Polytope of Type {2,45,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,45,4}*1440
if this polytope has a name.
Group : SmallGroup(1440,4575)
Rank : 4
Schlafli Type : {2,45,4}
Number of vertices, edges, etc : 2, 90, 180, 8
Order of s0s1s2s3 : 90
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,45,4}*720
3-fold quotients : {2,15,4}*480
4-fold quotients : {2,45,2}*360
5-fold quotients : {2,9,4}*288
6-fold quotients : {2,15,4}*240
10-fold quotients : {2,9,4}*144
12-fold quotients : {2,15,2}*120
15-fold quotients : {2,3,4}*96
20-fold quotients : {2,9,2}*72
30-fold quotients : {2,3,4}*48
36-fold quotients : {2,5,2}*40
60-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 51)( 16, 53)( 17, 52)
( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)( 25, 56)
( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)( 33, 48)
( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)( 65,128)
( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)( 73,132)
( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)( 81,172)
( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)( 89,164)
( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)( 97,168)
( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)(105,148)
(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)(113,140)
(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)(121,144)
(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)(196,233)
(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)(204,237)
(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)(212,229)
(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)(244,309)
(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)(252,313)
(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)(260,353)
(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)(268,345)
(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)(276,349)
(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)(284,329)
(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)(292,321)
(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)(300,325)
(301,324)(302,326);;
s2 := ( 3, 75)( 4, 76)( 5, 78)( 6, 77)( 7, 83)( 8, 84)( 9, 86)( 10, 85)
( 11, 79)( 12, 80)( 13, 82)( 14, 81)( 15, 63)( 16, 64)( 17, 66)( 18, 65)
( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27,111)( 28,112)( 29,114)( 30,113)( 31,119)( 32,120)( 33,122)( 34,121)
( 35,115)( 36,116)( 37,118)( 38,117)( 39, 99)( 40,100)( 41,102)( 42,101)
( 43,107)( 44,108)( 45,110)( 46,109)( 47,103)( 48,104)( 49,106)( 50,105)
( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 95)( 56, 96)( 57, 98)( 58, 97)
( 59, 91)( 60, 92)( 61, 94)( 62, 93)(123,139)(124,140)(125,142)(126,141)
(127,135)(128,136)(129,138)(130,137)(131,143)(132,144)(133,146)(134,145)
(147,175)(148,176)(149,178)(150,177)(151,171)(152,172)(153,174)(154,173)
(155,179)(156,180)(157,182)(158,181)(159,163)(160,164)(161,166)(162,165)
(169,170)(183,255)(184,256)(185,258)(186,257)(187,263)(188,264)(189,266)
(190,265)(191,259)(192,260)(193,262)(194,261)(195,243)(196,244)(197,246)
(198,245)(199,251)(200,252)(201,254)(202,253)(203,247)(204,248)(205,250)
(206,249)(207,291)(208,292)(209,294)(210,293)(211,299)(212,300)(213,302)
(214,301)(215,295)(216,296)(217,298)(218,297)(219,279)(220,280)(221,282)
(222,281)(223,287)(224,288)(225,290)(226,289)(227,283)(228,284)(229,286)
(230,285)(231,267)(232,268)(233,270)(234,269)(235,275)(236,276)(237,278)
(238,277)(239,271)(240,272)(241,274)(242,273)(303,319)(304,320)(305,322)
(306,321)(307,315)(308,316)(309,318)(310,317)(311,323)(312,324)(313,326)
(314,325)(327,355)(328,356)(329,358)(330,357)(331,351)(332,352)(333,354)
(334,353)(335,359)(336,360)(337,362)(338,361)(339,343)(340,344)(341,346)
(342,345)(349,350);;
s3 := ( 3,186)( 4,185)( 5,184)( 6,183)( 7,190)( 8,189)( 9,188)( 10,187)
( 11,194)( 12,193)( 13,192)( 14,191)( 15,198)( 16,197)( 17,196)( 18,195)
( 19,202)( 20,201)( 21,200)( 22,199)( 23,206)( 24,205)( 25,204)( 26,203)
( 27,210)( 28,209)( 29,208)( 30,207)( 31,214)( 32,213)( 33,212)( 34,211)
( 35,218)( 36,217)( 37,216)( 38,215)( 39,222)( 40,221)( 41,220)( 42,219)
( 43,226)( 44,225)( 45,224)( 46,223)( 47,230)( 48,229)( 49,228)( 50,227)
( 51,234)( 52,233)( 53,232)( 54,231)( 55,238)( 56,237)( 57,236)( 58,235)
( 59,242)( 60,241)( 61,240)( 62,239)( 63,246)( 64,245)( 65,244)( 66,243)
( 67,250)( 68,249)( 69,248)( 70,247)( 71,254)( 72,253)( 73,252)( 74,251)
( 75,258)( 76,257)( 77,256)( 78,255)( 79,262)( 80,261)( 81,260)( 82,259)
( 83,266)( 84,265)( 85,264)( 86,263)( 87,270)( 88,269)( 89,268)( 90,267)
( 91,274)( 92,273)( 93,272)( 94,271)( 95,278)( 96,277)( 97,276)( 98,275)
( 99,282)(100,281)(101,280)(102,279)(103,286)(104,285)(105,284)(106,283)
(107,290)(108,289)(109,288)(110,287)(111,294)(112,293)(113,292)(114,291)
(115,298)(116,297)(117,296)(118,295)(119,302)(120,301)(121,300)(122,299)
(123,306)(124,305)(125,304)(126,303)(127,310)(128,309)(129,308)(130,307)
(131,314)(132,313)(133,312)(134,311)(135,318)(136,317)(137,316)(138,315)
(139,322)(140,321)(141,320)(142,319)(143,326)(144,325)(145,324)(146,323)
(147,330)(148,329)(149,328)(150,327)(151,334)(152,333)(153,332)(154,331)
(155,338)(156,337)(157,336)(158,335)(159,342)(160,341)(161,340)(162,339)
(163,346)(164,345)(165,344)(166,343)(167,350)(168,349)(169,348)(170,347)
(171,354)(172,353)(173,352)(174,351)(175,358)(176,357)(177,356)(178,355)
(179,362)(180,361)(181,360)(182,359);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*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*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(362)!(1,2);
s1 := Sym(362)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 51)( 16, 53)
( 17, 52)( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)
( 25, 56)( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)
( 33, 48)( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)
( 65,128)( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)
( 73,132)( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)
( 81,172)( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)
( 89,164)( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)
( 97,168)( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)
(105,148)(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)
(113,140)(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)
(121,144)(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)
(196,233)(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)
(204,237)(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)
(212,229)(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)
(244,309)(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)
(252,313)(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)
(260,353)(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)
(268,345)(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)
(276,349)(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)
(284,329)(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)
(292,321)(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)
(300,325)(301,324)(302,326);
s2 := Sym(362)!( 3, 75)( 4, 76)( 5, 78)( 6, 77)( 7, 83)( 8, 84)( 9, 86)
( 10, 85)( 11, 79)( 12, 80)( 13, 82)( 14, 81)( 15, 63)( 16, 64)( 17, 66)
( 18, 65)( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)
( 26, 69)( 27,111)( 28,112)( 29,114)( 30,113)( 31,119)( 32,120)( 33,122)
( 34,121)( 35,115)( 36,116)( 37,118)( 38,117)( 39, 99)( 40,100)( 41,102)
( 42,101)( 43,107)( 44,108)( 45,110)( 46,109)( 47,103)( 48,104)( 49,106)
( 50,105)( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 95)( 56, 96)( 57, 98)
( 58, 97)( 59, 91)( 60, 92)( 61, 94)( 62, 93)(123,139)(124,140)(125,142)
(126,141)(127,135)(128,136)(129,138)(130,137)(131,143)(132,144)(133,146)
(134,145)(147,175)(148,176)(149,178)(150,177)(151,171)(152,172)(153,174)
(154,173)(155,179)(156,180)(157,182)(158,181)(159,163)(160,164)(161,166)
(162,165)(169,170)(183,255)(184,256)(185,258)(186,257)(187,263)(188,264)
(189,266)(190,265)(191,259)(192,260)(193,262)(194,261)(195,243)(196,244)
(197,246)(198,245)(199,251)(200,252)(201,254)(202,253)(203,247)(204,248)
(205,250)(206,249)(207,291)(208,292)(209,294)(210,293)(211,299)(212,300)
(213,302)(214,301)(215,295)(216,296)(217,298)(218,297)(219,279)(220,280)
(221,282)(222,281)(223,287)(224,288)(225,290)(226,289)(227,283)(228,284)
(229,286)(230,285)(231,267)(232,268)(233,270)(234,269)(235,275)(236,276)
(237,278)(238,277)(239,271)(240,272)(241,274)(242,273)(303,319)(304,320)
(305,322)(306,321)(307,315)(308,316)(309,318)(310,317)(311,323)(312,324)
(313,326)(314,325)(327,355)(328,356)(329,358)(330,357)(331,351)(332,352)
(333,354)(334,353)(335,359)(336,360)(337,362)(338,361)(339,343)(340,344)
(341,346)(342,345)(349,350);
s3 := Sym(362)!( 3,186)( 4,185)( 5,184)( 6,183)( 7,190)( 8,189)( 9,188)
( 10,187)( 11,194)( 12,193)( 13,192)( 14,191)( 15,198)( 16,197)( 17,196)
( 18,195)( 19,202)( 20,201)( 21,200)( 22,199)( 23,206)( 24,205)( 25,204)
( 26,203)( 27,210)( 28,209)( 29,208)( 30,207)( 31,214)( 32,213)( 33,212)
( 34,211)( 35,218)( 36,217)( 37,216)( 38,215)( 39,222)( 40,221)( 41,220)
( 42,219)( 43,226)( 44,225)( 45,224)( 46,223)( 47,230)( 48,229)( 49,228)
( 50,227)( 51,234)( 52,233)( 53,232)( 54,231)( 55,238)( 56,237)( 57,236)
( 58,235)( 59,242)( 60,241)( 61,240)( 62,239)( 63,246)( 64,245)( 65,244)
( 66,243)( 67,250)( 68,249)( 69,248)( 70,247)( 71,254)( 72,253)( 73,252)
( 74,251)( 75,258)( 76,257)( 77,256)( 78,255)( 79,262)( 80,261)( 81,260)
( 82,259)( 83,266)( 84,265)( 85,264)( 86,263)( 87,270)( 88,269)( 89,268)
( 90,267)( 91,274)( 92,273)( 93,272)( 94,271)( 95,278)( 96,277)( 97,276)
( 98,275)( 99,282)(100,281)(101,280)(102,279)(103,286)(104,285)(105,284)
(106,283)(107,290)(108,289)(109,288)(110,287)(111,294)(112,293)(113,292)
(114,291)(115,298)(116,297)(117,296)(118,295)(119,302)(120,301)(121,300)
(122,299)(123,306)(124,305)(125,304)(126,303)(127,310)(128,309)(129,308)
(130,307)(131,314)(132,313)(133,312)(134,311)(135,318)(136,317)(137,316)
(138,315)(139,322)(140,321)(141,320)(142,319)(143,326)(144,325)(145,324)
(146,323)(147,330)(148,329)(149,328)(150,327)(151,334)(152,333)(153,332)
(154,331)(155,338)(156,337)(157,336)(158,335)(159,342)(160,341)(161,340)
(162,339)(163,346)(164,345)(165,344)(166,343)(167,350)(168,349)(169,348)
(170,347)(171,354)(172,353)(173,352)(174,351)(175,358)(176,357)(177,356)
(178,355)(179,362)(180,361)(181,360)(182,359);
poly := sub<Sym(362)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*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 >;
to this polytope