Polytope of Type {4,180}

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

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

References : None.
to this polytope