Polytope of Type {4,30,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,6}*1440d
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) :
3-fold quotients : {4,30,2}*480b
5-fold quotients : {4,6,6}*288d
6-fold quotients : {4,15,2}*240
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)( 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);;
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 := ( 1,201)( 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, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)( 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);
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)!( 1,201)( 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,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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