Polytope of Type {180}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {180}*360
Also Known As : 180-gon, {180}. if this polytope has another name.
Group : SmallGroup(360,27)
Rank : 2
Schlafli Type : {180}
Number of vertices, edges, etc : 180, 180
Order of s0s1 : 180
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{180,2} of size 720
{180,4} of size 1440
{180,4} of size 1440
{180,4} of size 1440
Vertex Figure Of :
{2,180} of size 720
{4,180} of size 1440
{4,180} of size 1440
{4,180} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {90}*180
3-fold quotients : {60}*120
4-fold quotients : {45}*90
5-fold quotients : {36}*72
6-fold quotients : {30}*60
9-fold quotients : {20}*40
10-fold quotients : {18}*36
12-fold quotients : {15}*30
15-fold quotients : {12}*24
18-fold quotients : {10}*20
20-fold quotients : {9}*18
30-fold quotients : {6}*12
36-fold quotients : {5}*10
45-fold quotients : {4}*8
60-fold quotients : {3}*6
90-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {360}*720
3-fold covers : {540}*1080
4-fold covers : {720}*1440
5-fold covers : {900}*1800
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 13)( 5, 15)( 6, 14)( 7, 10)( 8, 12)( 9, 11)( 16, 32)
( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)( 24, 42)
( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 47, 48)( 49, 58)
( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 77)( 62, 76)( 63, 78)
( 64, 89)( 65, 88)( 66, 90)( 67, 86)( 68, 85)( 69, 87)( 70, 83)( 71, 82)
( 72, 84)( 73, 80)( 74, 79)( 75, 81)( 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)(106,167)(107,166)(108,168)(109,179)(110,178)
(111,180)(112,176)(113,175)(114,177)(115,173)(116,172)(117,174)(118,170)
(119,169)(120,171)(121,152)(122,151)(123,153)(124,164)(125,163)(126,165)
(127,161)(128,160)(129,162)(130,158)(131,157)(132,159)(133,155)(134,154)
(135,156);;
s1 := ( 1,109)( 2,111)( 3,110)( 4,106)( 5,108)( 6,107)( 7,118)( 8,120)
( 9,119)( 10,115)( 11,117)( 12,116)( 13,112)( 14,114)( 15,113)( 16, 94)
( 17, 96)( 18, 95)( 19, 91)( 20, 93)( 21, 92)( 22,103)( 23,105)( 24,104)
( 25,100)( 26,102)( 27,101)( 28, 97)( 29, 99)( 30, 98)( 31,125)( 32,124)
( 33,126)( 34,122)( 35,121)( 36,123)( 37,134)( 38,133)( 39,135)( 40,131)
( 41,130)( 42,132)( 43,128)( 44,127)( 45,129)( 46,154)( 47,156)( 48,155)
( 49,151)( 50,153)( 51,152)( 52,163)( 53,165)( 54,164)( 55,160)( 56,162)
( 57,161)( 58,157)( 59,159)( 60,158)( 61,139)( 62,141)( 63,140)( 64,136)
( 65,138)( 66,137)( 67,148)( 68,150)( 69,149)( 70,145)( 71,147)( 72,146)
( 73,142)( 74,144)( 75,143)( 76,170)( 77,169)( 78,171)( 79,167)( 80,166)
( 81,168)( 82,179)( 83,178)( 84,180)( 85,176)( 86,175)( 87,177)( 88,173)
( 89,172)( 90,174);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(180)!( 2, 3)( 4, 13)( 5, 15)( 6, 14)( 7, 10)( 8, 12)( 9, 11)
( 16, 32)( 17, 31)( 18, 33)( 19, 44)( 20, 43)( 21, 45)( 22, 41)( 23, 40)
( 24, 42)( 25, 38)( 26, 37)( 27, 39)( 28, 35)( 29, 34)( 30, 36)( 47, 48)
( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 77)( 62, 76)
( 63, 78)( 64, 89)( 65, 88)( 66, 90)( 67, 86)( 68, 85)( 69, 87)( 70, 83)
( 71, 82)( 72, 84)( 73, 80)( 74, 79)( 75, 81)( 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)(106,167)(107,166)(108,168)(109,179)
(110,178)(111,180)(112,176)(113,175)(114,177)(115,173)(116,172)(117,174)
(118,170)(119,169)(120,171)(121,152)(122,151)(123,153)(124,164)(125,163)
(126,165)(127,161)(128,160)(129,162)(130,158)(131,157)(132,159)(133,155)
(134,154)(135,156);
s1 := Sym(180)!( 1,109)( 2,111)( 3,110)( 4,106)( 5,108)( 6,107)( 7,118)
( 8,120)( 9,119)( 10,115)( 11,117)( 12,116)( 13,112)( 14,114)( 15,113)
( 16, 94)( 17, 96)( 18, 95)( 19, 91)( 20, 93)( 21, 92)( 22,103)( 23,105)
( 24,104)( 25,100)( 26,102)( 27,101)( 28, 97)( 29, 99)( 30, 98)( 31,125)
( 32,124)( 33,126)( 34,122)( 35,121)( 36,123)( 37,134)( 38,133)( 39,135)
( 40,131)( 41,130)( 42,132)( 43,128)( 44,127)( 45,129)( 46,154)( 47,156)
( 48,155)( 49,151)( 50,153)( 51,152)( 52,163)( 53,165)( 54,164)( 55,160)
( 56,162)( 57,161)( 58,157)( 59,159)( 60,158)( 61,139)( 62,141)( 63,140)
( 64,136)( 65,138)( 66,137)( 67,148)( 68,150)( 69,149)( 70,145)( 71,147)
( 72,146)( 73,142)( 74,144)( 75,143)( 76,170)( 77,169)( 78,171)( 79,167)
( 80,166)( 81,168)( 82,179)( 83,178)( 84,180)( 85,176)( 86,175)( 87,177)
( 88,173)( 89,172)( 90,174);
poly := sub<Sym(180)|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 >;
References : None.
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