211)(189,210)(190,220)
(191,219)(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)
(199,200)(201,209)(202,208)(203,207)(204,206)(221,288)(222,287)(223,297)
(224,296)(225,295)(226,294)(227,293)(228,292)(229,291)(230,290)(231,289)
(232,277)(233,276)(234,286)(235,285)(236,284)(237,283)(238,282)(239,281)
(240,280)(241,279)(242,278)(243,321)(244,320)(245,330)(246,329)(247,328)
(248,327)(249,326)(250,325)(251,324)(252,323)(253,322)(254,310)(255,309)
(256,319)(257,318)(258,317)(259,316)(260,315)(261,314)(262,313)(263,312)
(264,311)(265,299)(266,298)(267,308)(268,307)(269,306)(270,305)(271,304)
(272,303)(273,302)(274,301)(275,300)(331,398)(332,397)(333,407)(334,406)
(335,405)(336,404)(337,403)(338,402)(339,401)(340,400)(341,399)(342,387)
(343,386)(344,396)(345,395)(346,394)(347,393)(348,392)(349,391)(350,390)
(351,389)(352,388)(353,431)(354,430)(355,440)(356,439)(357,438)(358,437)
(359,436)(360,435)(361,434)(362,433)(363,432)(364,420)(365,419)(366,429)
(367,428)(368,427)(369,426)(370,425)(371,424)(372,423)(373,422)(374,421)
(375,409)(376,408)(377,418)(378,417)(379,416)(380,415)(381,414)(382,413)
(383,412)(384,411)(385,410);
poly := sub<Sym(440)|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*s0*s1*s0*s1*s0*s1,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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