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# Polytope of Type {4,15}

Atlas Canonical Name : {4,15}*480
if this polytope has a name.
Group : SmallGroup(480,1195)
Rank : 3
Schlafli Type : {4,15}
Number of vertices, edges, etc : 16, 120, 60
Order of s0s1s2 : 15
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,15,2} of size 960
{4,15,4} of size 1920
Vertex Figure Of :
{2,4,15} of size 960
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,15}*960b, {8,15}*960c, {4,15}*960, {4,30}*960c, {4,30}*960d
3-fold covers : {4,45}*1440
4-fold covers : {4,60}*1920f, {4,60}*1920g, {8,15}*1920b, {8,30}*1920h, {8,30}*1920i, {8,30}*1920j, {8,30}*1920k, {4,15}*1920a, {4,30}*1920c, {4,60}*1920h, {4,60}*1920i, {8,15}*1920c, {4,15}*1920b, {4,30}*1920d
Permutation Representation (GAP) :
```s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)
(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)
(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)
(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)
(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240);;
s1 := (  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)( 18, 75)
( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)( 26, 71)
( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)( 34, 59)
( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)( 42, 55)
( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 81,161)( 82,171)
( 83,174)( 84,168)( 85,165)( 86,175)( 87,170)( 88,164)( 89,173)( 90,167)
( 91,162)( 92,172)( 93,169)( 94,163)( 95,166)( 96,176)( 97,225)( 98,235)
( 99,238)(100,232)(101,229)(102,239)(103,234)(104,228)(105,237)(106,231)
(107,226)(108,236)(109,233)(110,227)(111,230)(112,240)(113,209)(114,219)
(115,222)(116,216)(117,213)(118,223)(119,218)(120,212)(121,221)(122,215)
(123,210)(124,220)(125,217)(126,211)(127,214)(128,224)(129,193)(130,203)
(131,206)(132,200)(133,197)(134,207)(135,202)(136,196)(137,205)(138,199)
(139,194)(140,204)(141,201)(142,195)(143,198)(144,208)(145,177)(146,187)
(147,190)(148,184)(149,181)(150,191)(151,186)(152,180)(153,189)(154,183)
(155,178)(156,188)(157,185)(158,179)(159,182)(160,192);;
s2 := (  1,113)(  2,125)(  3,120)(  4,124)(  5,122)(  6,118)(  7,127)(  8,115)
(  9,121)( 10,117)( 11,128)( 12,116)( 13,114)( 14,126)( 15,119)( 16,123)
( 17, 97)( 18,109)( 19,104)( 20,108)( 21,106)( 22,102)( 23,111)( 24, 99)
( 25,105)( 26,101)( 27,112)( 28,100)( 29, 98)( 30,110)( 31,103)( 32,107)
( 33, 81)( 34, 93)( 35, 88)( 36, 92)( 37, 90)( 38, 86)( 39, 95)( 40, 83)
( 41, 89)( 42, 85)( 43, 96)( 44, 84)( 45, 82)( 46, 94)( 47, 87)( 48, 91)
( 49,145)( 50,157)( 51,152)( 52,156)( 53,154)( 54,150)( 55,159)( 56,147)
( 57,153)( 58,149)( 59,160)( 60,148)( 61,146)( 62,158)( 63,151)( 64,155)
( 65,129)( 66,141)( 67,136)( 68,140)( 69,138)( 70,134)( 71,143)( 72,131)
( 73,137)( 74,133)( 75,144)( 76,132)( 77,130)( 78,142)( 79,135)( 80,139)
(161,193)(162,205)(163,200)(164,204)(165,202)(166,198)(167,207)(168,195)
(169,201)(170,197)(171,208)(172,196)(173,194)(174,206)(175,199)(176,203)
(178,189)(179,184)(180,188)(181,186)(183,191)(187,192)(209,225)(210,237)
(211,232)(212,236)(213,234)(214,230)(215,239)(216,227)(217,233)(218,229)
(219,240)(220,228)(221,226)(222,238)(223,231)(224,235);;
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*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*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(240)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)
(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)
(184,192)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)
(200,208)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)
(216,224)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)
(232,240);
s1 := Sym(240)!(  2, 11)(  3, 14)(  4,  8)(  6, 15)(  7, 10)(  9, 13)( 17, 65)
( 18, 75)( 19, 78)( 20, 72)( 21, 69)( 22, 79)( 23, 74)( 24, 68)( 25, 77)
( 26, 71)( 27, 66)( 28, 76)( 29, 73)( 30, 67)( 31, 70)( 32, 80)( 33, 49)
( 34, 59)( 35, 62)( 36, 56)( 37, 53)( 38, 63)( 39, 58)( 40, 52)( 41, 61)
( 42, 55)( 43, 50)( 44, 60)( 45, 57)( 46, 51)( 47, 54)( 48, 64)( 81,161)
( 82,171)( 83,174)( 84,168)( 85,165)( 86,175)( 87,170)( 88,164)( 89,173)
( 90,167)( 91,162)( 92,172)( 93,169)( 94,163)( 95,166)( 96,176)( 97,225)
( 98,235)( 99,238)(100,232)(101,229)(102,239)(103,234)(104,228)(105,237)
(106,231)(107,226)(108,236)(109,233)(110,227)(111,230)(112,240)(113,209)
(114,219)(115,222)(116,216)(117,213)(118,223)(119,218)(120,212)(121,221)
(122,215)(123,210)(124,220)(125,217)(126,211)(127,214)(128,224)(129,193)
(130,203)(131,206)(132,200)(133,197)(134,207)(135,202)(136,196)(137,205)
(138,199)(139,194)(140,204)(141,201)(142,195)(143,198)(144,208)(145,177)
(146,187)(147,190)(148,184)(149,181)(150,191)(151,186)(152,180)(153,189)
(154,183)(155,178)(156,188)(157,185)(158,179)(159,182)(160,192);
s2 := Sym(240)!(  1,113)(  2,125)(  3,120)(  4,124)(  5,122)(  6,118)(  7,127)
(  8,115)(  9,121)( 10,117)( 11,128)( 12,116)( 13,114)( 14,126)( 15,119)
( 16,123)( 17, 97)( 18,109)( 19,104)( 20,108)( 21,106)( 22,102)( 23,111)
( 24, 99)( 25,105)( 26,101)( 27,112)( 28,100)( 29, 98)( 30,110)( 31,103)
( 32,107)( 33, 81)( 34, 93)( 35, 88)( 36, 92)( 37, 90)( 38, 86)( 39, 95)
( 40, 83)( 41, 89)( 42, 85)( 43, 96)( 44, 84)( 45, 82)( 46, 94)( 47, 87)
( 48, 91)( 49,145)( 50,157)( 51,152)( 52,156)( 53,154)( 54,150)( 55,159)
( 56,147)( 57,153)( 58,149)( 59,160)( 60,148)( 61,146)( 62,158)( 63,151)
( 64,155)( 65,129)( 66,141)( 67,136)( 68,140)( 69,138)( 70,134)( 71,143)
( 72,131)( 73,137)( 74,133)( 75,144)( 76,132)( 77,130)( 78,142)( 79,135)
( 80,139)(161,193)(162,205)(163,200)(164,204)(165,202)(166,198)(167,207)
(168,195)(169,201)(170,197)(171,208)(172,196)(173,194)(174,206)(175,199)
(176,203)(178,189)(179,184)(180,188)(181,186)(183,191)(187,192)(209,225)
(210,237)(211,232)(212,236)(213,234)(214,230)(215,239)(216,227)(217,233)
(218,229)(219,240)(220,228)(221,226)(222,238)(223,231)(224,235);
poly := sub<Sym(240)|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*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*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