Polytope of Type {6,4,20}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,20}*1440
if this polytope has a name.
Group : SmallGroup(1440,4764)
Rank : 4
Schlafli Type : {6,4,20}
Number of vertices, edges, etc : 9, 18, 60, 20
Order of s0s1s2s3 : 20
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 : {6,4,10}*720
5-fold quotients : {6,4,4}*288
10-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 91)( 2, 92)( 3, 93)( 4, 94)( 5, 95)( 6,101)( 7,102)( 8,103)
( 9,104)( 10,105)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16,121)
( 17,122)( 18,123)( 19,124)( 20,125)( 21,131)( 22,132)( 23,133)( 24,134)
( 25,135)( 26,126)( 27,127)( 28,128)( 29,129)( 30,130)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)( 40,120)
( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,146)( 52,147)( 53,148)( 54,149)( 55,150)( 56,141)
( 57,142)( 58,143)( 59,144)( 60,145)( 61,166)( 62,167)( 63,168)( 64,169)
( 65,170)( 66,176)( 67,177)( 68,178)( 69,179)( 70,180)( 71,171)( 72,172)
( 73,173)( 74,174)( 75,175)( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)
( 81,161)( 82,162)( 83,163)( 84,164)( 85,165)( 86,156)( 87,157)( 88,158)
( 89,159)( 90,160)(181,271)(182,272)(183,273)(184,274)(185,275)(186,281)
(187,282)(188,283)(189,284)(190,285)(191,276)(192,277)(193,278)(194,279)
(195,280)(196,301)(197,302)(198,303)(199,304)(200,305)(201,311)(202,312)
(203,313)(204,314)(205,315)(206,306)(207,307)(208,308)(209,309)(210,310)
(211,286)(212,287)(213,288)(214,289)(215,290)(216,296)(217,297)(218,298)
(219,299)(220,300)(221,291)(222,292)(223,293)(224,294)(225,295)(226,316)
(227,317)(228,318)(229,319)(230,320)(231,326)(232,327)(233,328)(234,329)
(235,330)(236,321)(237,322)(238,323)(239,324)(240,325)(241,346)(242,347)
(243,348)(244,349)(245,350)(246,356)(247,357)(248,358)(249,359)(250,360)
(251,351)(252,352)(253,353)(254,354)(255,355)(256,331)(257,332)(258,333)
(259,334)(260,335)(261,341)(262,342)(263,343)(264,344)(265,345)(266,336)
(267,337)(268,338)(269,339)(270,340);;
s1 := ( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 11, 41)( 12, 42)( 13, 43)
( 14, 44)( 15, 45)( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)( 46, 61)
( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 56, 86)( 57, 87)( 58, 88)( 59, 89)
( 60, 90)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)( 91,106)( 92,107)
( 93,108)( 94,109)( 95,110)(101,131)(102,132)(103,133)(104,134)(105,135)
(111,126)(112,127)(113,128)(114,129)(115,130)(136,151)(137,152)(138,153)
(139,154)(140,155)(146,176)(147,177)(148,178)(149,179)(150,180)(156,171)
(157,172)(158,173)(159,174)(160,175)(181,286)(182,287)(183,288)(184,289)
(185,290)(186,276)(187,277)(188,278)(189,279)(190,280)(191,311)(192,312)
(193,313)(194,314)(195,315)(196,271)(197,272)(198,273)(199,274)(200,275)
(201,306)(202,307)(203,308)(204,309)(205,310)(206,296)(207,297)(208,298)
(209,299)(210,300)(211,301)(212,302)(213,303)(214,304)(215,305)(216,291)
(217,292)(218,293)(219,294)(220,295)(221,281)(222,282)(223,283)(224,284)
(225,285)(226,331)(227,332)(228,333)(229,334)(230,335)(231,321)(232,322)
(233,323)(234,324)(235,325)(236,356)(237,357)(238,358)(239,359)(240,360)
(241,316)(242,317)(243,318)(244,319)(245,320)(246,351)(247,352)(248,353)
(249,354)(250,355)(251,341)(252,342)(253,343)(254,344)(255,345)(256,346)
(257,347)(258,348)(259,349)(260,350)(261,336)(262,337)(263,338)(264,339)
(265,340)(266,326)(267,327)(268,328)(269,329)(270,330);;
s2 := ( 1,181)( 2,185)( 3,184)( 4,183)( 5,182)( 6,186)( 7,190)( 8,189)
( 9,188)( 10,187)( 11,191)( 12,195)( 13,194)( 14,193)( 15,192)( 16,221)
( 17,225)( 18,224)( 19,223)( 20,222)( 21,211)( 22,215)( 23,214)( 24,213)
( 25,212)( 26,216)( 27,220)( 28,219)( 29,218)( 30,217)( 31,201)( 32,205)
( 33,204)( 34,203)( 35,202)( 36,206)( 37,210)( 38,209)( 39,208)( 40,207)
( 41,196)( 42,200)( 43,199)( 44,198)( 45,197)( 46,226)( 47,230)( 48,229)
( 49,228)( 50,227)( 51,231)( 52,235)( 53,234)( 54,233)( 55,232)( 56,236)
( 57,240)( 58,239)( 59,238)( 60,237)( 61,266)( 62,270)( 63,269)( 64,268)
( 65,267)( 66,256)( 67,260)( 68,259)( 69,258)( 70,257)( 71,261)( 72,265)
( 73,264)( 74,263)( 75,262)( 76,246)( 77,250)( 78,249)( 79,248)( 80,247)
( 81,251)( 82,255)( 83,254)( 84,253)( 85,252)( 86,241)( 87,245)( 88,244)
( 89,243)( 90,242)( 91,271)( 92,275)( 93,274)( 94,273)( 95,272)( 96,276)
( 97,280)( 98,279)( 99,278)(100,277)(101,281)(102,285)(103,284)(104,283)
(105,282)(106,311)(107,315)(108,314)(109,313)(110,312)(111,301)(112,305)
(113,304)(114,303)(115,302)(116,306)(117,310)(118,309)(119,308)(120,307)
(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)(128,299)
(129,298)(130,297)(131,286)(132,290)(133,289)(134,288)(135,287)(136,316)
(137,320)(138,319)(139,318)(140,317)(141,321)(142,325)(143,324)(144,323)
(145,322)(146,326)(147,330)(148,329)(149,328)(150,327)(151,356)(152,360)
(153,359)(154,358)(155,357)(156,346)(157,350)(158,349)(159,348)(160,347)
(161,351)(162,355)(163,354)(164,353)(165,352)(166,336)(167,340)(168,339)
(169,338)(170,337)(171,341)(172,345)(173,344)(174,343)(175,342)(176,331)
(177,335)(178,334)(179,333)(180,332);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)
(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)(118,120)
(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)
(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)(158,160)
(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)(178,180)
(181,227)(182,226)(183,230)(184,229)(185,228)(186,232)(187,231)(188,235)
(189,234)(190,233)(191,237)(192,236)(193,240)(194,239)(195,238)(196,242)
(197,241)(198,245)(199,244)(200,243)(201,247)(202,246)(203,250)(204,249)
(205,248)(206,252)(207,251)(208,255)(209,254)(210,253)(211,257)(212,256)
(213,260)(214,259)(215,258)(216,262)(217,261)(218,265)(219,264)(220,263)
(221,267)(222,266)(223,270)(224,269)(225,268)(271,317)(272,316)(273,320)
(274,319)(275,318)(276,322)(277,321)(278,325)(279,324)(280,323)(281,327)
(282,326)(283,330)(284,329)(285,328)(286,332)(287,331)(288,335)(289,334)
(290,333)(291,337)(292,336)(293,340)(294,339)(295,338)(296,342)(297,341)
(298,345)(299,344)(300,343)(301,347)(302,346)(303,350)(304,349)(305,348)
(306,352)(307,351)(308,355)(309,354)(310,353)(311,357)(312,356)(313,360)
(314,359)(315,358);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1,
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)( 6,101)( 7,102)
( 8,103)( 9,104)( 10,105)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)
( 16,121)( 17,122)( 18,123)( 19,124)( 20,125)( 21,131)( 22,132)( 23,133)
( 24,134)( 25,135)( 26,126)( 27,127)( 28,128)( 29,129)( 30,130)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,116)( 37,117)( 38,118)( 39,119)
( 40,120)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,146)( 52,147)( 53,148)( 54,149)( 55,150)
( 56,141)( 57,142)( 58,143)( 59,144)( 60,145)( 61,166)( 62,167)( 63,168)
( 64,169)( 65,170)( 66,176)( 67,177)( 68,178)( 69,179)( 70,180)( 71,171)
( 72,172)( 73,173)( 74,174)( 75,175)( 76,151)( 77,152)( 78,153)( 79,154)
( 80,155)( 81,161)( 82,162)( 83,163)( 84,164)( 85,165)( 86,156)( 87,157)
( 88,158)( 89,159)( 90,160)(181,271)(182,272)(183,273)(184,274)(185,275)
(186,281)(187,282)(188,283)(189,284)(190,285)(191,276)(192,277)(193,278)
(194,279)(195,280)(196,301)(197,302)(198,303)(199,304)(200,305)(201,311)
(202,312)(203,313)(204,314)(205,315)(206,306)(207,307)(208,308)(209,309)
(210,310)(211,286)(212,287)(213,288)(214,289)(215,290)(216,296)(217,297)
(218,298)(219,299)(220,300)(221,291)(222,292)(223,293)(224,294)(225,295)
(226,316)(227,317)(228,318)(229,319)(230,320)(231,326)(232,327)(233,328)
(234,329)(235,330)(236,321)(237,322)(238,323)(239,324)(240,325)(241,346)
(242,347)(243,348)(244,349)(245,350)(246,356)(247,357)(248,358)(249,359)
(250,360)(251,351)(252,352)(253,353)(254,354)(255,355)(256,331)(257,332)
(258,333)(259,334)(260,335)(261,341)(262,342)(263,343)(264,344)(265,345)
(266,336)(267,337)(268,338)(269,339)(270,340);
s1 := Sym(360)!( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 11, 41)( 12, 42)
( 13, 43)( 14, 44)( 15, 45)( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)
( 46, 61)( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 56, 86)( 57, 87)( 58, 88)
( 59, 89)( 60, 90)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)( 91,106)
( 92,107)( 93,108)( 94,109)( 95,110)(101,131)(102,132)(103,133)(104,134)
(105,135)(111,126)(112,127)(113,128)(114,129)(115,130)(136,151)(137,152)
(138,153)(139,154)(140,155)(146,176)(147,177)(148,178)(149,179)(150,180)
(156,171)(157,172)(158,173)(159,174)(160,175)(181,286)(182,287)(183,288)
(184,289)(185,290)(186,276)(187,277)(188,278)(189,279)(190,280)(191,311)
(192,312)(193,313)(194,314)(195,315)(196,271)(197,272)(198,273)(199,274)
(200,275)(201,306)(202,307)(203,308)(204,309)(205,310)(206,296)(207,297)
(208,298)(209,299)(210,300)(211,301)(212,302)(213,303)(214,304)(215,305)
(216,291)(217,292)(218,293)(219,294)(220,295)(221,281)(222,282)(223,283)
(224,284)(225,285)(226,331)(227,332)(228,333)(229,334)(230,335)(231,321)
(232,322)(233,323)(234,324)(235,325)(236,356)(237,357)(238,358)(239,359)
(240,360)(241,316)(242,317)(243,318)(244,319)(245,320)(246,351)(247,352)
(248,353)(249,354)(250,355)(251,341)(252,342)(253,343)(254,344)(255,345)
(256,346)(257,347)(258,348)(259,349)(260,350)(261,336)(262,337)(263,338)
(264,339)(265,340)(266,326)(267,327)(268,328)(269,329)(270,330);
s2 := Sym(360)!( 1,181)( 2,185)( 3,184)( 4,183)( 5,182)( 6,186)( 7,190)
( 8,189)( 9,188)( 10,187)( 11,191)( 12,195)( 13,194)( 14,193)( 15,192)
( 16,221)( 17,225)( 18,224)( 19,223)( 20,222)( 21,211)( 22,215)( 23,214)
( 24,213)( 25,212)( 26,216)( 27,220)( 28,219)( 29,218)( 30,217)( 31,201)
( 32,205)( 33,204)( 34,203)( 35,202)( 36,206)( 37,210)( 38,209)( 39,208)
( 40,207)( 41,196)( 42,200)( 43,199)( 44,198)( 45,197)( 46,226)( 47,230)
( 48,229)( 49,228)( 50,227)( 51,231)( 52,235)( 53,234)( 54,233)( 55,232)
( 56,236)( 57,240)( 58,239)( 59,238)( 60,237)( 61,266)( 62,270)( 63,269)
( 64,268)( 65,267)( 66,256)( 67,260)( 68,259)( 69,258)( 70,257)( 71,261)
( 72,265)( 73,264)( 74,263)( 75,262)( 76,246)( 77,250)( 78,249)( 79,248)
( 80,247)( 81,251)( 82,255)( 83,254)( 84,253)( 85,252)( 86,241)( 87,245)
( 88,244)( 89,243)( 90,242)( 91,271)( 92,275)( 93,274)( 94,273)( 95,272)
( 96,276)( 97,280)( 98,279)( 99,278)(100,277)(101,281)(102,285)(103,284)
(104,283)(105,282)(106,311)(107,315)(108,314)(109,313)(110,312)(111,301)
(112,305)(113,304)(114,303)(115,302)(116,306)(117,310)(118,309)(119,308)
(120,307)(121,291)(122,295)(123,294)(124,293)(125,292)(126,296)(127,300)
(128,299)(129,298)(130,297)(131,286)(132,290)(133,289)(134,288)(135,287)
(136,316)(137,320)(138,319)(139,318)(140,317)(141,321)(142,325)(143,324)
(144,323)(145,322)(146,326)(147,330)(148,329)(149,328)(150,327)(151,356)
(152,360)(153,359)(154,358)(155,357)(156,346)(157,350)(158,349)(159,348)
(160,347)(161,351)(162,355)(163,354)(164,353)(165,352)(166,336)(167,340)
(168,339)(169,338)(170,337)(171,341)(172,345)(173,344)(174,343)(175,342)
(176,331)(177,335)(178,334)(179,333)(180,332);
s3 := Sym(360)!( 1, 2)( 3, 5)( 6, 7)( 8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)
( 98,100)(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)
(118,120)(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)
(138,140)(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)
(158,160)(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)
(178,180)(181,227)(182,226)(183,230)(184,229)(185,228)(186,232)(187,231)
(188,235)(189,234)(190,233)(191,237)(192,236)(193,240)(194,239)(195,238)
(196,242)(197,241)(198,245)(199,244)(200,243)(201,247)(202,246)(203,250)
(204,249)(205,248)(206,252)(207,251)(208,255)(209,254)(210,253)(211,257)
(212,256)(213,260)(214,259)(215,258)(216,262)(217,261)(218,265)(219,264)
(220,263)(221,267)(222,266)(223,270)(224,269)(225,268)(271,317)(272,316)
(273,320)(274,319)(275,318)(276,322)(277,321)(278,325)(279,324)(280,323)
(281,327)(282,326)(283,330)(284,329)(285,328)(286,332)(287,331)(288,335)
(289,334)(290,333)(291,337)(292,336)(293,340)(294,339)(295,338)(296,342)
(297,341)(298,345)(299,344)(300,343)(301,347)(302,346)(303,350)(304,349)
(305,348)(306,352)(307,351)(308,355)(309,354)(310,353)(311,357)(312,356)
(313,360)(314,359)(315,358);
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1,
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