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