Polytope of Type {4,6,30}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,30}*1440e
if this polytope has a name.
Group : SmallGroup(1440,5901)
Rank : 4
Schlafli Type : {4,6,30}
Number of vertices, edges, etc : 4, 12, 90, 30
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 : {4,6,10}*480b
5-fold quotients : {4,6,6}*288d
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)( 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);;
s2 := ( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 77)( 6, 80)( 7, 79)( 8, 78)
( 9, 73)( 10, 76)( 11, 75)( 12, 74)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21,101)( 22,104)( 23,103)( 24,102)
( 25,117)( 26,120)( 27,119)( 28,118)( 29,113)( 30,116)( 31,115)( 32,114)
( 33,109)( 34,112)( 35,111)( 36,110)( 37,105)( 38,108)( 39,107)( 40,106)
( 41, 81)( 42, 84)( 43, 83)( 44, 82)( 45, 97)( 46,100)( 47, 99)( 48, 98)
( 49, 93)( 50, 96)( 51, 95)( 52, 94)( 53, 89)( 54, 92)( 55, 91)( 56, 90)
( 57, 85)( 58, 88)( 59, 87)( 60, 86)(122,124)(125,137)(126,140)(127,139)
(128,138)(129,133)(130,136)(131,135)(132,134)(141,161)(142,164)(143,163)
(144,162)(145,177)(146,180)(147,179)(148,178)(149,173)(150,176)(151,175)
(152,174)(153,169)(154,172)(155,171)(156,170)(157,165)(158,168)(159,167)
(160,166)(181,241)(182,244)(183,243)(184,242)(185,257)(186,260)(187,259)
(188,258)(189,253)(190,256)(191,255)(192,254)(193,249)(194,252)(195,251)
(196,250)(197,245)(198,248)(199,247)(200,246)(201,281)(202,284)(203,283)
(204,282)(205,297)(206,300)(207,299)(208,298)(209,293)(210,296)(211,295)
(212,294)(213,289)(214,292)(215,291)(216,290)(217,285)(218,288)(219,287)
(220,286)(221,261)(222,264)(223,263)(224,262)(225,277)(226,280)(227,279)
(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)
(236,270)(237,265)(238,268)(239,267)(240,266)(302,304)(305,317)(306,320)
(307,319)(308,318)(309,313)(310,316)(311,315)(312,314)(321,341)(322,344)
(323,343)(324,342)(325,357)(326,360)(327,359)(328,358)(329,353)(330,356)
(331,355)(332,354)(333,349)(334,352)(335,351)(336,350)(337,345)(338,348)
(339,347)(340,346);;
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*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!( 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)( 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);
s2 := Sym(360)!( 1, 61)( 2, 64)( 3, 63)( 4, 62)( 5, 77)( 6, 80)( 7, 79)
( 8, 78)( 9, 73)( 10, 76)( 11, 75)( 12, 74)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21,101)( 22,104)( 23,103)
( 24,102)( 25,117)( 26,120)( 27,119)( 28,118)( 29,113)( 30,116)( 31,115)
( 32,114)( 33,109)( 34,112)( 35,111)( 36,110)( 37,105)( 38,108)( 39,107)
( 40,106)( 41, 81)( 42, 84)( 43, 83)( 44, 82)( 45, 97)( 46,100)( 47, 99)
( 48, 98)( 49, 93)( 50, 96)( 51, 95)( 52, 94)( 53, 89)( 54, 92)( 55, 91)
( 56, 90)( 57, 85)( 58, 88)( 59, 87)( 60, 86)(122,124)(125,137)(126,140)
(127,139)(128,138)(129,133)(130,136)(131,135)(132,134)(141,161)(142,164)
(143,163)(144,162)(145,177)(146,180)(147,179)(148,178)(149,173)(150,176)
(151,175)(152,174)(153,169)(154,172)(155,171)(156,170)(157,165)(158,168)
(159,167)(160,166)(181,241)(182,244)(183,243)(184,242)(185,257)(186,260)
(187,259)(188,258)(189,253)(190,256)(191,255)(192,254)(193,249)(194,252)
(195,251)(196,250)(197,245)(198,248)(199,247)(200,246)(201,281)(202,284)
(203,283)(204,282)(205,297)(206,300)(207,299)(208,298)(209,293)(210,296)
(211,295)(212,294)(213,289)(214,292)(215,291)(216,290)(217,285)(218,288)
(219,287)(220,286)(221,261)(222,264)(223,263)(224,262)(225,277)(226,280)
(227,279)(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)
(235,271)(236,270)(237,265)(238,268)(239,267)(240,266)(302,304)(305,317)
(306,320)(307,319)(308,318)(309,313)(310,316)(311,315)(312,314)(321,341)
(322,344)(323,343)(324,342)(325,357)(326,360)(327,359)(328,358)(329,353)
(330,356)(331,355)(332,354)(333,349)(334,352)(335,351)(336,350)(337,345)
(338,348)(339,347)(340,346);
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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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