Polytope of Type {4,4,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,15}*960b
if this polytope has a name.
Group : SmallGroup(960,11092)
Rank : 4
Schlafli Type : {4,4,15}
Number of vertices, edges, etc : 4, 16, 60, 30
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,15,2} of size 1920
Vertex Figure Of :
   {2,4,4,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,15}*480
   4-fold quotients : {4,2,15}*240, {2,4,15}*240
   5-fold quotients : {4,4,3}*192b
   8-fold quotients : {2,2,15}*120
   10-fold quotients : {2,4,3}*96
   12-fold quotients : {4,2,5}*80
   20-fold quotients : {4,2,3}*48, {2,4,3}*48
   24-fold quotients : {2,2,5}*40
   40-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8,15}*1920, {8,4,15}*1920, {4,4,30}*1920d
Permutation Representation (GAP) :

s0 := (121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)
(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)
(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)
(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)
(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)
(177,237)(178,238)(179,239)(180,240);;
s1 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)(  8,126)
(  9,131)( 10,132)( 11,129)( 12,130)( 13,135)( 14,136)( 15,133)( 16,134)
( 17,139)( 18,140)( 19,137)( 20,138)( 21,143)( 22,144)( 23,141)( 24,142)
( 25,147)( 26,148)( 27,145)( 28,146)( 29,151)( 30,152)( 31,149)( 32,150)
( 33,155)( 34,156)( 35,153)( 36,154)( 37,159)( 38,160)( 39,157)( 40,158)
( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,171)( 50,172)( 51,169)( 52,170)( 53,175)( 54,176)( 55,173)( 56,174)
( 57,179)( 58,180)( 59,177)( 60,178)( 61,183)( 62,184)( 63,181)( 64,182)
( 65,187)( 66,188)( 67,185)( 68,186)( 69,191)( 70,192)( 71,189)( 72,190)
( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)( 80,198)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,207)( 86,208)( 87,205)( 88,206)
( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)( 96,214)
( 97,219)( 98,220)( 99,217)(100,218)(101,223)(102,224)(103,221)(104,222)
(105,227)(106,228)(107,225)(108,226)(109,231)(110,232)(111,229)(112,230)
(113,235)(114,236)(115,233)(116,234)(117,239)(118,240)(119,237)(120,238);;
s2 := (  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);;
s3 := (  1, 45)(  2, 48)(  3, 47)(  4, 46)(  5, 41)(  6, 44)(  7, 43)(  8, 42)
(  9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 53)( 14, 56)( 15, 55)( 16, 54)
( 17, 49)( 18, 52)( 19, 51)( 20, 50)( 21, 25)( 22, 28)( 23, 27)( 24, 26)
( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 34, 36)( 61,105)( 62,108)( 63,107)
( 64,106)( 65,101)( 66,104)( 67,103)( 68,102)( 69,117)( 70,120)( 71,119)
( 72,118)( 73,113)( 74,116)( 75,115)( 76,114)( 77,109)( 78,112)( 79,111)
( 80,110)( 81, 85)( 82, 88)( 83, 87)( 84, 86)( 89, 97)( 90,100)( 91, 99)
( 92, 98)( 94, 96)(121,165)(122,168)(123,167)(124,166)(125,161)(126,164)
(127,163)(128,162)(129,177)(130,180)(131,179)(132,178)(133,173)(134,176)
(135,175)(136,174)(137,169)(138,172)(139,171)(140,170)(141,145)(142,148)
(143,147)(144,146)(149,157)(150,160)(151,159)(152,158)(154,156)(181,225)
(182,228)(183,227)(184,226)(185,221)(186,224)(187,223)(188,222)(189,237)
(190,240)(191,239)(192,238)(193,233)(194,236)(195,235)(196,234)(197,229)
(198,232)(199,231)(200,230)(201,205)(202,208)(203,207)(204,206)(209,217)
(210,220)(211,219)(212,218)(214,216);;
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)
(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)
(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)
(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)
(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)
(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)
(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)
(176,236)(177,237)(178,238)(179,239)(180,240);
s1 := Sym(240)!(  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)
(  8,126)(  9,131)( 10,132)( 11,129)( 12,130)( 13,135)( 14,136)( 15,133)
( 16,134)( 17,139)( 18,140)( 19,137)( 20,138)( 21,143)( 22,144)( 23,141)
( 24,142)( 25,147)( 26,148)( 27,145)( 28,146)( 29,151)( 30,152)( 31,149)
( 32,150)( 33,155)( 34,156)( 35,153)( 36,154)( 37,159)( 38,160)( 39,157)
( 40,158)( 41,163)( 42,164)( 43,161)( 44,162)( 45,167)( 46,168)( 47,165)
( 48,166)( 49,171)( 50,172)( 51,169)( 52,170)( 53,175)( 54,176)( 55,173)
( 56,174)( 57,179)( 58,180)( 59,177)( 60,178)( 61,183)( 62,184)( 63,181)
( 64,182)( 65,187)( 66,188)( 67,185)( 68,186)( 69,191)( 70,192)( 71,189)
( 72,190)( 73,195)( 74,196)( 75,193)( 76,194)( 77,199)( 78,200)( 79,197)
( 80,198)( 81,203)( 82,204)( 83,201)( 84,202)( 85,207)( 86,208)( 87,205)
( 88,206)( 89,211)( 90,212)( 91,209)( 92,210)( 93,215)( 94,216)( 95,213)
( 96,214)( 97,219)( 98,220)( 99,217)(100,218)(101,223)(102,224)(103,221)
(104,222)(105,227)(106,228)(107,225)(108,226)(109,231)(110,232)(111,229)
(112,230)(113,235)(114,236)(115,233)(116,234)(117,239)(118,240)(119,237)
(120,238);
s2 := 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);
s3 := Sym(240)!(  1, 45)(  2, 48)(  3, 47)(  4, 46)(  5, 41)(  6, 44)(  7, 43)
(  8, 42)(  9, 57)( 10, 60)( 11, 59)( 12, 58)( 13, 53)( 14, 56)( 15, 55)
( 16, 54)( 17, 49)( 18, 52)( 19, 51)( 20, 50)( 21, 25)( 22, 28)( 23, 27)
( 24, 26)( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 34, 36)( 61,105)( 62,108)
( 63,107)( 64,106)( 65,101)( 66,104)( 67,103)( 68,102)( 69,117)( 70,120)
( 71,119)( 72,118)( 73,113)( 74,116)( 75,115)( 76,114)( 77,109)( 78,112)
( 79,111)( 80,110)( 81, 85)( 82, 88)( 83, 87)( 84, 86)( 89, 97)( 90,100)
( 91, 99)( 92, 98)( 94, 96)(121,165)(122,168)(123,167)(124,166)(125,161)
(126,164)(127,163)(128,162)(129,177)(130,180)(131,179)(132,178)(133,173)
(134,176)(135,175)(136,174)(137,169)(138,172)(139,171)(140,170)(141,145)
(142,148)(143,147)(144,146)(149,157)(150,160)(151,159)(152,158)(154,156)
(181,225)(182,228)(183,227)(184,226)(185,221)(186,224)(187,223)(188,222)
(189,237)(190,240)(191,239)(192,238)(193,233)(194,236)(195,235)(196,234)
(197,229)(198,232)(199,231)(200,230)(201,205)(202,208)(203,207)(204,206)
(209,217)(210,220)(211,219)(212,218)(214,216);
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope