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Polytope of Type {4,6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,30}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5712)
Rank : 4
Schlafli Type : {4,6,30}
Number of vertices, edges, etc : 4, 12, 90, 30
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) :
   2-fold quotients : {4,6,15}*720, {2,6,30}*720c
   3-fold quotients : {4,2,30}*480
   4-fold quotients : {2,6,15}*360
   5-fold quotients : {4,6,6}*288c
   6-fold quotients : {4,2,15}*240, {2,2,30}*240
   9-fold quotients : {4,2,10}*160
   10-fold quotients : {4,6,3}*144, {2,6,6}*144b
   12-fold quotients : {2,2,15}*120
   15-fold quotients : {4,2,6}*96
   18-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {2,6,3}*72
   30-fold quotients : {4,2,3}*48, {2,2,6}*48
   36-fold quotients : {2,2,5}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {2,2,3}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)
(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)
( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)
( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180)(181,271)(182,272)(183,273)(184,274)(185,275)(186,276)
(187,277)(188,278)(189,279)(190,280)(191,281)(192,282)(193,283)(194,284)
(195,285)(196,286)(197,287)(198,288)(199,289)(200,290)(201,291)(202,292)
(203,293)(204,294)(205,295)(206,296)(207,297)(208,298)(209,299)(210,300)
(211,301)(212,302)(213,303)(214,304)(215,305)(216,306)(217,307)(218,308)
(219,309)(220,310)(221,311)(222,312)(223,313)(224,314)(225,315)(226,316)
(227,317)(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)
(235,325)(236,326)(237,327)(238,328)(239,329)(240,330)(241,331)(242,332)
(243,333)(244,334)(245,335)(246,336)(247,337)(248,338)(249,339)(250,340)
(251,341)(252,342)(253,343)(254,344)(255,345)(256,346)(257,347)(258,348)
(259,349)(260,350)(261,351)(262,352)(263,353)(264,354)(265,355)(266,356)
(267,357)(268,358)(269,359)(270,360);;
s1 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)
( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)
( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)(100,145)
(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)(108,168)
(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)
(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)(124,154)
(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)
(133,163)(134,164)(135,165)(196,211)(197,212)(198,213)(199,214)(200,215)
(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)(208,223)
(209,224)(210,225)(241,256)(242,257)(243,258)(244,259)(245,260)(246,261)
(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)(253,268)(254,269)
(255,270)(271,316)(272,317)(273,318)(274,319)(275,320)(276,321)(277,322)
(278,323)(279,324)(280,325)(281,326)(282,327)(283,328)(284,329)(285,330)
(286,346)(287,347)(288,348)(289,349)(290,350)(291,351)(292,352)(293,353)
(294,354)(295,355)(296,356)(297,357)(298,358)(299,359)(300,360)(301,331)
(302,332)(303,333)(304,334)(305,335)(306,336)(307,337)(308,338)(309,339)
(310,340)(311,341)(312,342)(313,343)(314,344)(315,345);;
s2 := (  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 26)(  7, 30)(  8, 29)
(  9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)( 55, 72)
( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)( 81, 86)
( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,106)( 92,110)( 93,109)( 94,108)
( 95,107)( 96,116)( 97,120)( 98,119)( 99,118)(100,117)(101,111)(102,115)
(103,114)(104,113)(105,112)(122,125)(123,124)(126,131)(127,135)(128,134)
(129,133)(130,132)(136,151)(137,155)(138,154)(139,153)(140,152)(141,161)
(142,165)(143,164)(144,163)(145,162)(146,156)(147,160)(148,159)(149,158)
(150,157)(167,170)(168,169)(171,176)(172,180)(173,179)(174,178)(175,177)
(181,196)(182,200)(183,199)(184,198)(185,197)(186,206)(187,210)(188,209)
(189,208)(190,207)(191,201)(192,205)(193,204)(194,203)(195,202)(212,215)
(213,214)(216,221)(217,225)(218,224)(219,223)(220,222)(226,241)(227,245)
(228,244)(229,243)(230,242)(231,251)(232,255)(233,254)(234,253)(235,252)
(236,246)(237,250)(238,249)(239,248)(240,247)(257,260)(258,259)(261,266)
(262,270)(263,269)(264,268)(265,267)(271,286)(272,290)(273,289)(274,288)
(275,287)(276,296)(277,300)(278,299)(279,298)(280,297)(281,291)(282,295)
(283,294)(284,293)(285,292)(302,305)(303,304)(306,311)(307,315)(308,314)
(309,313)(310,312)(316,331)(317,335)(318,334)(319,333)(320,332)(321,341)
(322,345)(323,344)(324,343)(325,342)(326,336)(327,340)(328,339)(329,338)
(330,337)(347,350)(348,349)(351,356)(352,360)(353,359)(354,358)(355,357);;
s3 := (  1,187)(  2,186)(  3,190)(  4,189)(  5,188)(  6,182)(  7,181)(  8,185)
(  9,184)( 10,183)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)( 16,217)
( 17,216)( 18,220)( 19,219)( 20,218)( 21,212)( 22,211)( 23,215)( 24,214)
( 25,213)( 26,222)( 27,221)( 28,225)( 29,224)( 30,223)( 31,202)( 32,201)
( 33,205)( 34,204)( 35,203)( 36,197)( 37,196)( 38,200)( 39,199)( 40,198)
( 41,207)( 42,206)( 43,210)( 44,209)( 45,208)( 46,232)( 47,231)( 48,235)
( 49,234)( 50,233)( 51,227)( 52,226)( 53,230)( 54,229)( 55,228)( 56,237)
( 57,236)( 58,240)( 59,239)( 60,238)( 61,262)( 62,261)( 63,265)( 64,264)
( 65,263)( 66,257)( 67,256)( 68,260)( 69,259)( 70,258)( 71,267)( 72,266)
( 73,270)( 74,269)( 75,268)( 76,247)( 77,246)( 78,250)( 79,249)( 80,248)
( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,252)( 87,251)( 88,255)
( 89,254)( 90,253)( 91,277)( 92,276)( 93,280)( 94,279)( 95,278)( 96,272)
( 97,271)( 98,275)( 99,274)(100,273)(101,282)(102,281)(103,285)(104,284)
(105,283)(106,307)(107,306)(108,310)(109,309)(110,308)(111,302)(112,301)
(113,305)(114,304)(115,303)(116,312)(117,311)(118,315)(119,314)(120,313)
(121,292)(122,291)(123,295)(124,294)(125,293)(126,287)(127,286)(128,290)
(129,289)(130,288)(131,297)(132,296)(133,300)(134,299)(135,298)(136,322)
(137,321)(138,325)(139,324)(140,323)(141,317)(142,316)(143,320)(144,319)
(145,318)(146,327)(147,326)(148,330)(149,329)(150,328)(151,352)(152,351)
(153,355)(154,354)(155,353)(156,347)(157,346)(158,350)(159,349)(160,348)
(161,357)(162,356)(163,360)(164,359)(165,358)(166,337)(167,336)(168,340)
(169,339)(170,338)(171,332)(172,331)(173,335)(174,334)(175,333)(176,342)
(177,341)(178,345)(179,344)(180,343);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)
(  8, 98)(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)
( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180)(181,271)(182,272)(183,273)(184,274)(185,275)
(186,276)(187,277)(188,278)(189,279)(190,280)(191,281)(192,282)(193,283)
(194,284)(195,285)(196,286)(197,287)(198,288)(199,289)(200,290)(201,291)
(202,292)(203,293)(204,294)(205,295)(206,296)(207,297)(208,298)(209,299)
(210,300)(211,301)(212,302)(213,303)(214,304)(215,305)(216,306)(217,307)
(218,308)(219,309)(220,310)(221,311)(222,312)(223,313)(224,314)(225,315)
(226,316)(227,317)(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)
(234,324)(235,325)(236,326)(237,327)(238,328)(239,329)(240,330)(241,331)
(242,332)(243,333)(244,334)(245,335)(246,336)(247,337)(248,338)(249,339)
(250,340)(251,341)(252,342)(253,343)(254,344)(255,345)(256,346)(257,347)
(258,348)(259,349)(260,350)(261,351)(262,352)(263,353)(264,354)(265,355)
(266,356)(267,357)(268,358)(269,359)(270,360);
s1 := Sym(360)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)
(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)
(108,168)(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)
(116,176)(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)
(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)
(132,162)(133,163)(134,164)(135,165)(196,211)(197,212)(198,213)(199,214)
(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)
(208,223)(209,224)(210,225)(241,256)(242,257)(243,258)(244,259)(245,260)
(246,261)(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)(253,268)
(254,269)(255,270)(271,316)(272,317)(273,318)(274,319)(275,320)(276,321)
(277,322)(278,323)(279,324)(280,325)(281,326)(282,327)(283,328)(284,329)
(285,330)(286,346)(287,347)(288,348)(289,349)(290,350)(291,351)(292,352)
(293,353)(294,354)(295,355)(296,356)(297,357)(298,358)(299,359)(300,360)
(301,331)(302,332)(303,333)(304,334)(305,335)(306,336)(307,337)(308,338)
(309,339)(310,340)(311,341)(312,342)(313,343)(314,344)(315,345);
s2 := Sym(360)!(  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 26)(  7, 30)
(  8, 29)(  9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)
( 32, 35)( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)
( 55, 72)( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)
( 81, 86)( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,106)( 92,110)( 93,109)
( 94,108)( 95,107)( 96,116)( 97,120)( 98,119)( 99,118)(100,117)(101,111)
(102,115)(103,114)(104,113)(105,112)(122,125)(123,124)(126,131)(127,135)
(128,134)(129,133)(130,132)(136,151)(137,155)(138,154)(139,153)(140,152)
(141,161)(142,165)(143,164)(144,163)(145,162)(146,156)(147,160)(148,159)
(149,158)(150,157)(167,170)(168,169)(171,176)(172,180)(173,179)(174,178)
(175,177)(181,196)(182,200)(183,199)(184,198)(185,197)(186,206)(187,210)
(188,209)(189,208)(190,207)(191,201)(192,205)(193,204)(194,203)(195,202)
(212,215)(213,214)(216,221)(217,225)(218,224)(219,223)(220,222)(226,241)
(227,245)(228,244)(229,243)(230,242)(231,251)(232,255)(233,254)(234,253)
(235,252)(236,246)(237,250)(238,249)(239,248)(240,247)(257,260)(258,259)
(261,266)(262,270)(263,269)(264,268)(265,267)(271,286)(272,290)(273,289)
(274,288)(275,287)(276,296)(277,300)(278,299)(279,298)(280,297)(281,291)
(282,295)(283,294)(284,293)(285,292)(302,305)(303,304)(306,311)(307,315)
(308,314)(309,313)(310,312)(316,331)(317,335)(318,334)(319,333)(320,332)
(321,341)(322,345)(323,344)(324,343)(325,342)(326,336)(327,340)(328,339)
(329,338)(330,337)(347,350)(348,349)(351,356)(352,360)(353,359)(354,358)
(355,357);
s3 := Sym(360)!(  1,187)(  2,186)(  3,190)(  4,189)(  5,188)(  6,182)(  7,181)
(  8,185)(  9,184)( 10,183)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)
( 16,217)( 17,216)( 18,220)( 19,219)( 20,218)( 21,212)( 22,211)( 23,215)
( 24,214)( 25,213)( 26,222)( 27,221)( 28,225)( 29,224)( 30,223)( 31,202)
( 32,201)( 33,205)( 34,204)( 35,203)( 36,197)( 37,196)( 38,200)( 39,199)
( 40,198)( 41,207)( 42,206)( 43,210)( 44,209)( 45,208)( 46,232)( 47,231)
( 48,235)( 49,234)( 50,233)( 51,227)( 52,226)( 53,230)( 54,229)( 55,228)
( 56,237)( 57,236)( 58,240)( 59,239)( 60,238)( 61,262)( 62,261)( 63,265)
( 64,264)( 65,263)( 66,257)( 67,256)( 68,260)( 69,259)( 70,258)( 71,267)
( 72,266)( 73,270)( 74,269)( 75,268)( 76,247)( 77,246)( 78,250)( 79,249)
( 80,248)( 81,242)( 82,241)( 83,245)( 84,244)( 85,243)( 86,252)( 87,251)
( 88,255)( 89,254)( 90,253)( 91,277)( 92,276)( 93,280)( 94,279)( 95,278)
( 96,272)( 97,271)( 98,275)( 99,274)(100,273)(101,282)(102,281)(103,285)
(104,284)(105,283)(106,307)(107,306)(108,310)(109,309)(110,308)(111,302)
(112,301)(113,305)(114,304)(115,303)(116,312)(117,311)(118,315)(119,314)
(120,313)(121,292)(122,291)(123,295)(124,294)(125,293)(126,287)(127,286)
(128,290)(129,289)(130,288)(131,297)(132,296)(133,300)(134,299)(135,298)
(136,322)(137,321)(138,325)(139,324)(140,323)(141,317)(142,316)(143,320)
(144,319)(145,318)(146,327)(147,326)(148,330)(149,329)(150,328)(151,352)
(152,351)(153,355)(154,354)(155,353)(156,347)(157,346)(158,350)(159,349)
(160,348)(161,357)(162,356)(163,360)(164,359)(165,358)(166,337)(167,336)
(168,340)(169,339)(170,338)(171,332)(172,331)(173,335)(174,334)(175,333)
(176,342)(177,341)(178,345)(179,344)(180,343);
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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