Polytope of Type {4,6,15}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,15}*960
if this polytope has a name.
Group : SmallGroup(960,10994)
Rank : 4
Schlafli Type : {4,6,15}
Number of vertices, edges, etc : 4, 16, 60, 20
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,15,2} of size 1920
Vertex Figure Of :
{2,4,6,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,15}*480
5-fold quotients : {4,6,3}*192
10-fold quotients : {2,6,3}*96
12-fold quotients : {4,2,5}*80
20-fold quotients : {2,3,3}*48
24-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,6,15}*1920, {4,12,15}*1920, {4,6,30}*1920
Permutation Representation (GAP) :
s0 := ( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)
( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)
( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)
( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)
( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)( 88,208)
( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)( 96,216)
( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)(104,224)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240);;
s1 := ( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120)(121,181)(122,183)(123,182)(124,184)(125,185)(126,187)
(127,186)(128,188)(129,189)(130,191)(131,190)(132,192)(133,193)(134,195)
(135,194)(136,196)(137,197)(138,199)(139,198)(140,200)(141,221)(142,223)
(143,222)(144,224)(145,225)(146,227)(147,226)(148,228)(149,229)(150,231)
(151,230)(152,232)(153,233)(154,235)(155,234)(156,236)(157,237)(158,239)
(159,238)(160,240)(161,201)(162,203)(163,202)(164,204)(165,205)(166,207)
(167,206)(168,208)(169,209)(170,211)(171,210)(172,212)(173,213)(174,215)
(175,214)(176,216)(177,217)(178,219)(179,218)(180,220);;
s2 := ( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 37)( 6, 38)( 7, 40)( 8, 39)
( 9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)( 47, 60)
( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61, 81)( 62, 82)( 63, 84)
( 64, 83)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69, 93)( 70, 94)( 71, 96)
( 72, 95)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)( 79, 88)
( 80, 87)(103,104)(105,117)(106,118)(107,120)(108,119)(109,113)(110,114)
(111,116)(112,115)(121,141)(122,142)(123,144)(124,143)(125,157)(126,158)
(127,160)(128,159)(129,153)(130,154)(131,156)(132,155)(133,149)(134,150)
(135,152)(136,151)(137,145)(138,146)(139,148)(140,147)(163,164)(165,177)
(166,178)(167,180)(168,179)(169,173)(170,174)(171,176)(172,175)(181,201)
(182,202)(183,204)(184,203)(185,217)(186,218)(187,220)(188,219)(189,213)
(190,214)(191,216)(192,215)(193,209)(194,210)(195,212)(196,211)(197,205)
(198,206)(199,208)(200,207)(223,224)(225,237)(226,238)(227,240)(228,239)
(229,233)(230,234)(231,236)(232,235);;
s3 := ( 1, 8)( 2, 6)( 3, 7)( 4, 5)( 9, 20)( 10, 18)( 11, 19)( 12, 17)
( 13, 16)( 21, 48)( 22, 46)( 23, 47)( 24, 45)( 25, 44)( 26, 42)( 27, 43)
( 28, 41)( 29, 60)( 30, 58)( 31, 59)( 32, 57)( 33, 56)( 34, 54)( 35, 55)
( 36, 53)( 37, 52)( 38, 50)( 39, 51)( 40, 49)( 61, 68)( 62, 66)( 63, 67)
( 64, 65)( 69, 80)( 70, 78)( 71, 79)( 72, 77)( 73, 76)( 81,108)( 82,106)
( 83,107)( 84,105)( 85,104)( 86,102)( 87,103)( 88,101)( 89,120)( 90,118)
( 91,119)( 92,117)( 93,116)( 94,114)( 95,115)( 96,113)( 97,112)( 98,110)
( 99,111)(100,109)(121,128)(122,126)(123,127)(124,125)(129,140)(130,138)
(131,139)(132,137)(133,136)(141,168)(142,166)(143,167)(144,165)(145,164)
(146,162)(147,163)(148,161)(149,180)(150,178)(151,179)(152,177)(153,176)
(154,174)(155,175)(156,173)(157,172)(158,170)(159,171)(160,169)(181,188)
(182,186)(183,187)(184,185)(189,200)(190,198)(191,199)(192,197)(193,196)
(201,228)(202,226)(203,227)(204,225)(205,224)(206,222)(207,223)(208,221)
(209,240)(210,238)(211,239)(212,237)(213,236)(214,234)(215,235)(216,233)
(217,232)(218,230)(219,231)(220,229);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)
( 8,128)( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)
( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)
( 24,144)( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)
( 32,152)( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)
( 80,200)( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)
( 88,208)( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)
( 96,216)( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)
(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)
(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)
(120,240);
s1 := Sym(240)!( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120)(121,181)(122,183)(123,182)(124,184)(125,185)
(126,187)(127,186)(128,188)(129,189)(130,191)(131,190)(132,192)(133,193)
(134,195)(135,194)(136,196)(137,197)(138,199)(139,198)(140,200)(141,221)
(142,223)(143,222)(144,224)(145,225)(146,227)(147,226)(148,228)(149,229)
(150,231)(151,230)(152,232)(153,233)(154,235)(155,234)(156,236)(157,237)
(158,239)(159,238)(160,240)(161,201)(162,203)(163,202)(164,204)(165,205)
(166,207)(167,206)(168,208)(169,209)(170,211)(171,210)(172,212)(173,213)
(174,215)(175,214)(176,216)(177,217)(178,219)(179,218)(180,220);
s2 := Sym(240)!( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 37)( 6, 38)( 7, 40)
( 8, 39)( 9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)
( 47, 60)( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61, 81)( 62, 82)
( 63, 84)( 64, 83)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69, 93)( 70, 94)
( 71, 96)( 72, 95)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 85)( 78, 86)
( 79, 88)( 80, 87)(103,104)(105,117)(106,118)(107,120)(108,119)(109,113)
(110,114)(111,116)(112,115)(121,141)(122,142)(123,144)(124,143)(125,157)
(126,158)(127,160)(128,159)(129,153)(130,154)(131,156)(132,155)(133,149)
(134,150)(135,152)(136,151)(137,145)(138,146)(139,148)(140,147)(163,164)
(165,177)(166,178)(167,180)(168,179)(169,173)(170,174)(171,176)(172,175)
(181,201)(182,202)(183,204)(184,203)(185,217)(186,218)(187,220)(188,219)
(189,213)(190,214)(191,216)(192,215)(193,209)(194,210)(195,212)(196,211)
(197,205)(198,206)(199,208)(200,207)(223,224)(225,237)(226,238)(227,240)
(228,239)(229,233)(230,234)(231,236)(232,235);
s3 := Sym(240)!( 1, 8)( 2, 6)( 3, 7)( 4, 5)( 9, 20)( 10, 18)( 11, 19)
( 12, 17)( 13, 16)( 21, 48)( 22, 46)( 23, 47)( 24, 45)( 25, 44)( 26, 42)
( 27, 43)( 28, 41)( 29, 60)( 30, 58)( 31, 59)( 32, 57)( 33, 56)( 34, 54)
( 35, 55)( 36, 53)( 37, 52)( 38, 50)( 39, 51)( 40, 49)( 61, 68)( 62, 66)
( 63, 67)( 64, 65)( 69, 80)( 70, 78)( 71, 79)( 72, 77)( 73, 76)( 81,108)
( 82,106)( 83,107)( 84,105)( 85,104)( 86,102)( 87,103)( 88,101)( 89,120)
( 90,118)( 91,119)( 92,117)( 93,116)( 94,114)( 95,115)( 96,113)( 97,112)
( 98,110)( 99,111)(100,109)(121,128)(122,126)(123,127)(124,125)(129,140)
(130,138)(131,139)(132,137)(133,136)(141,168)(142,166)(143,167)(144,165)
(145,164)(146,162)(147,163)(148,161)(149,180)(150,178)(151,179)(152,177)
(153,176)(154,174)(155,175)(156,173)(157,172)(158,170)(159,171)(160,169)
(181,188)(182,186)(183,187)(184,185)(189,200)(190,198)(191,199)(192,197)
(193,196)(201,228)(202,226)(203,227)(204,225)(205,224)(206,222)(207,223)
(208,221)(209,240)(210,238)(211,239)(212,237)(213,236)(214,234)(215,235)
(216,233)(217,232)(218,230)(219,231)(220,229);
poly := sub<Sym(240)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3 >;
References : None.
to this polytope