Polytope of Type {4,15,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,15,6}*1440b
if this polytope has a name.
Group : SmallGroup(1440,5900)
Rank : 4
Schlafli Type : {4,15,6}
Number of vertices, edges, etc : 8, 60, 90, 6
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,15,6}*720
   3-fold quotients : {4,15,2}*480
   4-fold quotients : {2,15,6}*360
   5-fold quotients : {4,3,6}*288
   6-fold quotients : {4,15,2}*240
   10-fold quotients : {4,3,6}*144
   12-fold quotients : {2,15,2}*120
   15-fold quotients : {4,3,2}*96
   20-fold quotients : {2,3,6}*72
   30-fold quotients : {4,3,2}*48
   36-fold quotients : {2,5,2}*40
   60-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  1,183)(  2,184)(  3,181)(  4,182)(  5,187)(  6,188)(  7,185)(  8,186)
(  9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)( 16,194)
( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)( 24,202)
( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)( 32,210)
( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)( 40,218)
( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)( 48,226)
( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)( 56,234)
( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)( 64,242)
( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)( 72,250)
( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)
( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)
( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)
( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)(104,282)
(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)(112,290)
(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)(120,298)
(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)(128,306)
(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)(136,314)
(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)(144,322)
(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)(152,330)
(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)(160,338)
(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)(168,346)
(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)(176,354)
(177,359)(178,360)(179,357)(180,358);;
s1 := (  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);;
s2 := (  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 81)(  6, 84)(  7, 83)(  8, 82)
(  9, 97)( 10,100)( 11, 99)( 12, 98)( 13, 93)( 14, 96)( 15, 95)( 16, 94)
( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 77)( 30, 80)( 31, 79)( 32, 78)
( 33, 73)( 34, 76)( 35, 75)( 36, 74)( 37, 69)( 38, 72)( 39, 71)( 40, 70)
( 41,105)( 42,108)( 43,107)( 44,106)( 45,101)( 46,104)( 47,103)( 48,102)
( 49,117)( 50,120)( 51,119)( 52,118)( 53,113)( 54,116)( 55,115)( 56,114)
( 57,109)( 58,112)( 59,111)( 60,110)(121,145)(122,148)(123,147)(124,146)
(125,141)(126,144)(127,143)(128,142)(129,157)(130,160)(131,159)(132,158)
(133,153)(134,156)(135,155)(136,154)(137,149)(138,152)(139,151)(140,150)
(161,165)(162,168)(163,167)(164,166)(169,177)(170,180)(171,179)(172,178)
(174,176)(181,265)(182,268)(183,267)(184,266)(185,261)(186,264)(187,263)
(188,262)(189,277)(190,280)(191,279)(192,278)(193,273)(194,276)(195,275)
(196,274)(197,269)(198,272)(199,271)(200,270)(201,245)(202,248)(203,247)
(204,246)(205,241)(206,244)(207,243)(208,242)(209,257)(210,260)(211,259)
(212,258)(213,253)(214,256)(215,255)(216,254)(217,249)(218,252)(219,251)
(220,250)(221,285)(222,288)(223,287)(224,286)(225,281)(226,284)(227,283)
(228,282)(229,297)(230,300)(231,299)(232,298)(233,293)(234,296)(235,295)
(236,294)(237,289)(238,292)(239,291)(240,290)(301,325)(302,328)(303,327)
(304,326)(305,321)(306,324)(307,323)(308,322)(309,337)(310,340)(311,339)
(312,338)(313,333)(314,336)(315,335)(316,334)(317,329)(318,332)(319,331)
(320,330)(341,345)(342,348)(343,347)(344,346)(349,357)(350,360)(351,359)
(352,358)(354,356);;
s3 := ( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)( 26, 46)( 27, 47)( 28, 48)
( 29, 49)( 30, 50)( 31, 51)( 32, 52)( 33, 53)( 34, 54)( 35, 55)( 36, 56)
( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 81,101)( 82,102)( 83,103)( 84,104)
( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,111)( 92,112)
( 93,113)( 94,114)( 95,115)( 96,116)( 97,117)( 98,118)( 99,119)(100,120)
(141,161)(142,162)(143,163)(144,164)(145,165)(146,166)(147,167)(148,168)
(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)(156,176)
(157,177)(158,178)(159,179)(160,180)(201,221)(202,222)(203,223)(204,224)
(205,225)(206,226)(207,227)(208,228)(209,229)(210,230)(211,231)(212,232)
(213,233)(214,234)(215,235)(216,236)(217,237)(218,238)(219,239)(220,240)
(261,281)(262,282)(263,283)(264,284)(265,285)(266,286)(267,287)(268,288)
(269,289)(270,290)(271,291)(272,292)(273,293)(274,294)(275,295)(276,296)
(277,297)(278,298)(279,299)(280,300)(321,341)(322,342)(323,343)(324,344)
(325,345)(326,346)(327,347)(328,348)(329,349)(330,350)(331,351)(332,352)
(333,353)(334,354)(335,355)(336,356)(337,357)(338,358)(339,359)(340,360);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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(360)!(  1,183)(  2,184)(  3,181)(  4,182)(  5,187)(  6,188)(  7,185)
(  8,186)(  9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)
( 16,194)( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)
( 24,202)( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)
( 32,210)( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)
( 40,218)( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)
( 48,226)( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)
( 56,234)( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)
( 64,242)( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)
( 72,250)( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)
( 80,258)( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)
( 88,266)( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)
( 96,274)( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)
(104,282)(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)
(112,290)(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)
(120,298)(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)
(128,306)(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)
(136,314)(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)
(144,322)(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)
(152,330)(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)
(160,338)(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)
(168,346)(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)
(176,354)(177,359)(178,360)(179,357)(180,358);
s1 := 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);
s2 := Sym(360)!(  1, 85)(  2, 88)(  3, 87)(  4, 86)(  5, 81)(  6, 84)(  7, 83)
(  8, 82)(  9, 97)( 10,100)( 11, 99)( 12, 98)( 13, 93)( 14, 96)( 15, 95)
( 16, 94)( 17, 89)( 18, 92)( 19, 91)( 20, 90)( 21, 65)( 22, 68)( 23, 67)
( 24, 66)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 77)( 30, 80)( 31, 79)
( 32, 78)( 33, 73)( 34, 76)( 35, 75)( 36, 74)( 37, 69)( 38, 72)( 39, 71)
( 40, 70)( 41,105)( 42,108)( 43,107)( 44,106)( 45,101)( 46,104)( 47,103)
( 48,102)( 49,117)( 50,120)( 51,119)( 52,118)( 53,113)( 54,116)( 55,115)
( 56,114)( 57,109)( 58,112)( 59,111)( 60,110)(121,145)(122,148)(123,147)
(124,146)(125,141)(126,144)(127,143)(128,142)(129,157)(130,160)(131,159)
(132,158)(133,153)(134,156)(135,155)(136,154)(137,149)(138,152)(139,151)
(140,150)(161,165)(162,168)(163,167)(164,166)(169,177)(170,180)(171,179)
(172,178)(174,176)(181,265)(182,268)(183,267)(184,266)(185,261)(186,264)
(187,263)(188,262)(189,277)(190,280)(191,279)(192,278)(193,273)(194,276)
(195,275)(196,274)(197,269)(198,272)(199,271)(200,270)(201,245)(202,248)
(203,247)(204,246)(205,241)(206,244)(207,243)(208,242)(209,257)(210,260)
(211,259)(212,258)(213,253)(214,256)(215,255)(216,254)(217,249)(218,252)
(219,251)(220,250)(221,285)(222,288)(223,287)(224,286)(225,281)(226,284)
(227,283)(228,282)(229,297)(230,300)(231,299)(232,298)(233,293)(234,296)
(235,295)(236,294)(237,289)(238,292)(239,291)(240,290)(301,325)(302,328)
(303,327)(304,326)(305,321)(306,324)(307,323)(308,322)(309,337)(310,340)
(311,339)(312,338)(313,333)(314,336)(315,335)(316,334)(317,329)(318,332)
(319,331)(320,330)(341,345)(342,348)(343,347)(344,346)(349,357)(350,360)
(351,359)(352,358)(354,356);
s3 := Sym(360)!( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)( 26, 46)( 27, 47)
( 28, 48)( 29, 49)( 30, 50)( 31, 51)( 32, 52)( 33, 53)( 34, 54)( 35, 55)
( 36, 56)( 37, 57)( 38, 58)( 39, 59)( 40, 60)( 81,101)( 82,102)( 83,103)
( 84,104)( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,111)
( 92,112)( 93,113)( 94,114)( 95,115)( 96,116)( 97,117)( 98,118)( 99,119)
(100,120)(141,161)(142,162)(143,163)(144,164)(145,165)(146,166)(147,167)
(148,168)(149,169)(150,170)(151,171)(152,172)(153,173)(154,174)(155,175)
(156,176)(157,177)(158,178)(159,179)(160,180)(201,221)(202,222)(203,223)
(204,224)(205,225)(206,226)(207,227)(208,228)(209,229)(210,230)(211,231)
(212,232)(213,233)(214,234)(215,235)(216,236)(217,237)(218,238)(219,239)
(220,240)(261,281)(262,282)(263,283)(264,284)(265,285)(266,286)(267,287)
(268,288)(269,289)(270,290)(271,291)(272,292)(273,293)(274,294)(275,295)
(276,296)(277,297)(278,298)(279,299)(280,300)(321,341)(322,342)(323,343)
(324,344)(325,345)(326,346)(327,347)(328,348)(329,349)(330,350)(331,351)
(332,352)(333,353)(334,354)(335,355)(336,356)(337,357)(338,358)(339,359)
(340,360);
poly := sub<Sym(360)|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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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