Polytope of Type {4,60}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,60}*960a
if this polytope has a name.
Group : SmallGroup(960,5037)
Rank : 3
Schlafli Type : {4,60}
Number of vertices, edges, etc : 8, 240, 120
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,60,2} of size 1920
Vertex Figure Of :
   {2,4,60} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,60}*480a
   3-fold quotients : {4,20}*320
   4-fold quotients : {2,60}*240, {4,30}*240a
   5-fold quotients : {4,12}*192a
   6-fold quotients : {4,20}*160
   8-fold quotients : {2,30}*120
   10-fold quotients : {4,12}*96a
   12-fold quotients : {2,20}*80, {4,10}*80
   15-fold quotients : {4,4}*64
   16-fold quotients : {2,15}*60
   20-fold quotients : {2,12}*48, {4,6}*48a
   24-fold quotients : {2,10}*40
   30-fold quotients : {4,4}*32
   40-fold quotients : {2,6}*24
   48-fold quotients : {2,5}*20
   60-fold quotients : {2,4}*16, {4,2}*16
   80-fold quotients : {2,3}*12
   120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,60}*1920a, {4,120}*1920a, {4,60}*1920a, {4,120}*1920b, {8,60}*1920b
Permutation Representation (GAP) :

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