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