Polytope of Type {6,30,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30,4}*1440e
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}*480b
5-fold quotients : {6,6,4}*288e
6-fold quotients : {2,15,4}*240
10-fold quotients : {6,3,4}*144
15-fold quotients : {2,6,4}*96c
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, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)
(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)
(353,354)(355,356)(357,358)(359,360);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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)!( 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, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)
(351,352)(353,354)(355,356)(357,358)(359,360);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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