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# Polytope of Type {6,20}

Atlas Canonical Name : {6,20}*1920b
if this polytope has a name.
Group : SmallGroup(1920,238657)
Rank : 3
Schlafli Type : {6,20}
Number of vertices, edges, etc : 48, 480, 160
Order of s0s1s2 : 120
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,20}*960e
4-fold quotients : {6,20}*480c
5-fold quotients : {6,4}*384b
8-fold quotients : {6,20}*240a, {6,20}*240b
10-fold quotients : {6,4}*192b
16-fold quotients : {6,10}*120
20-fold quotients : {6,4}*96
24-fold quotients : {2,20}*80
40-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {2,10}*40
80-fold quotients : {3,4}*24, {6,2}*24
96-fold quotients : {2,5}*20
120-fold quotients : {2,4}*16
160-fold quotients : {3,2}*12
240-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)( 41, 81)
( 42, 82)( 43, 84)( 44, 83)( 45, 87)( 46, 88)( 47, 85)( 48, 86)( 49, 89)
( 50, 90)( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 97)
( 58, 98)( 59,100)( 60, 99)( 61,103)( 62,104)( 63,101)( 64,102)( 65,105)
( 66,106)( 67,108)( 68,107)( 69,111)( 70,112)( 71,109)( 72,110)( 73,113)
( 74,114)( 75,116)( 76,115)( 77,119)( 78,120)( 79,117)( 80,118)(123,124)
(125,127)(126,128)(131,132)(133,135)(134,136)(139,140)(141,143)(142,144)
(147,148)(149,151)(150,152)(155,156)(157,159)(158,160)(161,201)(162,202)
(163,204)(164,203)(165,207)(166,208)(167,205)(168,206)(169,209)(170,210)
(171,212)(172,211)(173,215)(174,216)(175,213)(176,214)(177,217)(178,218)
(179,220)(180,219)(181,223)(182,224)(183,221)(184,222)(185,225)(186,226)
(187,228)(188,227)(189,231)(190,232)(191,229)(192,230)(193,233)(194,234)
(195,236)(196,235)(197,239)(198,240)(199,237)(200,238)(241,242)(245,248)
(246,247)(249,250)(253,256)(254,255)(257,258)(261,264)(262,263)(265,266)
(269,272)(270,271)(273,274)(277,280)(278,279)(281,322)(282,321)(283,323)
(284,324)(285,328)(286,327)(287,326)(288,325)(289,330)(290,329)(291,331)
(292,332)(293,336)(294,335)(295,334)(296,333)(297,338)(298,337)(299,339)
(300,340)(301,344)(302,343)(303,342)(304,341)(305,346)(306,345)(307,347)
(308,348)(309,352)(310,351)(311,350)(312,349)(313,354)(314,353)(315,355)
(316,356)(317,360)(318,359)(319,358)(320,357)(361,362)(365,368)(366,367)
(369,370)(373,376)(374,375)(377,378)(381,384)(382,383)(385,386)(389,392)
(390,391)(393,394)(397,400)(398,399)(401,442)(402,441)(403,443)(404,444)
(405,448)(406,447)(407,446)(408,445)(409,450)(410,449)(411,451)(412,452)
(413,456)(414,455)(415,454)(416,453)(417,458)(418,457)(419,459)(420,460)
(421,464)(422,463)(423,462)(424,461)(425,466)(426,465)(427,467)(428,468)
(429,472)(430,471)(431,470)(432,469)(433,474)(434,473)(435,475)(436,476)
(437,480)(438,479)(439,478)(440,477);;
s1 := (  1, 41)(  2, 42)(  3, 47)(  4, 48)(  5, 46)(  6, 45)(  7, 43)(  8, 44)
(  9, 73)( 10, 74)( 11, 79)( 12, 80)( 13, 78)( 14, 77)( 15, 75)( 16, 76)
( 17, 65)( 18, 66)( 19, 71)( 20, 72)( 21, 70)( 22, 69)( 23, 67)( 24, 68)
( 25, 57)( 26, 58)( 27, 63)( 28, 64)( 29, 62)( 30, 61)( 31, 59)( 32, 60)
( 33, 49)( 34, 50)( 35, 55)( 36, 56)( 37, 54)( 38, 53)( 39, 51)( 40, 52)
( 83, 87)( 84, 88)( 85, 86)( 89,113)( 90,114)( 91,119)( 92,120)( 93,118)
( 94,117)( 95,115)( 96,116)( 97,105)( 98,106)( 99,111)(100,112)(101,110)
(102,109)(103,107)(104,108)(121,162)(122,161)(123,168)(124,167)(125,165)
(126,166)(127,164)(128,163)(129,194)(130,193)(131,200)(132,199)(133,197)
(134,198)(135,196)(136,195)(137,186)(138,185)(139,192)(140,191)(141,189)
(142,190)(143,188)(144,187)(145,178)(146,177)(147,184)(148,183)(149,181)
(150,182)(151,180)(152,179)(153,170)(154,169)(155,176)(156,175)(157,173)
(158,174)(159,172)(160,171)(201,202)(203,208)(204,207)(209,234)(210,233)
(211,240)(212,239)(213,237)(214,238)(215,236)(216,235)(217,226)(218,225)
(219,232)(220,231)(221,229)(222,230)(223,228)(224,227)(241,402)(242,401)
(243,408)(244,407)(245,405)(246,406)(247,404)(248,403)(249,434)(250,433)
(251,440)(252,439)(253,437)(254,438)(255,436)(256,435)(257,426)(258,425)
(259,432)(260,431)(261,429)(262,430)(263,428)(264,427)(265,418)(266,417)
(267,424)(268,423)(269,421)(270,422)(271,420)(272,419)(273,410)(274,409)
(275,416)(276,415)(277,413)(278,414)(279,412)(280,411)(281,362)(282,361)
(283,368)(284,367)(285,365)(286,366)(287,364)(288,363)(289,394)(290,393)
(291,400)(292,399)(293,397)(294,398)(295,396)(296,395)(297,386)(298,385)
(299,392)(300,391)(301,389)(302,390)(303,388)(304,387)(305,378)(306,377)
(307,384)(308,383)(309,381)(310,382)(311,380)(312,379)(313,370)(314,369)
(315,376)(316,375)(317,373)(318,374)(319,372)(320,371)(321,442)(322,441)
(323,448)(324,447)(325,445)(326,446)(327,444)(328,443)(329,474)(330,473)
(331,480)(332,479)(333,477)(334,478)(335,476)(336,475)(337,466)(338,465)
(339,472)(340,471)(341,469)(342,470)(343,468)(344,467)(345,458)(346,457)
(347,464)(348,463)(349,461)(350,462)(351,460)(352,459)(353,450)(354,449)
(355,456)(356,455)(357,453)(358,454)(359,452)(360,451);;
s2 := (  1,251)(  2,252)(  3,250)(  4,249)(  5,255)(  6,256)(  7,254)(  8,253)
(  9,243)( 10,244)( 11,242)( 12,241)( 13,247)( 14,248)( 15,246)( 16,245)
( 17,275)( 18,276)( 19,274)( 20,273)( 21,279)( 22,280)( 23,278)( 24,277)
( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)( 32,269)
( 33,259)( 34,260)( 35,258)( 36,257)( 37,263)( 38,264)( 39,262)( 40,261)
( 41,291)( 42,292)( 43,290)( 44,289)( 45,295)( 46,296)( 47,294)( 48,293)
( 49,283)( 50,284)( 51,282)( 52,281)( 53,287)( 54,288)( 55,286)( 56,285)
( 57,315)( 58,316)( 59,314)( 60,313)( 61,319)( 62,320)( 63,318)( 64,317)
( 65,307)( 66,308)( 67,306)( 68,305)( 69,311)( 70,312)( 71,310)( 72,309)
( 73,299)( 74,300)( 75,298)( 76,297)( 77,303)( 78,304)( 79,302)( 80,301)
( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)( 88,333)
( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)( 96,325)
( 97,355)( 98,356)( 99,354)(100,353)(101,359)(102,360)(103,358)(104,357)
(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)(112,349)
(113,339)(114,340)(115,338)(116,337)(117,343)(118,344)(119,342)(120,341)
(121,372)(122,371)(123,369)(124,370)(125,376)(126,375)(127,373)(128,374)
(129,364)(130,363)(131,361)(132,362)(133,368)(134,367)(135,365)(136,366)
(137,396)(138,395)(139,393)(140,394)(141,400)(142,399)(143,397)(144,398)
(145,388)(146,387)(147,385)(148,386)(149,392)(150,391)(151,389)(152,390)
(153,380)(154,379)(155,377)(156,378)(157,384)(158,383)(159,381)(160,382)
(161,412)(162,411)(163,409)(164,410)(165,416)(166,415)(167,413)(168,414)
(169,404)(170,403)(171,401)(172,402)(173,408)(174,407)(175,405)(176,406)
(177,436)(178,435)(179,433)(180,434)(181,440)(182,439)(183,437)(184,438)
(185,428)(186,427)(187,425)(188,426)(189,432)(190,431)(191,429)(192,430)
(193,420)(194,419)(195,417)(196,418)(197,424)(198,423)(199,421)(200,422)
(201,452)(202,451)(203,449)(204,450)(205,456)(206,455)(207,453)(208,454)
(209,444)(210,443)(211,441)(212,442)(213,448)(214,447)(215,445)(216,446)
(217,476)(218,475)(219,473)(220,474)(221,480)(222,479)(223,477)(224,478)
(225,468)(226,467)(227,465)(228,466)(229,472)(230,471)(231,469)(232,470)
(233,460)(234,459)(235,457)(236,458)(237,464)(238,463)(239,461)(240,462);;
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*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(480)!(  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 27, 28)( 29, 31)( 30, 32)( 35, 36)( 37, 39)( 38, 40)
( 41, 81)( 42, 82)( 43, 84)( 44, 83)( 45, 87)( 46, 88)( 47, 85)( 48, 86)
( 49, 89)( 50, 90)( 51, 92)( 52, 91)( 53, 95)( 54, 96)( 55, 93)( 56, 94)
( 57, 97)( 58, 98)( 59,100)( 60, 99)( 61,103)( 62,104)( 63,101)( 64,102)
( 65,105)( 66,106)( 67,108)( 68,107)( 69,111)( 70,112)( 71,109)( 72,110)
( 73,113)( 74,114)( 75,116)( 76,115)( 77,119)( 78,120)( 79,117)( 80,118)
(123,124)(125,127)(126,128)(131,132)(133,135)(134,136)(139,140)(141,143)
(142,144)(147,148)(149,151)(150,152)(155,156)(157,159)(158,160)(161,201)
(162,202)(163,204)(164,203)(165,207)(166,208)(167,205)(168,206)(169,209)
(170,210)(171,212)(172,211)(173,215)(174,216)(175,213)(176,214)(177,217)
(178,218)(179,220)(180,219)(181,223)(182,224)(183,221)(184,222)(185,225)
(186,226)(187,228)(188,227)(189,231)(190,232)(191,229)(192,230)(193,233)
(194,234)(195,236)(196,235)(197,239)(198,240)(199,237)(200,238)(241,242)
(245,248)(246,247)(249,250)(253,256)(254,255)(257,258)(261,264)(262,263)
(265,266)(269,272)(270,271)(273,274)(277,280)(278,279)(281,322)(282,321)
(283,323)(284,324)(285,328)(286,327)(287,326)(288,325)(289,330)(290,329)
(291,331)(292,332)(293,336)(294,335)(295,334)(296,333)(297,338)(298,337)
(299,339)(300,340)(301,344)(302,343)(303,342)(304,341)(305,346)(306,345)
(307,347)(308,348)(309,352)(310,351)(311,350)(312,349)(313,354)(314,353)
(315,355)(316,356)(317,360)(318,359)(319,358)(320,357)(361,362)(365,368)
(366,367)(369,370)(373,376)(374,375)(377,378)(381,384)(382,383)(385,386)
(389,392)(390,391)(393,394)(397,400)(398,399)(401,442)(402,441)(403,443)
(404,444)(405,448)(406,447)(407,446)(408,445)(409,450)(410,449)(411,451)
(412,452)(413,456)(414,455)(415,454)(416,453)(417,458)(418,457)(419,459)
(420,460)(421,464)(422,463)(423,462)(424,461)(425,466)(426,465)(427,467)
(428,468)(429,472)(430,471)(431,470)(432,469)(433,474)(434,473)(435,475)
(436,476)(437,480)(438,479)(439,478)(440,477);
s1 := Sym(480)!(  1, 41)(  2, 42)(  3, 47)(  4, 48)(  5, 46)(  6, 45)(  7, 43)
(  8, 44)(  9, 73)( 10, 74)( 11, 79)( 12, 80)( 13, 78)( 14, 77)( 15, 75)
( 16, 76)( 17, 65)( 18, 66)( 19, 71)( 20, 72)( 21, 70)( 22, 69)( 23, 67)
( 24, 68)( 25, 57)( 26, 58)( 27, 63)( 28, 64)( 29, 62)( 30, 61)( 31, 59)
( 32, 60)( 33, 49)( 34, 50)( 35, 55)( 36, 56)( 37, 54)( 38, 53)( 39, 51)
( 40, 52)( 83, 87)( 84, 88)( 85, 86)( 89,113)( 90,114)( 91,119)( 92,120)
( 93,118)( 94,117)( 95,115)( 96,116)( 97,105)( 98,106)( 99,111)(100,112)
(101,110)(102,109)(103,107)(104,108)(121,162)(122,161)(123,168)(124,167)
(125,165)(126,166)(127,164)(128,163)(129,194)(130,193)(131,200)(132,199)
(133,197)(134,198)(135,196)(136,195)(137,186)(138,185)(139,192)(140,191)
(141,189)(142,190)(143,188)(144,187)(145,178)(146,177)(147,184)(148,183)
(149,181)(150,182)(151,180)(152,179)(153,170)(154,169)(155,176)(156,175)
(157,173)(158,174)(159,172)(160,171)(201,202)(203,208)(204,207)(209,234)
(210,233)(211,240)(212,239)(213,237)(214,238)(215,236)(216,235)(217,226)
(218,225)(219,232)(220,231)(221,229)(222,230)(223,228)(224,227)(241,402)
(242,401)(243,408)(244,407)(245,405)(246,406)(247,404)(248,403)(249,434)
(250,433)(251,440)(252,439)(253,437)(254,438)(255,436)(256,435)(257,426)
(258,425)(259,432)(260,431)(261,429)(262,430)(263,428)(264,427)(265,418)
(266,417)(267,424)(268,423)(269,421)(270,422)(271,420)(272,419)(273,410)
(274,409)(275,416)(276,415)(277,413)(278,414)(279,412)(280,411)(281,362)
(282,361)(283,368)(284,367)(285,365)(286,366)(287,364)(288,363)(289,394)
(290,393)(291,400)(292,399)(293,397)(294,398)(295,396)(296,395)(297,386)
(298,385)(299,392)(300,391)(301,389)(302,390)(303,388)(304,387)(305,378)
(306,377)(307,384)(308,383)(309,381)(310,382)(311,380)(312,379)(313,370)
(314,369)(315,376)(316,375)(317,373)(318,374)(319,372)(320,371)(321,442)
(322,441)(323,448)(324,447)(325,445)(326,446)(327,444)(328,443)(329,474)
(330,473)(331,480)(332,479)(333,477)(334,478)(335,476)(336,475)(337,466)
(338,465)(339,472)(340,471)(341,469)(342,470)(343,468)(344,467)(345,458)
(346,457)(347,464)(348,463)(349,461)(350,462)(351,460)(352,459)(353,450)
(354,449)(355,456)(356,455)(357,453)(358,454)(359,452)(360,451);
s2 := Sym(480)!(  1,251)(  2,252)(  3,250)(  4,249)(  5,255)(  6,256)(  7,254)
(  8,253)(  9,243)( 10,244)( 11,242)( 12,241)( 13,247)( 14,248)( 15,246)
( 16,245)( 17,275)( 18,276)( 19,274)( 20,273)( 21,279)( 22,280)( 23,278)
( 24,277)( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)
( 32,269)( 33,259)( 34,260)( 35,258)( 36,257)( 37,263)( 38,264)( 39,262)
( 40,261)( 41,291)( 42,292)( 43,290)( 44,289)( 45,295)( 46,296)( 47,294)
( 48,293)( 49,283)( 50,284)( 51,282)( 52,281)( 53,287)( 54,288)( 55,286)
( 56,285)( 57,315)( 58,316)( 59,314)( 60,313)( 61,319)( 62,320)( 63,318)
( 64,317)( 65,307)( 66,308)( 67,306)( 68,305)( 69,311)( 70,312)( 71,310)
( 72,309)( 73,299)( 74,300)( 75,298)( 76,297)( 77,303)( 78,304)( 79,302)
( 80,301)( 81,331)( 82,332)( 83,330)( 84,329)( 85,335)( 86,336)( 87,334)
( 88,333)( 89,323)( 90,324)( 91,322)( 92,321)( 93,327)( 94,328)( 95,326)
( 96,325)( 97,355)( 98,356)( 99,354)(100,353)(101,359)(102,360)(103,358)
(104,357)(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)
(112,349)(113,339)(114,340)(115,338)(116,337)(117,343)(118,344)(119,342)
(120,341)(121,372)(122,371)(123,369)(124,370)(125,376)(126,375)(127,373)
(128,374)(129,364)(130,363)(131,361)(132,362)(133,368)(134,367)(135,365)
(136,366)(137,396)(138,395)(139,393)(140,394)(141,400)(142,399)(143,397)
(144,398)(145,388)(146,387)(147,385)(148,386)(149,392)(150,391)(151,389)
(152,390)(153,380)(154,379)(155,377)(156,378)(157,384)(158,383)(159,381)
(160,382)(161,412)(162,411)(163,409)(164,410)(165,416)(166,415)(167,413)
(168,414)(169,404)(170,403)(171,401)(172,402)(173,408)(174,407)(175,405)
(176,406)(177,436)(178,435)(179,433)(180,434)(181,440)(182,439)(183,437)
(184,438)(185,428)(186,427)(187,425)(188,426)(189,432)(190,431)(191,429)
(192,430)(193,420)(194,419)(195,417)(196,418)(197,424)(198,423)(199,421)
(200,422)(201,452)(202,451)(203,449)(204,450)(205,456)(206,455)(207,453)
(208,454)(209,444)(210,443)(211,441)(212,442)(213,448)(214,447)(215,445)
(216,446)(217,476)(218,475)(219,473)(220,474)(221,480)(222,479)(223,477)
(224,478)(225,468)(226,467)(227,465)(228,466)(229,472)(230,471)(231,469)
(232,470)(233,460)(234,459)(235,457)(236,458)(237,464)(238,463)(239,461)
(240,462);
poly := sub<Sym(480)|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*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 >;

```
References : None.
to this polytope