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