Polytope of Type {3,6,30}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,30}*1440
Also Known As : {{3,6}4,{6,30|2}}. if this polytope has another name.
Group : SmallGroup(1440,5901)
Rank : 4
Schlafli Type : {3,6,30}
Number of vertices, edges, etc : 4, 12, 120, 30
Order of s0s1s2s3 : 60
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 : {3,6,10}*480
5-fold quotients : {3,6,6}*288
15-fold quotients : {3,6,2}*96
30-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 61,121)
( 62,122)( 63,124)( 64,123)( 65,125)( 66,126)( 67,128)( 68,127)( 69,129)
( 70,130)( 71,132)( 72,131)( 73,133)( 74,134)( 75,136)( 76,135)( 77,137)
( 78,138)( 79,140)( 80,139)( 81,141)( 82,142)( 83,144)( 84,143)( 85,145)
( 86,146)( 87,148)( 88,147)( 89,149)( 90,150)( 91,152)( 92,151)( 93,153)
( 94,154)( 95,156)( 96,155)( 97,157)( 98,158)( 99,160)(100,159)(101,161)
(102,162)(103,164)(104,163)(105,165)(106,166)(107,168)(108,167)(109,169)
(110,170)(111,172)(112,171)(113,173)(114,174)(115,176)(116,175)(117,177)
(118,178)(119,180)(120,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(203,204)(207,208)(211,212)(215,216)(219,220)(223,224)(227,228)(231,232)
(235,236)(239,240)(241,301)(242,302)(243,304)(244,303)(245,305)(246,306)
(247,308)(248,307)(249,309)(250,310)(251,312)(252,311)(253,313)(254,314)
(255,316)(256,315)(257,317)(258,318)(259,320)(260,319)(261,321)(262,322)
(263,324)(264,323)(265,325)(266,326)(267,328)(268,327)(269,329)(270,330)
(271,332)(272,331)(273,333)(274,334)(275,336)(276,335)(277,337)(278,338)
(279,340)(280,339)(281,341)(282,342)(283,344)(284,343)(285,345)(286,346)
(287,348)(288,347)(289,349)(290,350)(291,352)(292,351)(293,353)(294,354)
(295,356)(296,355)(297,357)(298,358)(299,360)(300,359);;
s1 := ( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 65)( 6, 68)( 7, 67)( 8, 66)
( 9, 69)( 10, 72)( 11, 71)( 12, 70)( 13, 73)( 14, 76)( 15, 75)( 16, 74)
( 17, 77)( 18, 80)( 19, 79)( 20, 78)( 21, 81)( 22, 84)( 23, 83)( 24, 82)
( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 89)( 30, 92)( 31, 91)( 32, 90)
( 33, 93)( 34, 96)( 35, 95)( 36, 94)( 37, 97)( 38,100)( 39, 99)( 40, 98)
( 41,101)( 42,104)( 43,103)( 44,102)( 45,105)( 46,108)( 47,107)( 48,106)
( 49,109)( 50,112)( 51,111)( 52,110)( 53,113)( 54,116)( 55,115)( 56,114)
( 57,117)( 58,120)( 59,119)( 60,118)(122,124)(126,128)(130,132)(134,136)
(138,140)(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)
(170,172)(174,176)(178,180)(181,241)(182,244)(183,243)(184,242)(185,245)
(186,248)(187,247)(188,246)(189,249)(190,252)(191,251)(192,250)(193,253)
(194,256)(195,255)(196,254)(197,257)(198,260)(199,259)(200,258)(201,261)
(202,264)(203,263)(204,262)(205,265)(206,268)(207,267)(208,266)(209,269)
(210,272)(211,271)(212,270)(213,273)(214,276)(215,275)(216,274)(217,277)
(218,280)(219,279)(220,278)(221,281)(222,284)(223,283)(224,282)(225,285)
(226,288)(227,287)(228,286)(229,289)(230,292)(231,291)(232,290)(233,293)
(234,296)(235,295)(236,294)(237,297)(238,300)(239,299)(240,298)(302,304)
(306,308)(310,312)(314,316)(318,320)(322,324)(326,328)(330,332)(334,336)
(338,340)(342,344)(346,348)(350,352)(354,356)(358,360);;
s2 := ( 1, 2)( 5, 18)( 6, 17)( 7, 19)( 8, 20)( 9, 14)( 10, 13)( 11, 15)
( 12, 16)( 21, 42)( 22, 41)( 23, 43)( 24, 44)( 25, 58)( 26, 57)( 27, 59)
( 28, 60)( 29, 54)( 30, 53)( 31, 55)( 32, 56)( 33, 50)( 34, 49)( 35, 51)
( 36, 52)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 61,122)( 62,121)( 63,123)
( 64,124)( 65,138)( 66,137)( 67,139)( 68,140)( 69,134)( 70,133)( 71,135)
( 72,136)( 73,130)( 74,129)( 75,131)( 76,132)( 77,126)( 78,125)( 79,127)
( 80,128)( 81,162)( 82,161)( 83,163)( 84,164)( 85,178)( 86,177)( 87,179)
( 88,180)( 89,174)( 90,173)( 91,175)( 92,176)( 93,170)( 94,169)( 95,171)
( 96,172)( 97,166)( 98,165)( 99,167)(100,168)(101,142)(102,141)(103,143)
(104,144)(105,158)(106,157)(107,159)(108,160)(109,154)(110,153)(111,155)
(112,156)(113,150)(114,149)(115,151)(116,152)(117,146)(118,145)(119,147)
(120,148)(181,182)(185,198)(186,197)(187,199)(188,200)(189,194)(190,193)
(191,195)(192,196)(201,222)(202,221)(203,223)(204,224)(205,238)(206,237)
(207,239)(208,240)(209,234)(210,233)(211,235)(212,236)(213,230)(214,229)
(215,231)(216,232)(217,226)(218,225)(219,227)(220,228)(241,302)(242,301)
(243,303)(244,304)(245,318)(246,317)(247,319)(248,320)(249,314)(250,313)
(251,315)(252,316)(253,310)(254,309)(255,311)(256,312)(257,306)(258,305)
(259,307)(260,308)(261,342)(262,341)(263,343)(264,344)(265,358)(266,357)
(267,359)(268,360)(269,354)(270,353)(271,355)(272,356)(273,350)(274,349)
(275,351)(276,352)(277,346)(278,345)(279,347)(280,348)(281,322)(282,321)
(283,323)(284,324)(285,338)(286,337)(287,339)(288,340)(289,334)(290,333)
(291,335)(292,336)(293,330)(294,329)(295,331)(296,332)(297,326)(298,325)
(299,327)(300,328);;
s3 := ( 1,205)( 2,206)( 3,207)( 4,208)( 5,201)( 6,202)( 7,203)( 8,204)
( 9,217)( 10,218)( 11,219)( 12,220)( 13,213)( 14,214)( 15,215)( 16,216)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,185)( 22,186)( 23,187)( 24,188)
( 25,181)( 26,182)( 27,183)( 28,184)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,193)( 34,194)( 35,195)( 36,196)( 37,189)( 38,190)( 39,191)( 40,192)
( 41,225)( 42,226)( 43,227)( 44,228)( 45,221)( 46,222)( 47,223)( 48,224)
( 49,237)( 50,238)( 51,239)( 52,240)( 53,233)( 54,234)( 55,235)( 56,236)
( 57,229)( 58,230)( 59,231)( 60,232)( 61,265)( 62,266)( 63,267)( 64,268)
( 65,261)( 66,262)( 67,263)( 68,264)( 69,277)( 70,278)( 71,279)( 72,280)
( 73,273)( 74,274)( 75,275)( 76,276)( 77,269)( 78,270)( 79,271)( 80,272)
( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)( 88,244)
( 89,257)( 90,258)( 91,259)( 92,260)( 93,253)( 94,254)( 95,255)( 96,256)
( 97,249)( 98,250)( 99,251)(100,252)(101,285)(102,286)(103,287)(104,288)
(105,281)(106,282)(107,283)(108,284)(109,297)(110,298)(111,299)(112,300)
(113,293)(114,294)(115,295)(116,296)(117,289)(118,290)(119,291)(120,292)
(121,325)(122,326)(123,327)(124,328)(125,321)(126,322)(127,323)(128,324)
(129,337)(130,338)(131,339)(132,340)(133,333)(134,334)(135,335)(136,336)
(137,329)(138,330)(139,331)(140,332)(141,305)(142,306)(143,307)(144,308)
(145,301)(146,302)(147,303)(148,304)(149,317)(150,318)(151,319)(152,320)
(153,313)(154,314)(155,315)(156,316)(157,309)(158,310)(159,311)(160,312)
(161,345)(162,346)(163,347)(164,348)(165,341)(166,342)(167,343)(168,344)
(169,357)(170,358)(171,359)(172,360)(173,353)(174,354)(175,355)(176,356)
(177,349)(178,350)(179,351)(180,352);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 61,121)( 62,122)( 63,124)( 64,123)( 65,125)( 66,126)( 67,128)( 68,127)
( 69,129)( 70,130)( 71,132)( 72,131)( 73,133)( 74,134)( 75,136)( 76,135)
( 77,137)( 78,138)( 79,140)( 80,139)( 81,141)( 82,142)( 83,144)( 84,143)
( 85,145)( 86,146)( 87,148)( 88,147)( 89,149)( 90,150)( 91,152)( 92,151)
( 93,153)( 94,154)( 95,156)( 96,155)( 97,157)( 98,158)( 99,160)(100,159)
(101,161)(102,162)(103,164)(104,163)(105,165)(106,166)(107,168)(108,167)
(109,169)(110,170)(111,172)(112,171)(113,173)(114,174)(115,176)(116,175)
(117,177)(118,178)(119,180)(120,179)(183,184)(187,188)(191,192)(195,196)
(199,200)(203,204)(207,208)(211,212)(215,216)(219,220)(223,224)(227,228)
(231,232)(235,236)(239,240)(241,301)(242,302)(243,304)(244,303)(245,305)
(246,306)(247,308)(248,307)(249,309)(250,310)(251,312)(252,311)(253,313)
(254,314)(255,316)(256,315)(257,317)(258,318)(259,320)(260,319)(261,321)
(262,322)(263,324)(264,323)(265,325)(266,326)(267,328)(268,327)(269,329)
(270,330)(271,332)(272,331)(273,333)(274,334)(275,336)(276,335)(277,337)
(278,338)(279,340)(280,339)(281,341)(282,342)(283,344)(284,343)(285,345)
(286,346)(287,348)(288,347)(289,349)(290,350)(291,352)(292,351)(293,353)
(294,354)(295,356)(296,355)(297,357)(298,358)(299,360)(300,359);
s1 := Sym(360)!( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 65)( 6, 68)( 7, 67)
( 8, 66)( 9, 69)( 10, 72)( 11, 71)( 12, 70)( 13, 73)( 14, 76)( 15, 75)
( 16, 74)( 17, 77)( 18, 80)( 19, 79)( 20, 78)( 21, 81)( 22, 84)( 23, 83)
( 24, 82)( 25, 85)( 26, 88)( 27, 87)( 28, 86)( 29, 89)( 30, 92)( 31, 91)
( 32, 90)( 33, 93)( 34, 96)( 35, 95)( 36, 94)( 37, 97)( 38,100)( 39, 99)
( 40, 98)( 41,101)( 42,104)( 43,103)( 44,102)( 45,105)( 46,108)( 47,107)
( 48,106)( 49,109)( 50,112)( 51,111)( 52,110)( 53,113)( 54,116)( 55,115)
( 56,114)( 57,117)( 58,120)( 59,119)( 60,118)(122,124)(126,128)(130,132)
(134,136)(138,140)(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)
(166,168)(170,172)(174,176)(178,180)(181,241)(182,244)(183,243)(184,242)
(185,245)(186,248)(187,247)(188,246)(189,249)(190,252)(191,251)(192,250)
(193,253)(194,256)(195,255)(196,254)(197,257)(198,260)(199,259)(200,258)
(201,261)(202,264)(203,263)(204,262)(205,265)(206,268)(207,267)(208,266)
(209,269)(210,272)(211,271)(212,270)(213,273)(214,276)(215,275)(216,274)
(217,277)(218,280)(219,279)(220,278)(221,281)(222,284)(223,283)(224,282)
(225,285)(226,288)(227,287)(228,286)(229,289)(230,292)(231,291)(232,290)
(233,293)(234,296)(235,295)(236,294)(237,297)(238,300)(239,299)(240,298)
(302,304)(306,308)(310,312)(314,316)(318,320)(322,324)(326,328)(330,332)
(334,336)(338,340)(342,344)(346,348)(350,352)(354,356)(358,360);
s2 := Sym(360)!( 1, 2)( 5, 18)( 6, 17)( 7, 19)( 8, 20)( 9, 14)( 10, 13)
( 11, 15)( 12, 16)( 21, 42)( 22, 41)( 23, 43)( 24, 44)( 25, 58)( 26, 57)
( 27, 59)( 28, 60)( 29, 54)( 30, 53)( 31, 55)( 32, 56)( 33, 50)( 34, 49)
( 35, 51)( 36, 52)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 61,122)( 62,121)
( 63,123)( 64,124)( 65,138)( 66,137)( 67,139)( 68,140)( 69,134)( 70,133)
( 71,135)( 72,136)( 73,130)( 74,129)( 75,131)( 76,132)( 77,126)( 78,125)
( 79,127)( 80,128)( 81,162)( 82,161)( 83,163)( 84,164)( 85,178)( 86,177)
( 87,179)( 88,180)( 89,174)( 90,173)( 91,175)( 92,176)( 93,170)( 94,169)
( 95,171)( 96,172)( 97,166)( 98,165)( 99,167)(100,168)(101,142)(102,141)
(103,143)(104,144)(105,158)(106,157)(107,159)(108,160)(109,154)(110,153)
(111,155)(112,156)(113,150)(114,149)(115,151)(116,152)(117,146)(118,145)
(119,147)(120,148)(181,182)(185,198)(186,197)(187,199)(188,200)(189,194)
(190,193)(191,195)(192,196)(201,222)(202,221)(203,223)(204,224)(205,238)
(206,237)(207,239)(208,240)(209,234)(210,233)(211,235)(212,236)(213,230)
(214,229)(215,231)(216,232)(217,226)(218,225)(219,227)(220,228)(241,302)
(242,301)(243,303)(244,304)(245,318)(246,317)(247,319)(248,320)(249,314)
(250,313)(251,315)(252,316)(253,310)(254,309)(255,311)(256,312)(257,306)
(258,305)(259,307)(260,308)(261,342)(262,341)(263,343)(264,344)(265,358)
(266,357)(267,359)(268,360)(269,354)(270,353)(271,355)(272,356)(273,350)
(274,349)(275,351)(276,352)(277,346)(278,345)(279,347)(280,348)(281,322)
(282,321)(283,323)(284,324)(285,338)(286,337)(287,339)(288,340)(289,334)
(290,333)(291,335)(292,336)(293,330)(294,329)(295,331)(296,332)(297,326)
(298,325)(299,327)(300,328);
s3 := Sym(360)!( 1,205)( 2,206)( 3,207)( 4,208)( 5,201)( 6,202)( 7,203)
( 8,204)( 9,217)( 10,218)( 11,219)( 12,220)( 13,213)( 14,214)( 15,215)
( 16,216)( 17,209)( 18,210)( 19,211)( 20,212)( 21,185)( 22,186)( 23,187)
( 24,188)( 25,181)( 26,182)( 27,183)( 28,184)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,193)( 34,194)( 35,195)( 36,196)( 37,189)( 38,190)( 39,191)
( 40,192)( 41,225)( 42,226)( 43,227)( 44,228)( 45,221)( 46,222)( 47,223)
( 48,224)( 49,237)( 50,238)( 51,239)( 52,240)( 53,233)( 54,234)( 55,235)
( 56,236)( 57,229)( 58,230)( 59,231)( 60,232)( 61,265)( 62,266)( 63,267)
( 64,268)( 65,261)( 66,262)( 67,263)( 68,264)( 69,277)( 70,278)( 71,279)
( 72,280)( 73,273)( 74,274)( 75,275)( 76,276)( 77,269)( 78,270)( 79,271)
( 80,272)( 81,245)( 82,246)( 83,247)( 84,248)( 85,241)( 86,242)( 87,243)
( 88,244)( 89,257)( 90,258)( 91,259)( 92,260)( 93,253)( 94,254)( 95,255)
( 96,256)( 97,249)( 98,250)( 99,251)(100,252)(101,285)(102,286)(103,287)
(104,288)(105,281)(106,282)(107,283)(108,284)(109,297)(110,298)(111,299)
(112,300)(113,293)(114,294)(115,295)(116,296)(117,289)(118,290)(119,291)
(120,292)(121,325)(122,326)(123,327)(124,328)(125,321)(126,322)(127,323)
(128,324)(129,337)(130,338)(131,339)(132,340)(133,333)(134,334)(135,335)
(136,336)(137,329)(138,330)(139,331)(140,332)(141,305)(142,306)(143,307)
(144,308)(145,301)(146,302)(147,303)(148,304)(149,317)(150,318)(151,319)
(152,320)(153,313)(154,314)(155,315)(156,316)(157,309)(158,310)(159,311)
(160,312)(161,345)(162,346)(163,347)(164,348)(165,341)(166,342)(167,343)
(168,344)(169,357)(170,358)(171,359)(172,360)(173,353)(174,354)(175,355)
(176,356)(177,349)(178,350)(179,351)(180,352);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope