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# Polytope of Type {8,15}

Atlas Canonical Name : {8,15}*960b
if this polytope has a name.
Group : SmallGroup(960,10999)
Rank : 3
Schlafli Type : {8,15}
Number of vertices, edges, etc : 32, 240, 60
Order of s0s1s2 : 15
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,15,2} of size 1920
Vertex Figure Of :
{2,8,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,15}*480
3-fold quotients : {8,5}*320b
6-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,15}*1920b, {8,30}*1920i, {8,30}*1920k
Permutation Representation (GAP) :
```s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 14)(  6, 13)(  7, 16)(  8, 15)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 23, 32)( 24, 31)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)( 40, 47)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 64)( 56, 63)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 78)( 70, 77)( 71, 80)( 72, 79)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 96)( 88, 95)
( 97,105)( 98,106)( 99,107)(100,108)(101,110)(102,109)(103,112)(104,111)
(113,121)(114,122)(115,123)(116,124)(117,126)(118,125)(119,128)(120,127)
(129,137)(130,138)(131,139)(132,140)(133,142)(134,141)(135,144)(136,143)
(145,153)(146,154)(147,155)(148,156)(149,158)(150,157)(151,160)(152,159)
(161,169)(162,170)(163,171)(164,172)(165,174)(166,173)(167,176)(168,175)
(177,185)(178,186)(179,187)(180,188)(181,190)(182,189)(183,192)(184,191)
(193,201)(194,202)(195,203)(196,204)(197,206)(198,205)(199,208)(200,207)
(209,217)(210,218)(211,219)(212,220)(213,222)(214,221)(215,224)(216,223)
(225,233)(226,234)(227,235)(228,236)(229,238)(230,237)(231,240)(232,239)
(241,249)(242,250)(243,251)(244,252)(245,254)(246,253)(247,256)(248,255)
(257,265)(258,266)(259,267)(260,268)(261,270)(262,269)(263,272)(264,271)
(273,281)(274,282)(275,283)(276,284)(277,286)(278,285)(279,288)(280,287)
(289,297)(290,298)(291,299)(292,300)(293,302)(294,301)(295,304)(296,303)
(305,313)(306,314)(307,315)(308,316)(309,318)(310,317)(311,320)(312,319)
(321,329)(322,330)(323,331)(324,332)(325,334)(326,333)(327,336)(328,335)
(337,345)(338,346)(339,347)(340,348)(341,350)(342,349)(343,352)(344,351)
(353,361)(354,362)(355,363)(356,364)(357,366)(358,365)(359,368)(360,367)
(369,377)(370,378)(371,379)(372,380)(373,382)(374,381)(375,384)(376,383)
(385,393)(386,394)(387,395)(388,396)(389,398)(390,397)(391,400)(392,399)
(401,409)(402,410)(403,411)(404,412)(405,414)(406,413)(407,416)(408,415)
(417,425)(418,426)(419,427)(420,428)(421,430)(422,429)(423,432)(424,431)
(433,441)(434,442)(435,443)(436,444)(437,446)(438,445)(439,448)(440,447)
(449,457)(450,458)(451,459)(452,460)(453,462)(454,461)(455,464)(456,463)
(465,473)(466,474)(467,475)(468,476)(469,478)(470,477)(471,480)(472,479);;
s1 := (  3, 27)(  4, 28)(  5, 18)(  6, 17)(  7, 11)(  8, 12)(  9, 30)( 10, 29)
( 15, 23)( 16, 24)( 19, 32)( 20, 31)( 21, 22)( 25, 26)( 33,129)( 34,130)
( 35,155)( 36,156)( 37,146)( 38,145)( 39,139)( 40,140)( 41,158)( 42,157)
( 43,135)( 44,136)( 45,141)( 46,142)( 47,151)( 48,152)( 49,134)( 50,133)
( 51,160)( 52,159)( 53,150)( 54,149)( 55,143)( 56,144)( 57,154)( 58,153)
( 59,131)( 60,132)( 61,138)( 62,137)( 63,148)( 64,147)( 65, 97)( 66, 98)
( 67,123)( 68,124)( 69,114)( 70,113)( 71,107)( 72,108)( 73,126)( 74,125)
( 75,103)( 76,104)( 77,109)( 78,110)( 79,119)( 80,120)( 81,102)( 82,101)
( 83,128)( 84,127)( 85,118)( 86,117)( 87,111)( 88,112)( 89,122)( 90,121)
( 91, 99)( 92,100)( 93,106)( 94,105)( 95,116)( 96,115)(161,321)(162,322)
(163,347)(164,348)(165,338)(166,337)(167,331)(168,332)(169,350)(170,349)
(171,327)(172,328)(173,333)(174,334)(175,343)(176,344)(177,326)(178,325)
(179,352)(180,351)(181,342)(182,341)(183,335)(184,336)(185,346)(186,345)
(187,323)(188,324)(189,330)(190,329)(191,340)(192,339)(193,449)(194,450)
(195,475)(196,476)(197,466)(198,465)(199,459)(200,460)(201,478)(202,477)
(203,455)(204,456)(205,461)(206,462)(207,471)(208,472)(209,454)(210,453)
(211,480)(212,479)(213,470)(214,469)(215,463)(216,464)(217,474)(218,473)
(219,451)(220,452)(221,458)(222,457)(223,468)(224,467)(225,417)(226,418)
(227,443)(228,444)(229,434)(230,433)(231,427)(232,428)(233,446)(234,445)
(235,423)(236,424)(237,429)(238,430)(239,439)(240,440)(241,422)(242,421)
(243,448)(244,447)(245,438)(246,437)(247,431)(248,432)(249,442)(250,441)
(251,419)(252,420)(253,426)(254,425)(255,436)(256,435)(257,385)(258,386)
(259,411)(260,412)(261,402)(262,401)(263,395)(264,396)(265,414)(266,413)
(267,391)(268,392)(269,397)(270,398)(271,407)(272,408)(273,390)(274,389)
(275,416)(276,415)(277,406)(278,405)(279,399)(280,400)(281,410)(282,409)
(283,387)(284,388)(285,394)(286,393)(287,404)(288,403)(289,353)(290,354)
(291,379)(292,380)(293,370)(294,369)(295,363)(296,364)(297,382)(298,381)
(299,359)(300,360)(301,365)(302,366)(303,375)(304,376)(305,358)(306,357)
(307,384)(308,383)(309,374)(310,373)(311,367)(312,368)(313,378)(314,377)
(315,355)(316,356)(317,362)(318,361)(319,372)(320,371);;
s2 := (  1,289)(  2,290)(  3,292)(  4,291)(  5,295)(  6,296)(  7,293)(  8,294)
(  9,297)( 10,298)( 11,300)( 12,299)( 13,303)( 14,304)( 15,301)( 16,302)
( 17,314)( 18,313)( 19,315)( 20,316)( 21,320)( 22,319)( 23,318)( 24,317)
( 25,306)( 26,305)( 27,307)( 28,308)( 29,312)( 30,311)( 31,310)( 32,309)
( 33,257)( 34,258)( 35,260)( 36,259)( 37,263)( 38,264)( 39,261)( 40,262)
( 41,265)( 42,266)( 43,268)( 44,267)( 45,271)( 46,272)( 47,269)( 48,270)
( 49,282)( 50,281)( 51,283)( 52,284)( 53,288)( 54,287)( 55,286)( 56,285)
( 57,274)( 58,273)( 59,275)( 60,276)( 61,280)( 62,279)( 63,278)( 64,277)
( 65,225)( 66,226)( 67,228)( 68,227)( 69,231)( 70,232)( 71,229)( 72,230)
( 73,233)( 74,234)( 75,236)( 76,235)( 77,239)( 78,240)( 79,237)( 80,238)
( 81,250)( 82,249)( 83,251)( 84,252)( 85,256)( 86,255)( 87,254)( 88,253)
( 89,242)( 90,241)( 91,243)( 92,244)( 93,248)( 94,247)( 95,246)( 96,245)
( 97,193)( 98,194)( 99,196)(100,195)(101,199)(102,200)(103,197)(104,198)
(105,201)(106,202)(107,204)(108,203)(109,207)(110,208)(111,205)(112,206)
(113,218)(114,217)(115,219)(116,220)(117,224)(118,223)(119,222)(120,221)
(121,210)(122,209)(123,211)(124,212)(125,216)(126,215)(127,214)(128,213)
(129,161)(130,162)(131,164)(132,163)(133,167)(134,168)(135,165)(136,166)
(137,169)(138,170)(139,172)(140,171)(141,175)(142,176)(143,173)(144,174)
(145,186)(146,185)(147,187)(148,188)(149,192)(150,191)(151,190)(152,189)
(153,178)(154,177)(155,179)(156,180)(157,184)(158,183)(159,182)(160,181)
(321,449)(322,450)(323,452)(324,451)(325,455)(326,456)(327,453)(328,454)
(329,457)(330,458)(331,460)(332,459)(333,463)(334,464)(335,461)(336,462)
(337,474)(338,473)(339,475)(340,476)(341,480)(342,479)(343,478)(344,477)
(345,466)(346,465)(347,467)(348,468)(349,472)(350,471)(351,470)(352,469)
(353,417)(354,418)(355,420)(356,419)(357,423)(358,424)(359,421)(360,422)
(361,425)(362,426)(363,428)(364,427)(365,431)(366,432)(367,429)(368,430)
(369,442)(370,441)(371,443)(372,444)(373,448)(374,447)(375,446)(376,445)
(377,434)(378,433)(379,435)(380,436)(381,440)(382,439)(383,438)(384,437)
(387,388)(389,391)(390,392)(395,396)(397,399)(398,400)(401,410)(402,409)
(403,411)(404,412)(405,416)(406,415)(407,414)(408,413);;
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*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0,
s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope