Polytope of Type {45,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,4}*720
if this polytope has a name.
Group : SmallGroup(720,397)
Rank : 3
Schlafli Type : {45,4}
Number of vertices, edges, etc : 90, 180, 8
Order of s0s1s2 : 90
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {45,4,2} of size 1440
Vertex Figure Of :
   {2,45,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {45,4}*360
   3-fold quotients : {15,4}*240
   4-fold quotients : {45,2}*180
   5-fold quotients : {9,4}*144
   6-fold quotients : {15,4}*120
   10-fold quotients : {9,4}*72
   12-fold quotients : {15,2}*60
   15-fold quotients : {3,4}*48
   20-fold quotients : {9,2}*36
   30-fold quotients : {3,4}*24
   36-fold quotients : {5,2}*20
   60-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {45,8}*1440, {90,4}*1440
Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)
( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)
( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)
( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)
( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)
(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)
(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)
(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)(194,231)
(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)(202,235)
(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)(210,227)
(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)(242,307)
(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)(250,311)
(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)(258,351)
(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)(266,343)
(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)(274,347)
(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)(282,327)
(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)(290,319)
(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)(298,323)
(299,322)(300,324);;
s1 := (  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)(  8, 83)
(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)
( 25,109)( 26,110)( 27,112)( 28,111)( 29,117)( 30,118)( 31,120)( 32,119)
( 33,113)( 34,114)( 35,116)( 36,115)( 37, 97)( 38, 98)( 39,100)( 40, 99)
( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)
( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)
( 57, 89)( 58, 90)( 59, 92)( 60, 91)(121,137)(122,138)(123,140)(124,139)
(125,133)(126,134)(127,136)(128,135)(129,141)(130,142)(131,144)(132,143)
(145,173)(146,174)(147,176)(148,175)(149,169)(150,170)(151,172)(152,171)
(153,177)(154,178)(155,180)(156,179)(157,161)(158,162)(159,164)(160,163)
(167,168)(181,253)(182,254)(183,256)(184,255)(185,261)(186,262)(187,264)
(188,263)(189,257)(190,258)(191,260)(192,259)(193,241)(194,242)(195,244)
(196,243)(197,249)(198,250)(199,252)(200,251)(201,245)(202,246)(203,248)
(204,247)(205,289)(206,290)(207,292)(208,291)(209,297)(210,298)(211,300)
(212,299)(213,293)(214,294)(215,296)(216,295)(217,277)(218,278)(219,280)
(220,279)(221,285)(222,286)(223,288)(224,287)(225,281)(226,282)(227,284)
(228,283)(229,265)(230,266)(231,268)(232,267)(233,273)(234,274)(235,276)
(236,275)(237,269)(238,270)(239,272)(240,271)(301,317)(302,318)(303,320)
(304,319)(305,313)(306,314)(307,316)(308,315)(309,321)(310,322)(311,324)
(312,323)(325,353)(326,354)(327,356)(328,355)(329,349)(330,350)(331,352)
(332,351)(333,357)(334,358)(335,360)(336,359)(337,341)(338,342)(339,344)
(340,343)(347,348);;
s2 := (  1,184)(  2,183)(  3,182)(  4,181)(  5,188)(  6,187)(  7,186)(  8,185)
(  9,192)( 10,191)( 11,190)( 12,189)( 13,196)( 14,195)( 15,194)( 16,193)
( 17,200)( 18,199)( 19,198)( 20,197)( 21,204)( 22,203)( 23,202)( 24,201)
( 25,208)( 26,207)( 27,206)( 28,205)( 29,212)( 30,211)( 31,210)( 32,209)
( 33,216)( 34,215)( 35,214)( 36,213)( 37,220)( 38,219)( 39,218)( 40,217)
( 41,224)( 42,223)( 43,222)( 44,221)( 45,228)( 46,227)( 47,226)( 48,225)
( 49,232)( 50,231)( 51,230)( 52,229)( 53,236)( 54,235)( 55,234)( 56,233)
( 57,240)( 58,239)( 59,238)( 60,237)( 61,244)( 62,243)( 63,242)( 64,241)
( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)( 72,249)
( 73,256)( 74,255)( 75,254)( 76,253)( 77,260)( 78,259)( 79,258)( 80,257)
( 81,264)( 82,263)( 83,262)( 84,261)( 85,268)( 86,267)( 87,266)( 88,265)
( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)( 96,273)
( 97,280)( 98,279)( 99,278)(100,277)(101,284)(102,283)(103,282)(104,281)
(105,288)(106,287)(107,286)(108,285)(109,292)(110,291)(111,290)(112,289)
(113,296)(114,295)(115,294)(116,293)(117,300)(118,299)(119,298)(120,297)
(121,304)(122,303)(123,302)(124,301)(125,308)(126,307)(127,306)(128,305)
(129,312)(130,311)(131,310)(132,309)(133,316)(134,315)(135,314)(136,313)
(137,320)(138,319)(139,318)(140,317)(141,324)(142,323)(143,322)(144,321)
(145,328)(146,327)(147,326)(148,325)(149,332)(150,331)(151,330)(152,329)
(153,336)(154,335)(155,334)(156,333)(157,340)(158,339)(159,338)(160,337)
(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)(168,345)
(169,352)(170,351)(171,350)(172,349)(173,356)(174,355)(175,354)(176,353)
(177,360)(178,359)(179,358)(180,357);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(360)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)
( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)
( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)
( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)
( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)
( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)
( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)
( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)
( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)
(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)
(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)
(119,142)(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)
(194,231)(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)
(202,235)(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)
(210,227)(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)
(242,307)(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)
(250,311)(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)
(258,351)(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)
(266,343)(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)
(274,347)(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)
(282,327)(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)
(290,319)(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)
(298,323)(299,322)(300,324);
s1 := Sym(360)!(  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)
(  8, 83)(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)
( 24, 67)( 25,109)( 26,110)( 27,112)( 28,111)( 29,117)( 30,118)( 31,120)
( 32,119)( 33,113)( 34,114)( 35,116)( 36,115)( 37, 97)( 38, 98)( 39,100)
( 40, 99)( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)
( 48,103)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)
( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)(121,137)(122,138)(123,140)
(124,139)(125,133)(126,134)(127,136)(128,135)(129,141)(130,142)(131,144)
(132,143)(145,173)(146,174)(147,176)(148,175)(149,169)(150,170)(151,172)
(152,171)(153,177)(154,178)(155,180)(156,179)(157,161)(158,162)(159,164)
(160,163)(167,168)(181,253)(182,254)(183,256)(184,255)(185,261)(186,262)
(187,264)(188,263)(189,257)(190,258)(191,260)(192,259)(193,241)(194,242)
(195,244)(196,243)(197,249)(198,250)(199,252)(200,251)(201,245)(202,246)
(203,248)(204,247)(205,289)(206,290)(207,292)(208,291)(209,297)(210,298)
(211,300)(212,299)(213,293)(214,294)(215,296)(216,295)(217,277)(218,278)
(219,280)(220,279)(221,285)(222,286)(223,288)(224,287)(225,281)(226,282)
(227,284)(228,283)(229,265)(230,266)(231,268)(232,267)(233,273)(234,274)
(235,276)(236,275)(237,269)(238,270)(239,272)(240,271)(301,317)(302,318)
(303,320)(304,319)(305,313)(306,314)(307,316)(308,315)(309,321)(310,322)
(311,324)(312,323)(325,353)(326,354)(327,356)(328,355)(329,349)(330,350)
(331,352)(332,351)(333,357)(334,358)(335,360)(336,359)(337,341)(338,342)
(339,344)(340,343)(347,348);
s2 := Sym(360)!(  1,184)(  2,183)(  3,182)(  4,181)(  5,188)(  6,187)(  7,186)
(  8,185)(  9,192)( 10,191)( 11,190)( 12,189)( 13,196)( 14,195)( 15,194)
( 16,193)( 17,200)( 18,199)( 19,198)( 20,197)( 21,204)( 22,203)( 23,202)
( 24,201)( 25,208)( 26,207)( 27,206)( 28,205)( 29,212)( 30,211)( 31,210)
( 32,209)( 33,216)( 34,215)( 35,214)( 36,213)( 37,220)( 38,219)( 39,218)
( 40,217)( 41,224)( 42,223)( 43,222)( 44,221)( 45,228)( 46,227)( 47,226)
( 48,225)( 49,232)( 50,231)( 51,230)( 52,229)( 53,236)( 54,235)( 55,234)
( 56,233)( 57,240)( 58,239)( 59,238)( 60,237)( 61,244)( 62,243)( 63,242)
( 64,241)( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)
( 72,249)( 73,256)( 74,255)( 75,254)( 76,253)( 77,260)( 78,259)( 79,258)
( 80,257)( 81,264)( 82,263)( 83,262)( 84,261)( 85,268)( 86,267)( 87,266)
( 88,265)( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)
( 96,273)( 97,280)( 98,279)( 99,278)(100,277)(101,284)(102,283)(103,282)
(104,281)(105,288)(106,287)(107,286)(108,285)(109,292)(110,291)(111,290)
(112,289)(113,296)(114,295)(115,294)(116,293)(117,300)(118,299)(119,298)
(120,297)(121,304)(122,303)(123,302)(124,301)(125,308)(126,307)(127,306)
(128,305)(129,312)(130,311)(131,310)(132,309)(133,316)(134,315)(135,314)
(136,313)(137,320)(138,319)(139,318)(140,317)(141,324)(142,323)(143,322)
(144,321)(145,328)(146,327)(147,326)(148,325)(149,332)(150,331)(151,330)
(152,329)(153,336)(154,335)(155,334)(156,333)(157,340)(158,339)(159,338)
(160,337)(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)
(168,345)(169,352)(170,351)(171,350)(172,349)(173,356)(174,355)(175,354)
(176,353)(177,360)(178,359)(179,358)(180,357);
poly := sub<Sym(360)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope