Polytope of Type {360}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {360}*720
Also Known As : 360-gon, {360}. if this polytope has another name.
Group : SmallGroup(720,69)
Rank : 2
Schlafli Type : {360}
Number of vertices, edges, etc : 360, 360
Order of s₀s₁ : 360
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {360,2} of size 1440
Vertex Figure Of :
   {2,360} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {180}*360
   3-fold quotients : {120}*240
   4-fold quotients : {90}*180
   5-fold quotients : {72}*144
   6-fold quotients : {60}*120
   8-fold quotients : {45}*90
   9-fold quotients : {40}*80
   10-fold quotients : {36}*72
   12-fold quotients : {30}*60
   15-fold quotients : {24}*48
   18-fold quotients : {20}*40
   20-fold quotients : {18}*36
   24-fold quotients : {15}*30
   30-fold quotients : {12}*24
   36-fold quotients : {10}*20
   40-fold quotients : {9}*18
   45-fold quotients : {8}*16
   60-fold quotients : {6}*12
   72-fold quotients : {5}*10
   90-fold quotients : {4}*8
   120-fold quotients : {3}*6
   180-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {720}*1440
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)( 91,136)( 92,138)( 93,137)( 94,148)
( 95,150)( 96,149)( 97,145)( 98,147)( 99,146)(100,142)(101,144)(102,143)
(103,139)(104,141)(105,140)(106,168)(107,167)(108,166)(109,180)(110,179)
(111,178)(112,177)(113,176)(114,175)(115,174)(116,173)(117,172)(118,171)
(119,170)(120,169)(121,153)(122,152)(123,151)(124,165)(125,164)(126,163)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)
(135,154)(181,271)(182,273)(183,272)(184,283)(185,285)(186,284)(187,280)
(188,282)(189,281)(190,277)(191,279)(192,278)(193,274)(194,276)(195,275)
(196,303)(197,302)(198,301)(199,315)(200,314)(201,313)(202,312)(203,311)
(204,310)(205,309)(206,308)(207,307)(208,306)(209,305)(210,304)(211,288)
(212,287)(213,286)(214,300)(215,299)(216,298)(217,297)(218,296)(219,295)
(220,294)(221,293)(222,292)(223,291)(224,290)(225,289)(226,316)(227,318)
(228,317)(229,328)(230,330)(231,329)(232,325)(233,327)(234,326)(235,322)
(236,324)(237,323)(238,319)(239,321)(240,320)(241,348)(242,347)(243,346)
(244,360)(245,359)(246,358)(247,357)(248,356)(249,355)(250,354)(251,353)
(252,352)(253,351)(254,350)(255,349)(256,333)(257,332)(258,331)(259,345)
(260,344)(261,343)(262,342)(263,341)(264,340)(265,339)(266,338)(267,337)
(268,336)(269,335)(270,334);;
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)( 91,334)( 92,336)( 93,335)( 94,331)( 95,333)( 96,332)
( 97,343)( 98,345)( 99,344)(100,340)(101,342)(102,341)(103,337)(104,339)
(105,338)(106,319)(107,321)(108,320)(109,316)(110,318)(111,317)(112,328)
(113,330)(114,329)(115,325)(116,327)(117,326)(118,322)(119,324)(120,323)
(121,351)(122,350)(123,349)(124,348)(125,347)(126,346)(127,360)(128,359)
(129,358)(130,357)(131,356)(132,355)(133,354)(134,353)(135,352)(136,289)
(137,291)(138,290)(139,286)(140,288)(141,287)(142,298)(143,300)(144,299)
(145,295)(146,297)(147,296)(148,292)(149,294)(150,293)(151,274)(152,276)
(153,275)(154,271)(155,273)(156,272)(157,283)(158,285)(159,284)(160,280)
(161,282)(162,281)(163,277)(164,279)(165,278)(166,306)(167,305)(168,304)
(169,303)(170,302)(171,301)(172,315)(173,314)(174,313)(175,312)(176,311)
(177,310)(178,309)(179,308)(180,307);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 91,136)( 92,138)( 93,137)
( 94,148)( 95,150)( 96,149)( 97,145)( 98,147)( 99,146)(100,142)(101,144)
(102,143)(103,139)(104,141)(105,140)(106,168)(107,167)(108,166)(109,180)
(110,179)(111,178)(112,177)(113,176)(114,175)(115,174)(116,173)(117,172)
(118,171)(119,170)(120,169)(121,153)(122,152)(123,151)(124,165)(125,164)
(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)
(134,155)(135,154)(181,271)(182,273)(183,272)(184,283)(185,285)(186,284)
(187,280)(188,282)(189,281)(190,277)(191,279)(192,278)(193,274)(194,276)
(195,275)(196,303)(197,302)(198,301)(199,315)(200,314)(201,313)(202,312)
(203,311)(204,310)(205,309)(206,308)(207,307)(208,306)(209,305)(210,304)
(211,288)(212,287)(213,286)(214,300)(215,299)(216,298)(217,297)(218,296)
(219,295)(220,294)(221,293)(222,292)(223,291)(224,290)(225,289)(226,316)
(227,318)(228,317)(229,328)(230,330)(231,329)(232,325)(233,327)(234,326)
(235,322)(236,324)(237,323)(238,319)(239,321)(240,320)(241,348)(242,347)
(243,346)(244,360)(245,359)(246,358)(247,357)(248,356)(249,355)(250,354)
(251,353)(252,352)(253,351)(254,350)(255,349)(256,333)(257,332)(258,331)
(259,345)(260,344)(261,343)(262,342)(263,341)(264,340)(265,339)(266,338)
(267,337)(268,336)(269,335)(270,334);
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)( 91,334)( 92,336)( 93,335)( 94,331)( 95,333)
( 96,332)( 97,343)( 98,345)( 99,344)(100,340)(101,342)(102,341)(103,337)
(104,339)(105,338)(106,319)(107,321)(108,320)(109,316)(110,318)(111,317)
(112,328)(113,330)(114,329)(115,325)(116,327)(117,326)(118,322)(119,324)
(120,323)(121,351)(122,350)(123,349)(124,348)(125,347)(126,346)(127,360)
(128,359)(129,358)(130,357)(131,356)(132,355)(133,354)(134,353)(135,352)
(136,289)(137,291)(138,290)(139,286)(140,288)(141,287)(142,298)(143,300)
(144,299)(145,295)(146,297)(147,296)(148,292)(149,294)(150,293)(151,274)
(152,276)(153,275)(154,271)(155,273)(156,272)(157,283)(158,285)(159,284)
(160,280)(161,282)(162,281)(163,277)(164,279)(165,278)(166,306)(167,305)
(168,304)(169,303)(170,302)(171,301)(172,315)(173,314)(174,313)(175,312)
(176,311)(177,310)(178,309)(179,308)(180,307);
poly := sub<Sym(360)|s0,s1>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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