# Polytope of Type {12,15}

Atlas Canonical Name : {12,15}*480
if this polytope has a name.
Group : SmallGroup(480,970)
Rank : 3
Schlafli Type : {12,15}
Number of vertices, edges, etc : 16, 120, 20
Order of s0s1s2 : 40
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,15,2} of size 960
Vertex Figure Of :
{2,12,15} of size 960
{4,12,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,15}*240
5-fold quotients : {12,3}*96
10-fold quotients : {6,3}*48
20-fold quotients : {3,3}*24
24-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,30}*960a
3-fold covers : {12,15}*1440c
4-fold covers : {12,15}*1920, {12,60}*1920a, {12,60}*1920b, {12,30}*1920
Permutation Representation (GAP) :
s0 := (  1,121)(  2,122)(  3,125)(  4,126)(  5,123)(  6,124)(  7,128)(  8,127)
(  9,129)( 10,130)( 11,133)( 12,134)( 13,131)( 14,132)( 15,136)( 16,135)
( 17,137)( 18,138)( 19,141)( 20,142)( 21,139)( 22,140)( 23,144)( 24,143)
( 25,145)( 26,146)( 27,149)( 28,150)( 29,147)( 30,148)( 31,152)( 32,151)
( 33,153)( 34,154)( 35,157)( 36,158)( 37,155)( 38,156)( 39,160)( 40,159)
( 41,201)( 42,202)( 43,205)( 44,206)( 45,203)( 46,204)( 47,208)( 48,207)
( 49,209)( 50,210)( 51,213)( 52,214)( 53,211)( 54,212)( 55,216)( 56,215)
( 57,217)( 58,218)( 59,221)( 60,222)( 61,219)( 62,220)( 63,224)( 64,223)
( 65,225)( 66,226)( 67,229)( 68,230)( 69,227)( 70,228)( 71,232)( 72,231)
( 73,233)( 74,234)( 75,237)( 76,238)( 77,235)( 78,236)( 79,240)( 80,239)
( 81,161)( 82,162)( 83,165)( 84,166)( 85,163)( 86,164)( 87,168)( 88,167)
( 89,169)( 90,170)( 91,173)( 92,174)( 93,171)( 94,172)( 95,176)( 96,175)
( 97,177)( 98,178)( 99,181)(100,182)(101,179)(102,180)(103,184)(104,183)
(105,185)(106,186)(107,189)(108,190)(109,187)(110,188)(111,192)(112,191)
(113,193)(114,194)(115,197)(116,198)(117,195)(118,196)(119,200)(120,199)
(241,361)(242,362)(243,365)(244,366)(245,363)(246,364)(247,368)(248,367)
(249,369)(250,370)(251,373)(252,374)(253,371)(254,372)(255,376)(256,375)
(257,377)(258,378)(259,381)(260,382)(261,379)(262,380)(263,384)(264,383)
(265,385)(266,386)(267,389)(268,390)(269,387)(270,388)(271,392)(272,391)
(273,393)(274,394)(275,397)(276,398)(277,395)(278,396)(279,400)(280,399)
(281,441)(282,442)(283,445)(284,446)(285,443)(286,444)(287,448)(288,447)
(289,449)(290,450)(291,453)(292,454)(293,451)(294,452)(295,456)(296,455)
(297,457)(298,458)(299,461)(300,462)(301,459)(302,460)(303,464)(304,463)
(305,465)(306,466)(307,469)(308,470)(309,467)(310,468)(311,472)(312,471)
(313,473)(314,474)(315,477)(316,478)(317,475)(318,476)(319,480)(320,479)
(321,401)(322,402)(323,405)(324,406)(325,403)(326,404)(327,408)(328,407)
(329,409)(330,410)(331,413)(332,414)(333,411)(334,412)(335,416)(336,415)
(337,417)(338,418)(339,421)(340,422)(341,419)(342,420)(343,424)(344,423)
(345,425)(346,426)(347,429)(348,430)(349,427)(350,428)(351,432)(352,431)
(353,433)(354,434)(355,437)(356,438)(357,435)(358,436)(359,440)(360,439);;
s1 := (  1,401)(  2,402)(  3,404)(  4,403)(  5,407)(  6,408)(  7,405)(  8,406)
(  9,433)( 10,434)( 11,436)( 12,435)( 13,439)( 14,440)( 15,437)( 16,438)
( 17,425)( 18,426)( 19,428)( 20,427)( 21,431)( 22,432)( 23,429)( 24,430)
( 25,417)( 26,418)( 27,420)( 28,419)( 29,423)( 30,424)( 31,421)( 32,422)
( 33,409)( 34,410)( 35,412)( 36,411)( 37,415)( 38,416)( 39,413)( 40,414)
( 41,361)( 42,362)( 43,364)( 44,363)( 45,367)( 46,368)( 47,365)( 48,366)
( 49,393)( 50,394)( 51,396)( 52,395)( 53,399)( 54,400)( 55,397)( 56,398)
( 57,385)( 58,386)( 59,388)( 60,387)( 61,391)( 62,392)( 63,389)( 64,390)
( 65,377)( 66,378)( 67,380)( 68,379)( 69,383)( 70,384)( 71,381)( 72,382)
( 73,369)( 74,370)( 75,372)( 76,371)( 77,375)( 78,376)( 79,373)( 80,374)
( 81,441)( 82,442)( 83,444)( 84,443)( 85,447)( 86,448)( 87,445)( 88,446)
( 89,473)( 90,474)( 91,476)( 92,475)( 93,479)( 94,480)( 95,477)( 96,478)
( 97,465)( 98,466)( 99,468)(100,467)(101,471)(102,472)(103,469)(104,470)
(105,457)(106,458)(107,460)(108,459)(109,463)(110,464)(111,461)(112,462)
(113,449)(114,450)(115,452)(116,451)(117,455)(118,456)(119,453)(120,454)
(121,282)(122,281)(123,283)(124,284)(125,288)(126,287)(127,286)(128,285)
(129,314)(130,313)(131,315)(132,316)(133,320)(134,319)(135,318)(136,317)
(137,306)(138,305)(139,307)(140,308)(141,312)(142,311)(143,310)(144,309)
(145,298)(146,297)(147,299)(148,300)(149,304)(150,303)(151,302)(152,301)
(153,290)(154,289)(155,291)(156,292)(157,296)(158,295)(159,294)(160,293)
(161,242)(162,241)(163,243)(164,244)(165,248)(166,247)(167,246)(168,245)
(169,274)(170,273)(171,275)(172,276)(173,280)(174,279)(175,278)(176,277)
(177,266)(178,265)(179,267)(180,268)(181,272)(182,271)(183,270)(184,269)
(185,258)(186,257)(187,259)(188,260)(189,264)(190,263)(191,262)(192,261)
(193,250)(194,249)(195,251)(196,252)(197,256)(198,255)(199,254)(200,253)
(201,322)(202,321)(203,323)(204,324)(205,328)(206,327)(207,326)(208,325)
(209,354)(210,353)(211,355)(212,356)(213,360)(214,359)(215,358)(216,357)
(217,346)(218,345)(219,347)(220,348)(221,352)(222,351)(223,350)(224,349)
(225,338)(226,337)(227,339)(228,340)(229,344)(230,343)(231,342)(232,341)
(233,330)(234,329)(235,331)(236,332)(237,336)(238,335)(239,334)(240,333);;
s2 := (  1,375)(  2,376)(  3,372)(  4,371)(  5,373)(  6,374)(  7,369)(  8,370)
(  9,367)( 10,368)( 11,364)( 12,363)( 13,365)( 14,366)( 15,361)( 16,362)
( 17,399)( 18,400)( 19,396)( 20,395)( 21,397)( 22,398)( 23,393)( 24,394)
( 25,391)( 26,392)( 27,388)( 28,387)( 29,389)( 30,390)( 31,385)( 32,386)
( 33,383)( 34,384)( 35,380)( 36,379)( 37,381)( 38,382)( 39,377)( 40,378)
( 41,455)( 42,456)( 43,452)( 44,451)( 45,453)( 46,454)( 47,449)( 48,450)
( 49,447)( 50,448)( 51,444)( 52,443)( 53,445)( 54,446)( 55,441)( 56,442)
( 57,479)( 58,480)( 59,476)( 60,475)( 61,477)( 62,478)( 63,473)( 64,474)
( 65,471)( 66,472)( 67,468)( 68,467)( 69,469)( 70,470)( 71,465)( 72,466)
( 73,463)( 74,464)( 75,460)( 76,459)( 77,461)( 78,462)( 79,457)( 80,458)
( 81,415)( 82,416)( 83,412)( 84,411)( 85,413)( 86,414)( 87,409)( 88,410)
( 89,407)( 90,408)( 91,404)( 92,403)( 93,405)( 94,406)( 95,401)( 96,402)
( 97,439)( 98,440)( 99,436)(100,435)(101,437)(102,438)(103,433)(104,434)
(105,431)(106,432)(107,428)(108,427)(109,429)(110,430)(111,425)(112,426)
(113,423)(114,424)(115,420)(116,419)(117,421)(118,422)(119,417)(120,418)
(121,256)(122,255)(123,251)(124,252)(125,254)(126,253)(127,250)(128,249)
(129,248)(130,247)(131,243)(132,244)(133,246)(134,245)(135,242)(136,241)
(137,280)(138,279)(139,275)(140,276)(141,278)(142,277)(143,274)(144,273)
(145,272)(146,271)(147,267)(148,268)(149,270)(150,269)(151,266)(152,265)
(153,264)(154,263)(155,259)(156,260)(157,262)(158,261)(159,258)(160,257)
(161,336)(162,335)(163,331)(164,332)(165,334)(166,333)(167,330)(168,329)
(169,328)(170,327)(171,323)(172,324)(173,326)(174,325)(175,322)(176,321)
(177,360)(178,359)(179,355)(180,356)(181,358)(182,357)(183,354)(184,353)
(185,352)(186,351)(187,347)(188,348)(189,350)(190,349)(191,346)(192,345)
(193,344)(194,343)(195,339)(196,340)(197,342)(198,341)(199,338)(200,337)
(201,296)(202,295)(203,291)(204,292)(205,294)(206,293)(207,290)(208,289)
(209,288)(210,287)(211,283)(212,284)(213,286)(214,285)(215,282)(216,281)
(217,320)(218,319)(219,315)(220,316)(221,318)(222,317)(223,314)(224,313)
(225,312)(226,311)(227,307)(228,308)(229,310)(230,309)(231,306)(232,305)
(233,304)(234,303)(235,299)(236,300)(237,302)(238,301)(239,298)(240,297);;
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*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope