Polytope of Type {6,30,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30,4}*1440f
if this polytope has a name.
Group : SmallGroup(1440,5900)
Rank : 4
Schlafli Type : {6,30,4}
Number of vertices, edges, etc : 6, 90, 60, 4
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 : {6,15,4}*720
3-fold quotients : {2,30,4}*480c
5-fold quotients : {6,6,4}*288f
6-fold quotients : {2,15,4}*240
10-fold quotients : {6,3,4}*144
15-fold quotients : {2,6,4}*96b
30-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 37)( 6, 38)( 7, 40)( 8, 39)
( 9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)( 47, 60)
( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61,141)( 62,142)( 63,144)
( 64,143)( 65,157)( 66,158)( 67,160)( 68,159)( 69,153)( 70,154)( 71,156)
( 72,155)( 73,149)( 74,150)( 75,152)( 76,151)( 77,145)( 78,146)( 79,148)
( 80,147)( 81,121)( 82,122)( 83,124)( 84,123)( 85,137)( 86,138)( 87,140)
( 88,139)( 89,133)( 90,134)( 91,136)( 92,135)( 93,129)( 94,130)( 95,132)
( 96,131)( 97,125)( 98,126)( 99,128)(100,127)(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)(181,201)(182,202)(183,204)(184,203)(185,217)(186,218)(187,220)
(188,219)(189,213)(190,214)(191,216)(192,215)(193,209)(194,210)(195,212)
(196,211)(197,205)(198,206)(199,208)(200,207)(223,224)(225,237)(226,238)
(227,240)(228,239)(229,233)(230,234)(231,236)(232,235)(241,321)(242,322)
(243,324)(244,323)(245,337)(246,338)(247,340)(248,339)(249,333)(250,334)
(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,325)(258,326)
(259,328)(260,327)(261,301)(262,302)(263,304)(264,303)(265,317)(266,318)
(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,309)(274,310)
(275,312)(276,311)(277,305)(278,306)(279,308)(280,307)(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 := ( 1,245)( 2,248)( 3,247)( 4,246)( 5,241)( 6,244)( 7,243)( 8,242)
( 9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,249)( 18,252)( 19,251)( 20,250)( 21,285)( 22,288)( 23,287)( 24,286)
( 25,281)( 26,284)( 27,283)( 28,282)( 29,297)( 30,300)( 31,299)( 32,298)
( 33,293)( 34,296)( 35,295)( 36,294)( 37,289)( 38,292)( 39,291)( 40,290)
( 41,265)( 42,268)( 43,267)( 44,266)( 45,261)( 46,264)( 47,263)( 48,262)
( 49,277)( 50,280)( 51,279)( 52,278)( 53,273)( 54,276)( 55,275)( 56,274)
( 57,269)( 58,272)( 59,271)( 60,270)( 61,185)( 62,188)( 63,187)( 64,186)
( 65,181)( 66,184)( 67,183)( 68,182)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,225)( 82,228)( 83,227)( 84,226)( 85,221)( 86,224)( 87,223)( 88,222)
( 89,237)( 90,240)( 91,239)( 92,238)( 93,233)( 94,236)( 95,235)( 96,234)
( 97,229)( 98,232)( 99,231)(100,230)(101,205)(102,208)(103,207)(104,206)
(105,201)(106,204)(107,203)(108,202)(109,217)(110,220)(111,219)(112,218)
(113,213)(114,216)(115,215)(116,214)(117,209)(118,212)(119,211)(120,210)
(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)(128,302)
(129,317)(130,320)(131,319)(132,318)(133,313)(134,316)(135,315)(136,314)
(137,309)(138,312)(139,311)(140,310)(141,345)(142,348)(143,347)(144,346)
(145,341)(146,344)(147,343)(148,342)(149,357)(150,360)(151,359)(152,358)
(153,353)(154,356)(155,355)(156,354)(157,349)(158,352)(159,351)(160,350)
(161,325)(162,328)(163,327)(164,326)(165,321)(166,324)(167,323)(168,322)
(169,337)(170,340)(171,339)(172,338)(173,333)(174,336)(175,335)(176,334)
(177,329)(178,332)(179,331)(180,330);;
s3 := ( 1,182)( 2,181)( 3,184)( 4,183)( 5,186)( 6,185)( 7,188)( 8,187)
( 9,190)( 10,189)( 11,192)( 12,191)( 13,194)( 14,193)( 15,196)( 16,195)
( 17,198)( 18,197)( 19,200)( 20,199)( 21,202)( 22,201)( 23,204)( 24,203)
( 25,206)( 26,205)( 27,208)( 28,207)( 29,210)( 30,209)( 31,212)( 32,211)
( 33,214)( 34,213)( 35,216)( 36,215)( 37,218)( 38,217)( 39,220)( 40,219)
( 41,222)( 42,221)( 43,224)( 44,223)( 45,226)( 46,225)( 47,228)( 48,227)
( 49,230)( 50,229)( 51,232)( 52,231)( 53,234)( 54,233)( 55,236)( 56,235)
( 57,238)( 58,237)( 59,240)( 60,239)( 61,242)( 62,241)( 63,244)( 64,243)
( 65,246)( 66,245)( 67,248)( 68,247)( 69,250)( 70,249)( 71,252)( 72,251)
( 73,254)( 74,253)( 75,256)( 76,255)( 77,258)( 78,257)( 79,260)( 80,259)
( 81,262)( 82,261)( 83,264)( 84,263)( 85,266)( 86,265)( 87,268)( 88,267)
( 89,270)( 90,269)( 91,272)( 92,271)( 93,274)( 94,273)( 95,276)( 96,275)
( 97,278)( 98,277)( 99,280)(100,279)(101,282)(102,281)(103,284)(104,283)
(105,286)(106,285)(107,288)(108,287)(109,290)(110,289)(111,292)(112,291)
(113,294)(114,293)(115,296)(116,295)(117,298)(118,297)(119,300)(120,299)
(121,302)(122,301)(123,304)(124,303)(125,306)(126,305)(127,308)(128,307)
(129,310)(130,309)(131,312)(132,311)(133,314)(134,313)(135,316)(136,315)
(137,318)(138,317)(139,320)(140,319)(141,322)(142,321)(143,324)(144,323)
(145,326)(146,325)(147,328)(148,327)(149,330)(150,329)(151,332)(152,331)
(153,334)(154,333)(155,336)(156,335)(157,338)(158,337)(159,340)(160,339)
(161,342)(162,341)(163,344)(164,343)(165,346)(166,345)(167,348)(168,347)
(169,350)(170,349)(171,352)(172,351)(173,354)(174,353)(175,356)(176,355)
(177,358)(178,357)(179,360)(180,359);;
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*s2*s3,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(360)!( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 37)( 6, 38)( 7, 40)
( 8, 39)( 9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)
( 47, 60)( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61,141)( 62,142)
( 63,144)( 64,143)( 65,157)( 66,158)( 67,160)( 68,159)( 69,153)( 70,154)
( 71,156)( 72,155)( 73,149)( 74,150)( 75,152)( 76,151)( 77,145)( 78,146)
( 79,148)( 80,147)( 81,121)( 82,122)( 83,124)( 84,123)( 85,137)( 86,138)
( 87,140)( 88,139)( 89,133)( 90,134)( 91,136)( 92,135)( 93,129)( 94,130)
( 95,132)( 96,131)( 97,125)( 98,126)( 99,128)(100,127)(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)(181,201)(182,202)(183,204)(184,203)(185,217)(186,218)
(187,220)(188,219)(189,213)(190,214)(191,216)(192,215)(193,209)(194,210)
(195,212)(196,211)(197,205)(198,206)(199,208)(200,207)(223,224)(225,237)
(226,238)(227,240)(228,239)(229,233)(230,234)(231,236)(232,235)(241,321)
(242,322)(243,324)(244,323)(245,337)(246,338)(247,340)(248,339)(249,333)
(250,334)(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,325)
(258,326)(259,328)(260,327)(261,301)(262,302)(263,304)(264,303)(265,317)
(266,318)(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,309)
(274,310)(275,312)(276,311)(277,305)(278,306)(279,308)(280,307)(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)!( 1,245)( 2,248)( 3,247)( 4,246)( 5,241)( 6,244)( 7,243)
( 8,242)( 9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,249)( 18,252)( 19,251)( 20,250)( 21,285)( 22,288)( 23,287)
( 24,286)( 25,281)( 26,284)( 27,283)( 28,282)( 29,297)( 30,300)( 31,299)
( 32,298)( 33,293)( 34,296)( 35,295)( 36,294)( 37,289)( 38,292)( 39,291)
( 40,290)( 41,265)( 42,268)( 43,267)( 44,266)( 45,261)( 46,264)( 47,263)
( 48,262)( 49,277)( 50,280)( 51,279)( 52,278)( 53,273)( 54,276)( 55,275)
( 56,274)( 57,269)( 58,272)( 59,271)( 60,270)( 61,185)( 62,188)( 63,187)
( 64,186)( 65,181)( 66,184)( 67,183)( 68,182)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)
( 80,190)( 81,225)( 82,228)( 83,227)( 84,226)( 85,221)( 86,224)( 87,223)
( 88,222)( 89,237)( 90,240)( 91,239)( 92,238)( 93,233)( 94,236)( 95,235)
( 96,234)( 97,229)( 98,232)( 99,231)(100,230)(101,205)(102,208)(103,207)
(104,206)(105,201)(106,204)(107,203)(108,202)(109,217)(110,220)(111,219)
(112,218)(113,213)(114,216)(115,215)(116,214)(117,209)(118,212)(119,211)
(120,210)(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)
(128,302)(129,317)(130,320)(131,319)(132,318)(133,313)(134,316)(135,315)
(136,314)(137,309)(138,312)(139,311)(140,310)(141,345)(142,348)(143,347)
(144,346)(145,341)(146,344)(147,343)(148,342)(149,357)(150,360)(151,359)
(152,358)(153,353)(154,356)(155,355)(156,354)(157,349)(158,352)(159,351)
(160,350)(161,325)(162,328)(163,327)(164,326)(165,321)(166,324)(167,323)
(168,322)(169,337)(170,340)(171,339)(172,338)(173,333)(174,336)(175,335)
(176,334)(177,329)(178,332)(179,331)(180,330);
s3 := Sym(360)!( 1,182)( 2,181)( 3,184)( 4,183)( 5,186)( 6,185)( 7,188)
( 8,187)( 9,190)( 10,189)( 11,192)( 12,191)( 13,194)( 14,193)( 15,196)
( 16,195)( 17,198)( 18,197)( 19,200)( 20,199)( 21,202)( 22,201)( 23,204)
( 24,203)( 25,206)( 26,205)( 27,208)( 28,207)( 29,210)( 30,209)( 31,212)
( 32,211)( 33,214)( 34,213)( 35,216)( 36,215)( 37,218)( 38,217)( 39,220)
( 40,219)( 41,222)( 42,221)( 43,224)( 44,223)( 45,226)( 46,225)( 47,228)
( 48,227)( 49,230)( 50,229)( 51,232)( 52,231)( 53,234)( 54,233)( 55,236)
( 56,235)( 57,238)( 58,237)( 59,240)( 60,239)( 61,242)( 62,241)( 63,244)
( 64,243)( 65,246)( 66,245)( 67,248)( 68,247)( 69,250)( 70,249)( 71,252)
( 72,251)( 73,254)( 74,253)( 75,256)( 76,255)( 77,258)( 78,257)( 79,260)
( 80,259)( 81,262)( 82,261)( 83,264)( 84,263)( 85,266)( 86,265)( 87,268)
( 88,267)( 89,270)( 90,269)( 91,272)( 92,271)( 93,274)( 94,273)( 95,276)
( 96,275)( 97,278)( 98,277)( 99,280)(100,279)(101,282)(102,281)(103,284)
(104,283)(105,286)(106,285)(107,288)(108,287)(109,290)(110,289)(111,292)
(112,291)(113,294)(114,293)(115,296)(116,295)(117,298)(118,297)(119,300)
(120,299)(121,302)(122,301)(123,304)(124,303)(125,306)(126,305)(127,308)
(128,307)(129,310)(130,309)(131,312)(132,311)(133,314)(134,313)(135,316)
(136,315)(137,318)(138,317)(139,320)(140,319)(141,322)(142,321)(143,324)
(144,323)(145,326)(146,325)(147,328)(148,327)(149,330)(150,329)(151,332)
(152,331)(153,334)(154,333)(155,336)(156,335)(157,338)(158,337)(159,340)
(160,339)(161,342)(162,341)(163,344)(164,343)(165,346)(166,345)(167,348)
(168,347)(169,350)(170,349)(171,352)(172,351)(173,354)(174,353)(175,356)
(176,355)(177,358)(178,357)(179,360)(180,359);
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*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1 >;
References : None.
to this polytope