Polytope of Type {8,15}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,15}*480
if this polytope has a name.
Group : SmallGroup(480,1032)
Rank : 3
Schlafli Type : {8,15}
Number of vertices, edges, etc : 16, 120, 30
Order of s0s1s2 : 60
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{8,15,2} of size 960
{8,15,4} of size 1920
Vertex Figure Of :
{2,8,15} of size 960
{4,8,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,15}*240
4-fold quotients : {4,15}*120
5-fold quotients : {8,3}*96
8-fold quotients : {2,15}*60
10-fold quotients : {4,3}*48
20-fold quotients : {4,3}*24
24-fold quotients : {2,5}*20
40-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,30}*960b
3-fold covers : {8,45}*1440, {24,15}*1440
4-fold covers : {8,15}*1920a, {8,60}*1920e, {8,30}*1920f, {8,60}*1920h
Permutation Representation (GAP) :
s0 := ( 1,125)( 2,126)( 3,128)( 4,127)( 5,122)( 6,121)( 7,123)( 8,124)
( 9,133)( 10,134)( 11,136)( 12,135)( 13,130)( 14,129)( 15,131)( 16,132)
( 17,141)( 18,142)( 19,144)( 20,143)( 21,138)( 22,137)( 23,139)( 24,140)
( 25,149)( 26,150)( 27,152)( 28,151)( 29,146)( 30,145)( 31,147)( 32,148)
( 33,157)( 34,158)( 35,160)( 36,159)( 37,154)( 38,153)( 39,155)( 40,156)
( 41,165)( 42,166)( 43,168)( 44,167)( 45,162)( 46,161)( 47,163)( 48,164)
( 49,173)( 50,174)( 51,176)( 52,175)( 53,170)( 54,169)( 55,171)( 56,172)
( 57,181)( 58,182)( 59,184)( 60,183)( 61,178)( 62,177)( 63,179)( 64,180)
( 65,189)( 66,190)( 67,192)( 68,191)( 69,186)( 70,185)( 71,187)( 72,188)
( 73,197)( 74,198)( 75,200)( 76,199)( 77,194)( 78,193)( 79,195)( 80,196)
( 81,205)( 82,206)( 83,208)( 84,207)( 85,202)( 86,201)( 87,203)( 88,204)
( 89,213)( 90,214)( 91,216)( 92,215)( 93,210)( 94,209)( 95,211)( 96,212)
( 97,221)( 98,222)( 99,224)(100,223)(101,218)(102,217)(103,219)(104,220)
(105,229)(106,230)(107,232)(108,231)(109,226)(110,225)(111,227)(112,228)
(113,237)(114,238)(115,240)(116,239)(117,234)(118,233)(119,235)(120,236);;
s1 := ( 3, 5)( 4, 6)( 7, 8)( 9, 33)( 10, 34)( 11, 37)( 12, 38)( 13, 35)
( 14, 36)( 15, 40)( 16, 39)( 17, 25)( 18, 26)( 19, 29)( 20, 30)( 21, 27)
( 22, 28)( 23, 32)( 24, 31)( 41, 81)( 42, 82)( 43, 85)( 44, 86)( 45, 83)
( 46, 84)( 47, 88)( 48, 87)( 49,113)( 50,114)( 51,117)( 52,118)( 53,115)
( 54,116)( 55,120)( 56,119)( 57,105)( 58,106)( 59,109)( 60,110)( 61,107)
( 62,108)( 63,112)( 64,111)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)
( 70,100)( 71,104)( 72,103)( 73, 89)( 74, 90)( 75, 93)( 76, 94)( 77, 91)
( 78, 92)( 79, 96)( 80, 95)(121,122)(123,126)(124,125)(129,154)(130,153)
(131,158)(132,157)(133,156)(134,155)(135,159)(136,160)(137,146)(138,145)
(139,150)(140,149)(141,148)(142,147)(143,151)(144,152)(161,202)(162,201)
(163,206)(164,205)(165,204)(166,203)(167,207)(168,208)(169,234)(170,233)
(171,238)(172,237)(173,236)(174,235)(175,239)(176,240)(177,226)(178,225)
(179,230)(180,229)(181,228)(182,227)(183,231)(184,232)(185,218)(186,217)
(187,222)(188,221)(189,220)(190,219)(191,223)(192,224)(193,210)(194,209)
(195,214)(196,213)(197,212)(198,211)(199,215)(200,216);;
s2 := ( 1, 89)( 2, 90)( 3, 96)( 4, 95)( 5, 94)( 6, 93)( 7, 92)( 8, 91)
( 9, 81)( 10, 82)( 11, 88)( 12, 87)( 13, 86)( 14, 85)( 15, 84)( 16, 83)
( 17,113)( 18,114)( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)( 24,115)
( 25,105)( 26,106)( 27,112)( 28,111)( 29,110)( 30,109)( 31,108)( 32,107)
( 33, 97)( 34, 98)( 35,104)( 36,103)( 37,102)( 38,101)( 39,100)( 40, 99)
( 41, 49)( 42, 50)( 43, 56)( 44, 55)( 45, 54)( 46, 53)( 47, 52)( 48, 51)
( 57, 73)( 58, 74)( 59, 80)( 60, 79)( 61, 78)( 62, 77)( 63, 76)( 64, 75)
( 67, 72)( 68, 71)( 69, 70)(121,210)(122,209)(123,215)(124,216)(125,213)
(126,214)(127,211)(128,212)(129,202)(130,201)(131,207)(132,208)(133,205)
(134,206)(135,203)(136,204)(137,234)(138,233)(139,239)(140,240)(141,237)
(142,238)(143,235)(144,236)(145,226)(146,225)(147,231)(148,232)(149,229)
(150,230)(151,227)(152,228)(153,218)(154,217)(155,223)(156,224)(157,221)
(158,222)(159,219)(160,220)(161,170)(162,169)(163,175)(164,176)(165,173)
(166,174)(167,171)(168,172)(177,194)(178,193)(179,199)(180,200)(181,197)
(182,198)(183,195)(184,196)(185,186)(187,191)(188,192);;
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,
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*s2*s1*s2*s1*s2*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)!( 1,125)( 2,126)( 3,128)( 4,127)( 5,122)( 6,121)( 7,123)
( 8,124)( 9,133)( 10,134)( 11,136)( 12,135)( 13,130)( 14,129)( 15,131)
( 16,132)( 17,141)( 18,142)( 19,144)( 20,143)( 21,138)( 22,137)( 23,139)
( 24,140)( 25,149)( 26,150)( 27,152)( 28,151)( 29,146)( 30,145)( 31,147)
( 32,148)( 33,157)( 34,158)( 35,160)( 36,159)( 37,154)( 38,153)( 39,155)
( 40,156)( 41,165)( 42,166)( 43,168)( 44,167)( 45,162)( 46,161)( 47,163)
( 48,164)( 49,173)( 50,174)( 51,176)( 52,175)( 53,170)( 54,169)( 55,171)
( 56,172)( 57,181)( 58,182)( 59,184)( 60,183)( 61,178)( 62,177)( 63,179)
( 64,180)( 65,189)( 66,190)( 67,192)( 68,191)( 69,186)( 70,185)( 71,187)
( 72,188)( 73,197)( 74,198)( 75,200)( 76,199)( 77,194)( 78,193)( 79,195)
( 80,196)( 81,205)( 82,206)( 83,208)( 84,207)( 85,202)( 86,201)( 87,203)
( 88,204)( 89,213)( 90,214)( 91,216)( 92,215)( 93,210)( 94,209)( 95,211)
( 96,212)( 97,221)( 98,222)( 99,224)(100,223)(101,218)(102,217)(103,219)
(104,220)(105,229)(106,230)(107,232)(108,231)(109,226)(110,225)(111,227)
(112,228)(113,237)(114,238)(115,240)(116,239)(117,234)(118,233)(119,235)
(120,236);
s1 := Sym(240)!( 3, 5)( 4, 6)( 7, 8)( 9, 33)( 10, 34)( 11, 37)( 12, 38)
( 13, 35)( 14, 36)( 15, 40)( 16, 39)( 17, 25)( 18, 26)( 19, 29)( 20, 30)
( 21, 27)( 22, 28)( 23, 32)( 24, 31)( 41, 81)( 42, 82)( 43, 85)( 44, 86)
( 45, 83)( 46, 84)( 47, 88)( 48, 87)( 49,113)( 50,114)( 51,117)( 52,118)
( 53,115)( 54,116)( 55,120)( 56,119)( 57,105)( 58,106)( 59,109)( 60,110)
( 61,107)( 62,108)( 63,112)( 64,111)( 65, 97)( 66, 98)( 67,101)( 68,102)
( 69, 99)( 70,100)( 71,104)( 72,103)( 73, 89)( 74, 90)( 75, 93)( 76, 94)
( 77, 91)( 78, 92)( 79, 96)( 80, 95)(121,122)(123,126)(124,125)(129,154)
(130,153)(131,158)(132,157)(133,156)(134,155)(135,159)(136,160)(137,146)
(138,145)(139,150)(140,149)(141,148)(142,147)(143,151)(144,152)(161,202)
(162,201)(163,206)(164,205)(165,204)(166,203)(167,207)(168,208)(169,234)
(170,233)(171,238)(172,237)(173,236)(174,235)(175,239)(176,240)(177,226)
(178,225)(179,230)(180,229)(181,228)(182,227)(183,231)(184,232)(185,218)
(186,217)(187,222)(188,221)(189,220)(190,219)(191,223)(192,224)(193,210)
(194,209)(195,214)(196,213)(197,212)(198,211)(199,215)(200,216);
s2 := Sym(240)!( 1, 89)( 2, 90)( 3, 96)( 4, 95)( 5, 94)( 6, 93)( 7, 92)
( 8, 91)( 9, 81)( 10, 82)( 11, 88)( 12, 87)( 13, 86)( 14, 85)( 15, 84)
( 16, 83)( 17,113)( 18,114)( 19,120)( 20,119)( 21,118)( 22,117)( 23,116)
( 24,115)( 25,105)( 26,106)( 27,112)( 28,111)( 29,110)( 30,109)( 31,108)
( 32,107)( 33, 97)( 34, 98)( 35,104)( 36,103)( 37,102)( 38,101)( 39,100)
( 40, 99)( 41, 49)( 42, 50)( 43, 56)( 44, 55)( 45, 54)( 46, 53)( 47, 52)
( 48, 51)( 57, 73)( 58, 74)( 59, 80)( 60, 79)( 61, 78)( 62, 77)( 63, 76)
( 64, 75)( 67, 72)( 68, 71)( 69, 70)(121,210)(122,209)(123,215)(124,216)
(125,213)(126,214)(127,211)(128,212)(129,202)(130,201)(131,207)(132,208)
(133,205)(134,206)(135,203)(136,204)(137,234)(138,233)(139,239)(140,240)
(141,237)(142,238)(143,235)(144,236)(145,226)(146,225)(147,231)(148,232)
(149,229)(150,230)(151,227)(152,228)(153,218)(154,217)(155,223)(156,224)
(157,221)(158,222)(159,219)(160,220)(161,170)(162,169)(163,175)(164,176)
(165,173)(166,174)(167,171)(168,172)(177,194)(178,193)(179,199)(180,200)
(181,197)(182,198)(183,195)(184,196)(185,186)(187,191)(188,192);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
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*s2*s1*s2*s1*s2*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