Polytope of Type {30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1440h
if this polytope has a name.
Group : SmallGroup(1440,5901)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 120, 360, 24
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {15,6}*720e
   3-fold quotients : {30,6}*480
   4-fold quotients : {30,6}*360c
   5-fold quotients : {6,6}*288b
   6-fold quotients : {15,6}*240
   8-fold quotients : {15,6}*180
   10-fold quotients : {3,6}*144
   12-fold quotients : {30,2}*120
   15-fold quotients : {6,6}*96
   20-fold quotients : {6,6}*72c
   24-fold quotients : {15,2}*60
   30-fold quotients : {3,6}*48, {6,3}*48
   36-fold quotients : {10,2}*40
   40-fold quotients : {3,6}*36
   60-fold quotients : {3,3}*24, {6,2}*24
   72-fold quotients : {5,2}*20
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5, 17)(  6, 18)(  7, 20)(  8, 19)(  9, 13)( 10, 14)( 11, 16)
( 12, 15)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 57)( 26, 58)( 27, 60)
( 28, 59)( 29, 53)( 30, 54)( 31, 56)( 32, 55)( 33, 49)( 34, 50)( 35, 52)
( 36, 51)( 37, 45)( 38, 46)( 39, 48)( 40, 47)( 61,121)( 62,122)( 63,124)
( 64,123)( 65,137)( 66,138)( 67,140)( 68,139)( 69,133)( 70,134)( 71,136)
( 72,135)( 73,129)( 74,130)( 75,132)( 76,131)( 77,125)( 78,126)( 79,128)
( 80,127)( 81,161)( 82,162)( 83,164)( 84,163)( 85,177)( 86,178)( 87,180)
( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)
( 96,171)( 97,165)( 98,166)( 99,168)(100,167)(101,141)(102,142)(103,144)
(104,143)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)(111,156)
(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)(119,148)
(120,147)(183,184)(185,197)(186,198)(187,200)(188,199)(189,193)(190,194)
(191,196)(192,195)(201,221)(202,222)(203,224)(204,223)(205,237)(206,238)
(207,240)(208,239)(209,233)(210,234)(211,236)(212,235)(213,229)(214,230)
(215,232)(216,231)(217,225)(218,226)(219,228)(220,227)(241,301)(242,302)
(243,304)(244,303)(245,317)(246,318)(247,320)(248,319)(249,313)(250,314)
(251,316)(252,315)(253,309)(254,310)(255,312)(256,311)(257,305)(258,306)
(259,308)(260,307)(261,341)(262,342)(263,344)(264,343)(265,357)(266,358)
(267,360)(268,359)(269,353)(270,354)(271,356)(272,355)(273,349)(274,350)
(275,352)(276,351)(277,345)(278,346)(279,348)(280,347)(281,321)(282,322)
(283,324)(284,323)(285,337)(286,338)(287,340)(288,339)(289,333)(290,334)
(291,336)(292,335)(293,329)(294,330)(295,332)(296,331)(297,325)(298,326)
(299,328)(300,327);;
s1 := (  1,265)(  2,268)(  3,267)(  4,266)(  5,261)(  6,264)(  7,263)(  8,262)
(  9,277)( 10,280)( 11,279)( 12,278)( 13,273)( 14,276)( 15,275)( 16,274)
( 17,269)( 18,272)( 19,271)( 20,270)( 21,245)( 22,248)( 23,247)( 24,246)
( 25,241)( 26,244)( 27,243)( 28,242)( 29,257)( 30,260)( 31,259)( 32,258)
( 33,253)( 34,256)( 35,255)( 36,254)( 37,249)( 38,252)( 39,251)( 40,250)
( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)( 48,282)
( 49,297)( 50,300)( 51,299)( 52,298)( 53,293)( 54,296)( 55,295)( 56,294)
( 57,289)( 58,292)( 59,291)( 60,290)( 61,205)( 62,208)( 63,207)( 64,206)
( 65,201)( 66,204)( 67,203)( 68,202)( 69,217)( 70,220)( 71,219)( 72,218)
( 73,213)( 74,216)( 75,215)( 76,214)( 77,209)( 78,212)( 79,211)( 80,210)
( 81,185)( 82,188)( 83,187)( 84,186)( 85,181)( 86,184)( 87,183)( 88,182)
( 89,197)( 90,200)( 91,199)( 92,198)( 93,193)( 94,196)( 95,195)( 96,194)
( 97,189)( 98,192)( 99,191)(100,190)(101,225)(102,228)(103,227)(104,226)
(105,221)(106,224)(107,223)(108,222)(109,237)(110,240)(111,239)(112,238)
(113,233)(114,236)(115,235)(116,234)(117,229)(118,232)(119,231)(120,230)
(121,325)(122,328)(123,327)(124,326)(125,321)(126,324)(127,323)(128,322)
(129,337)(130,340)(131,339)(132,338)(133,333)(134,336)(135,335)(136,334)
(137,329)(138,332)(139,331)(140,330)(141,305)(142,308)(143,307)(144,306)
(145,301)(146,304)(147,303)(148,302)(149,317)(150,320)(151,319)(152,318)
(153,313)(154,316)(155,315)(156,314)(157,309)(158,312)(159,311)(160,310)
(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)(168,342)
(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)
(177,349)(178,352)(179,351)(180,350);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 38)( 41, 42)( 45, 46)( 49, 50)( 53, 54)( 57, 58)( 61,122)
( 62,121)( 63,123)( 64,124)( 65,126)( 66,125)( 67,127)( 68,128)( 69,130)
( 70,129)( 71,131)( 72,132)( 73,134)( 74,133)( 75,135)( 76,136)( 77,138)
( 78,137)( 79,139)( 80,140)( 81,142)( 82,141)( 83,143)( 84,144)( 85,146)
( 86,145)( 87,147)( 88,148)( 89,150)( 90,149)( 91,151)( 92,152)( 93,154)
( 94,153)( 95,155)( 96,156)( 97,158)( 98,157)( 99,159)(100,160)(101,162)
(102,161)(103,163)(104,164)(105,166)(106,165)(107,167)(108,168)(109,170)
(110,169)(111,171)(112,172)(113,174)(114,173)(115,175)(116,176)(117,178)
(118,177)(119,179)(120,180)(181,182)(185,186)(189,190)(193,194)(197,198)
(201,202)(205,206)(209,210)(213,214)(217,218)(221,222)(225,226)(229,230)
(233,234)(237,238)(241,302)(242,301)(243,303)(244,304)(245,306)(246,305)
(247,307)(248,308)(249,310)(250,309)(251,311)(252,312)(253,314)(254,313)
(255,315)(256,316)(257,318)(258,317)(259,319)(260,320)(261,322)(262,321)
(263,323)(264,324)(265,326)(266,325)(267,327)(268,328)(269,330)(270,329)
(271,331)(272,332)(273,334)(274,333)(275,335)(276,336)(277,338)(278,337)
(279,339)(280,340)(281,342)(282,341)(283,343)(284,344)(285,346)(286,345)
(287,347)(288,348)(289,350)(290,349)(291,351)(292,352)(293,354)(294,353)
(295,355)(296,356)(297,358)(298,357)(299,359)(300,360);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*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*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(360)!(  3,  4)(  5, 17)(  6, 18)(  7, 20)(  8, 19)(  9, 13)( 10, 14)
( 11, 16)( 12, 15)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 57)( 26, 58)
( 27, 60)( 28, 59)( 29, 53)( 30, 54)( 31, 56)( 32, 55)( 33, 49)( 34, 50)
( 35, 52)( 36, 51)( 37, 45)( 38, 46)( 39, 48)( 40, 47)( 61,121)( 62,122)
( 63,124)( 64,123)( 65,137)( 66,138)( 67,140)( 68,139)( 69,133)( 70,134)
( 71,136)( 72,135)( 73,129)( 74,130)( 75,132)( 76,131)( 77,125)( 78,126)
( 79,128)( 80,127)( 81,161)( 82,162)( 83,164)( 84,163)( 85,177)( 86,178)
( 87,180)( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)
( 95,172)( 96,171)( 97,165)( 98,166)( 99,168)(100,167)(101,141)(102,142)
(103,144)(104,143)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)
(111,156)(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)
(119,148)(120,147)(183,184)(185,197)(186,198)(187,200)(188,199)(189,193)
(190,194)(191,196)(192,195)(201,221)(202,222)(203,224)(204,223)(205,237)
(206,238)(207,240)(208,239)(209,233)(210,234)(211,236)(212,235)(213,229)
(214,230)(215,232)(216,231)(217,225)(218,226)(219,228)(220,227)(241,301)
(242,302)(243,304)(244,303)(245,317)(246,318)(247,320)(248,319)(249,313)
(250,314)(251,316)(252,315)(253,309)(254,310)(255,312)(256,311)(257,305)
(258,306)(259,308)(260,307)(261,341)(262,342)(263,344)(264,343)(265,357)
(266,358)(267,360)(268,359)(269,353)(270,354)(271,356)(272,355)(273,349)
(274,350)(275,352)(276,351)(277,345)(278,346)(279,348)(280,347)(281,321)
(282,322)(283,324)(284,323)(285,337)(286,338)(287,340)(288,339)(289,333)
(290,334)(291,336)(292,335)(293,329)(294,330)(295,332)(296,331)(297,325)
(298,326)(299,328)(300,327);
s1 := Sym(360)!(  1,265)(  2,268)(  3,267)(  4,266)(  5,261)(  6,264)(  7,263)
(  8,262)(  9,277)( 10,280)( 11,279)( 12,278)( 13,273)( 14,276)( 15,275)
( 16,274)( 17,269)( 18,272)( 19,271)( 20,270)( 21,245)( 22,248)( 23,247)
( 24,246)( 25,241)( 26,244)( 27,243)( 28,242)( 29,257)( 30,260)( 31,259)
( 32,258)( 33,253)( 34,256)( 35,255)( 36,254)( 37,249)( 38,252)( 39,251)
( 40,250)( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)
( 48,282)( 49,297)( 50,300)( 51,299)( 52,298)( 53,293)( 54,296)( 55,295)
( 56,294)( 57,289)( 58,292)( 59,291)( 60,290)( 61,205)( 62,208)( 63,207)
( 64,206)( 65,201)( 66,204)( 67,203)( 68,202)( 69,217)( 70,220)( 71,219)
( 72,218)( 73,213)( 74,216)( 75,215)( 76,214)( 77,209)( 78,212)( 79,211)
( 80,210)( 81,185)( 82,188)( 83,187)( 84,186)( 85,181)( 86,184)( 87,183)
( 88,182)( 89,197)( 90,200)( 91,199)( 92,198)( 93,193)( 94,196)( 95,195)
( 96,194)( 97,189)( 98,192)( 99,191)(100,190)(101,225)(102,228)(103,227)
(104,226)(105,221)(106,224)(107,223)(108,222)(109,237)(110,240)(111,239)
(112,238)(113,233)(114,236)(115,235)(116,234)(117,229)(118,232)(119,231)
(120,230)(121,325)(122,328)(123,327)(124,326)(125,321)(126,324)(127,323)
(128,322)(129,337)(130,340)(131,339)(132,338)(133,333)(134,336)(135,335)
(136,334)(137,329)(138,332)(139,331)(140,330)(141,305)(142,308)(143,307)
(144,306)(145,301)(146,304)(147,303)(148,302)(149,317)(150,320)(151,319)
(152,318)(153,313)(154,316)(155,315)(156,314)(157,309)(158,312)(159,311)
(160,310)(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)
(168,342)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)
(176,354)(177,349)(178,352)(179,351)(180,350);
s2 := Sym(360)!(  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)
( 29, 30)( 33, 34)( 37, 38)( 41, 42)( 45, 46)( 49, 50)( 53, 54)( 57, 58)
( 61,122)( 62,121)( 63,123)( 64,124)( 65,126)( 66,125)( 67,127)( 68,128)
( 69,130)( 70,129)( 71,131)( 72,132)( 73,134)( 74,133)( 75,135)( 76,136)
( 77,138)( 78,137)( 79,139)( 80,140)( 81,142)( 82,141)( 83,143)( 84,144)
( 85,146)( 86,145)( 87,147)( 88,148)( 89,150)( 90,149)( 91,151)( 92,152)
( 93,154)( 94,153)( 95,155)( 96,156)( 97,158)( 98,157)( 99,159)(100,160)
(101,162)(102,161)(103,163)(104,164)(105,166)(106,165)(107,167)(108,168)
(109,170)(110,169)(111,171)(112,172)(113,174)(114,173)(115,175)(116,176)
(117,178)(118,177)(119,179)(120,180)(181,182)(185,186)(189,190)(193,194)
(197,198)(201,202)(205,206)(209,210)(213,214)(217,218)(221,222)(225,226)
(229,230)(233,234)(237,238)(241,302)(242,301)(243,303)(244,304)(245,306)
(246,305)(247,307)(248,308)(249,310)(250,309)(251,311)(252,312)(253,314)
(254,313)(255,315)(256,316)(257,318)(258,317)(259,319)(260,320)(261,322)
(262,321)(263,323)(264,324)(265,326)(266,325)(267,327)(268,328)(269,330)
(270,329)(271,331)(272,332)(273,334)(274,333)(275,335)(276,336)(277,338)
(278,337)(279,339)(280,340)(281,342)(282,341)(283,343)(284,344)(285,346)
(286,345)(287,347)(288,348)(289,350)(290,349)(291,351)(292,352)(293,354)
(294,353)(295,355)(296,356)(297,358)(298,357)(299,359)(300,360);
poly := sub<Sym(360)|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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*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*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