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