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