Polytope of Type {15,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,4,4}*960b
if this polytope has a name.
Group : SmallGroup(960,11092)
Rank : 4
Schlafli Type : {15,4,4}
Number of vertices, edges, etc : 30, 60, 16, 4
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{15,4,4,2} of size 1920
Vertex Figure Of :
{2,15,4,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {15,4,2}*480
4-fold quotients : {15,2,4}*240, {15,4,2}*240
5-fold quotients : {3,4,4}*192b
8-fold quotients : {15,2,2}*120
10-fold quotients : {3,4,2}*96
12-fold quotients : {5,2,4}*80
20-fold quotients : {3,2,4}*48, {3,4,2}*48
24-fold quotients : {5,2,2}*40
40-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {15,8,4}*1920, {15,4,8}*1920, {30,4,4}*1920d
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2> of order 2.
4 facets:
4 of 2-fold non-regular quotient of {15,4}*240
20 vertex figures:
10 of {2,4}*16
10 of {4,4}*32
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 17)( 6, 19)( 7, 18)( 8, 20)( 9, 13)( 10, 15)( 11, 14)( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)( 27, 58)( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)( 66, 79)( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)( 82,103)( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)( 90,115)( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(122,123)(125,137)(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(141,161)(142,163)(143,162)(144,164)(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)(152,176)(153,169)(154,171)(155,170)(156,172)(157,165)(158,167)(159,166)(160,168)(182,183)(185,197)(186,199)(187,198)(188,200)(189,193)(190,195)(191,194)(192,196)(201,221)(202,223)(203,222)(204,224)(205,237)(206,239)(207,238)(208,240)(209,233)(210,235)(211,234)(212,236)(213,229)(214,231)(215,230)(216,232)(217,225)(218,227)(219,226)(220,228);;
s1 := ( 1, 25)( 2, 26)( 3, 28)( 4, 27)( 5, 21)( 6, 22)( 7, 24)( 8, 23)( 9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 55, 56)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 97)( 70, 98)( 71,100)( 72, 99)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)(101,105)(102,106)(103,108)(104,107)(109,117)(110,118)(111,120)(112,119)(115,116)(121,145)(122,146)(123,148)(124,147)(125,141)(126,142)(127,144)(128,143)(129,157)(130,158)(131,160)(132,159)(133,153)(134,154)(135,156)(136,155)(137,149)(138,150)(139,152)(140,151)(161,165)(162,166)(163,168)(164,167)(169,177)(170,178)(171,180)(172,179)(175,176)(181,205)(182,206)(183,208)(184,207)(185,201)(186,202)(187,204)(188,203)(189,217)(190,218)(191,220)(192,219)(193,213)(194,214)(195,216)(196,215)(197,209)(198,210)(199,212)(200,211)(221,225)(222,226)(223,228)(224,227)(229,237)(230,238)(231,240)(232,239)(235,236);;
s2 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,184)(122,183)(123,182)(124,181)(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)(132,189)(133,196)(134,195)(135,194)(136,193)(137,200)(138,199)(139,198)(140,197)(141,204)(142,203)(143,202)(144,201)(145,208)(146,207)(147,206)(148,205)(149,212)(150,211)(151,210)(152,209)(153,216)(154,215)(155,214)(156,213)(157,220)(158,219)(159,218)(160,217)(161,224)(162,223)(163,222)(164,221)(165,228)(166,227)(167,226)(168,225)(169,232)(170,231)(171,230)(172,229)(173,236)(174,235)(175,234)(176,233)(177,240)(178,239)(179,238)(180,237);;
s3 := ( 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);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 2, 3)( 5, 17)( 6, 19)( 7, 18)( 8, 20)( 9, 13)( 10, 15)( 11, 14)( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)( 27, 58)( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)( 35, 50)( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)( 66, 79)( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)( 82,103)( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)( 90,115)( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(122,123)(125,137)(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(141,161)(142,163)(143,162)(144,164)(145,177)(146,179)(147,178)(148,180)(149,173)(150,175)(151,174)(152,176)(153,169)(154,171)(155,170)(156,172)(157,165)(158,167)(159,166)(160,168)(182,183)(185,197)(186,199)(187,198)(188,200)(189,193)(190,195)(191,194)(192,196)(201,221)(202,223)(203,222)(204,224)(205,237)(206,239)(207,238)(208,240)(209,233)(210,235)(211,234)(212,236)(213,229)(214,231)(215,230)(216,232)(217,225)(218,227)(219,226)(220,228);
s1 := Sym(240)!( 1, 25)( 2, 26)( 3, 28)( 4, 27)( 5, 21)( 6, 22)( 7, 24)( 8, 23)( 9, 37)( 10, 38)( 11, 40)( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 55, 56)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 97)( 70, 98)( 71,100)( 72, 99)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)(101,105)(102,106)(103,108)(104,107)(109,117)(110,118)(111,120)(112,119)(115,116)(121,145)(122,146)(123,148)(124,147)(125,141)(126,142)(127,144)(128,143)(129,157)(130,158)(131,160)(132,159)(133,153)(134,154)(135,156)(136,155)(137,149)(138,150)(139,152)(140,151)(161,165)(162,166)(163,168)(164,167)(169,177)(170,178)(171,180)(172,179)(175,176)(181,205)(182,206)(183,208)(184,207)(185,201)(186,202)(187,204)(188,203)(189,217)(190,218)(191,220)(192,219)(193,213)(194,214)(195,216)(196,215)(197,209)(198,210)(199,212)(200,211)(221,225)(222,226)(223,228)(224,227)(229,237)(230,238)(231,240)(232,239)(235,236);
s2 := Sym(240)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)(117,120)(118,119)(121,184)(122,183)(123,182)(124,181)(125,188)(126,187)(127,186)(128,185)(129,192)(130,191)(131,190)(132,189)(133,196)(134,195)(135,194)(136,193)(137,200)(138,199)(139,198)(140,197)(141,204)(142,203)(143,202)(144,201)(145,208)(146,207)(147,206)(148,205)(149,212)(150,211)(151,210)(152,209)(153,216)(154,215)(155,214)(156,213)(157,220)(158,219)(159,218)(160,217)(161,224)(162,223)(163,222)(164,221)(165,228)(166,227)(167,226)(168,225)(169,232)(170,231)(171,230)(172,229)(173,236)(174,235)(175,234)(176,233)(177,240)(178,239)(179,238)(180,237);
s3 := 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);
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope