Polytope of Type {180,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {180,4}*1440a
Also Known As : {180,4|2}. if this polytope has another name.
Group : SmallGroup(1440,848)
Rank : 3
Schlafli Type : {180,4}
Number of vertices, edges, etc : 180, 360, 4
Order of s₀s₁s₂ : 180
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {180,2}*720, {90,4}*720a
   3-fold quotients : {60,4}*480a
   4-fold quotients : {90,2}*360
   5-fold quotients : {36,4}*288a
   6-fold quotients : {60,2}*240, {30,4}*240a
   8-fold quotients : {45,2}*180
   9-fold quotients : {20,4}*160
   10-fold quotients : {36,2}*144, {18,4}*144a
   12-fold quotients : {30,2}*120
   15-fold quotients : {12,4}*96a
   18-fold quotients : {20,2}*80, {10,4}*80
   20-fold quotients : {18,2}*72
   24-fold quotients : {15,2}*60
   30-fold quotients : {12,2}*48, {6,4}*48a
   36-fold quotients : {10,2}*40
   40-fold quotients : {9,2}*36
   45-fold quotients : {4,4}*32
   60-fold quotients : {6,2}*24
   72-fold quotients : {5,2}*20
   90-fold quotients : {2,4}*16, {4,2}*16
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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