Polytope of Type {4,45,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,45,2}*1440
if this polytope has a name.
Group : SmallGroup(1440,4575)
Rank : 4
Schlafli Type : {4,45,2}
Number of vertices, edges, etc : 8, 180, 90, 2
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 : {4,45,2}*720
   3-fold quotients : {4,15,2}*480
   4-fold quotients : {2,45,2}*360
   5-fold quotients : {4,9,2}*288
   6-fold quotients : {4,15,2}*240
   10-fold quotients : {4,9,2}*144
   12-fold quotients : {2,15,2}*120
   15-fold quotients : {4,3,2}*96
   20-fold quotients : {2,9,2}*72
   30-fold quotients : {4,3,2}*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 := (  1,183)(  2,184)(  3,181)(  4,182)(  5,187)(  6,188)(  7,185)(  8,186)(  9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)( 16,194)( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)( 24,202)( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)( 32,210)( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)( 40,218)( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)( 48,226)( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)( 56,234)( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)( 64,242)( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)( 72,250)( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)(104,282)(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)(112,290)(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)(120,298)(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)(128,306)(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)(136,314)(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)(144,322)(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)(152,330)(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)(160,338)(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)(168,346)(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)(176,354)(177,359)(178,360)(179,357)(180,358);;
s1 := (  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);;
s2 := (  1,133)(  2,136)(  3,135)(  4,134)(  5,141)(  6,144)(  7,143)(  8,142)(  9,137)( 10,140)( 11,139)( 12,138)( 13,121)( 14,124)( 15,123)( 16,122)( 17,129)( 18,132)( 19,131)( 20,130)( 21,125)( 22,128)( 23,127)( 24,126)( 25,169)( 26,172)( 27,171)( 28,170)( 29,177)( 30,180)( 31,179)( 32,178)( 33,173)( 34,176)( 35,175)( 36,174)( 37,157)( 38,160)( 39,159)( 40,158)( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)( 57,149)( 58,152)( 59,151)( 60,150)( 61, 73)( 62, 76)( 63, 75)( 64, 74)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 98,100)(101,105)(102,108)(103,107)(104,106)(181,313)(182,316)(183,315)(184,314)(185,321)(186,324)(187,323)(188,322)(189,317)(190,320)(191,319)(192,318)(193,301)(194,304)(195,303)(196,302)(197,309)(198,312)(199,311)(200,310)(201,305)(202,308)(203,307)(204,306)(205,349)(206,352)(207,351)(208,350)(209,357)(210,360)(211,359)(212,358)(213,353)(214,356)(215,355)(216,354)(217,337)(218,340)(219,339)(220,338)(221,345)(222,348)(223,347)(224,346)(225,341)(226,344)(227,343)(228,342)(229,325)(230,328)(231,327)(232,326)(233,333)(234,336)(235,335)(236,334)(237,329)(238,332)(239,331)(240,330)(241,253)(242,256)(243,255)(244,254)(245,261)(246,264)(247,263)(248,262)(249,257)(250,260)(251,259)(252,258)(265,289)(266,292)(267,291)(268,290)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(278,280)(281,285)(282,288)(283,287)(284,286);;
s3 := (361,362);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*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)!(  1,183)(  2,184)(  3,181)(  4,182)(  5,187)(  6,188)(  7,185)(  8,186)(  9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)( 16,194)( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)( 24,202)( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)( 32,210)( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)( 40,218)( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)( 48,226)( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)( 56,234)( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)( 64,242)( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)( 72,250)( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)(104,282)(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)(112,290)(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)(120,298)(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)(128,306)(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)(136,314)(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)(144,322)(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)(152,330)(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)(160,338)(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)(168,346)(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)(176,354)(177,359)(178,360)(179,357)(180,358);
s1 := Sym(362)!(  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);
s2 := Sym(362)!(  1,133)(  2,136)(  3,135)(  4,134)(  5,141)(  6,144)(  7,143)(  8,142)(  9,137)( 10,140)( 11,139)( 12,138)( 13,121)( 14,124)( 15,123)( 16,122)( 17,129)( 18,132)( 19,131)( 20,130)( 21,125)( 22,128)( 23,127)( 24,126)( 25,169)( 26,172)( 27,171)( 28,170)( 29,177)( 30,180)( 31,179)( 32,178)( 33,173)( 34,176)( 35,175)( 36,174)( 37,157)( 38,160)( 39,159)( 40,158)( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)( 57,149)( 58,152)( 59,151)( 60,150)( 61, 73)( 62, 76)( 63, 75)( 64, 74)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 98,100)(101,105)(102,108)(103,107)(104,106)(181,313)(182,316)(183,315)(184,314)(185,321)(186,324)(187,323)(188,322)(189,317)(190,320)(191,319)(192,318)(193,301)(194,304)(195,303)(196,302)(197,309)(198,312)(199,311)(200,310)(201,305)(202,308)(203,307)(204,306)(205,349)(206,352)(207,351)(208,350)(209,357)(210,360)(211,359)(212,358)(213,353)(214,356)(215,355)(216,354)(217,337)(218,340)(219,339)(220,338)(221,345)(222,348)(223,347)(224,346)(225,341)(226,344)(227,343)(228,342)(229,325)(230,328)(231,327)(232,326)(233,333)(234,336)(235,335)(236,334)(237,329)(238,332)(239,331)(240,330)(241,253)(242,256)(243,255)(244,254)(245,261)(246,264)(247,263)(248,262)(249,257)(250,260)(251,259)(252,258)(265,289)(266,292)(267,291)(268,290)(269,297)(270,300)(271,299)(272,298)(273,293)(274,296)(275,295)(276,294)(278,280)(281,285)(282,288)(283,287)(284,286);
s3 := Sym(362)!(361,362);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*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