Polytope of Type {186}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {186}*372
Also Known As : 186-gon, {186}. if this polytope has another name.
Group : SmallGroup(372,14)
Rank : 2
Schlafli Type : {186}
Number of vertices, edges, etc : 186, 186
Order of s0s1 : 186
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {186,2} of size 744
   {186,4} of size 1488
   {186,4} of size 1488
   {186,4} of size 1488
Vertex Figure Of :
   {2,186} of size 744
   {4,186} of size 1488
   {4,186} of size 1488
   {4,186} of size 1488
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {93}*186
   3-fold quotients : {62}*124
   6-fold quotients : {31}*62
   31-fold quotients : {6}*12
   62-fold quotients : {3}*6
   93-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {372}*744
   3-fold covers : {558}*1116
   4-fold covers : {744}*1488
   5-fold covers : {930}*1860
Permutation Representation (GAP) :
s0 := (  2, 31)(  3, 30)(  4, 29)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)
( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 16, 17)( 32, 63)
( 33, 93)( 34, 92)( 35, 91)( 36, 90)( 37, 89)( 38, 88)( 39, 87)( 40, 86)
( 41, 85)( 42, 84)( 43, 83)( 44, 82)( 45, 81)( 46, 80)( 47, 79)( 48, 78)
( 49, 77)( 50, 76)( 51, 75)( 52, 74)( 53, 73)( 54, 72)( 55, 71)( 56, 70)
( 57, 69)( 58, 68)( 59, 67)( 60, 66)( 61, 65)( 62, 64)( 95,124)( 96,123)
( 97,122)( 98,121)( 99,120)(100,119)(101,118)(102,117)(103,116)(104,115)
(105,114)(106,113)(107,112)(108,111)(109,110)(125,156)(126,186)(127,185)
(128,184)(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)(135,177)
(136,176)(137,175)(138,174)(139,173)(140,172)(141,171)(142,170)(143,169)
(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)
(152,160)(153,159)(154,158)(155,157);;
s1 := (  1,126)(  2,125)(  3,155)(  4,154)(  5,153)(  6,152)(  7,151)(  8,150)
(  9,149)( 10,148)( 11,147)( 12,146)( 13,145)( 14,144)( 15,143)( 16,142)
( 17,141)( 18,140)( 19,139)( 20,138)( 21,137)( 22,136)( 23,135)( 24,134)
( 25,133)( 26,132)( 27,131)( 28,130)( 29,129)( 30,128)( 31,127)( 32, 95)
( 33, 94)( 34,124)( 35,123)( 36,122)( 37,121)( 38,120)( 39,119)( 40,118)
( 41,117)( 42,116)( 43,115)( 44,114)( 45,113)( 46,112)( 47,111)( 48,110)
( 49,109)( 50,108)( 51,107)( 52,106)( 53,105)( 54,104)( 55,103)( 56,102)
( 57,101)( 58,100)( 59, 99)( 60, 98)( 61, 97)( 62, 96)( 63,157)( 64,156)
( 65,186)( 66,185)( 67,184)( 68,183)( 69,182)( 70,181)( 71,180)( 72,179)
( 73,178)( 74,177)( 75,176)( 76,175)( 77,174)( 78,173)( 79,172)( 80,171)
( 81,170)( 82,169)( 83,168)( 84,167)( 85,166)( 86,165)( 87,164)( 88,163)
( 89,162)( 90,161)( 91,160)( 92,159)( 93,158);;
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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(186)!(  2, 31)(  3, 30)(  4, 29)(  5, 28)(  6, 27)(  7, 26)(  8, 25)
(  9, 24)( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 16, 17)
( 32, 63)( 33, 93)( 34, 92)( 35, 91)( 36, 90)( 37, 89)( 38, 88)( 39, 87)
( 40, 86)( 41, 85)( 42, 84)( 43, 83)( 44, 82)( 45, 81)( 46, 80)( 47, 79)
( 48, 78)( 49, 77)( 50, 76)( 51, 75)( 52, 74)( 53, 73)( 54, 72)( 55, 71)
( 56, 70)( 57, 69)( 58, 68)( 59, 67)( 60, 66)( 61, 65)( 62, 64)( 95,124)
( 96,123)( 97,122)( 98,121)( 99,120)(100,119)(101,118)(102,117)(103,116)
(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)(125,156)(126,186)
(127,185)(128,184)(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)
(135,177)(136,176)(137,175)(138,174)(139,173)(140,172)(141,171)(142,170)
(143,169)(144,168)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)
(151,161)(152,160)(153,159)(154,158)(155,157);
s1 := Sym(186)!(  1,126)(  2,125)(  3,155)(  4,154)(  5,153)(  6,152)(  7,151)
(  8,150)(  9,149)( 10,148)( 11,147)( 12,146)( 13,145)( 14,144)( 15,143)
( 16,142)( 17,141)( 18,140)( 19,139)( 20,138)( 21,137)( 22,136)( 23,135)
( 24,134)( 25,133)( 26,132)( 27,131)( 28,130)( 29,129)( 30,128)( 31,127)
( 32, 95)( 33, 94)( 34,124)( 35,123)( 36,122)( 37,121)( 38,120)( 39,119)
( 40,118)( 41,117)( 42,116)( 43,115)( 44,114)( 45,113)( 46,112)( 47,111)
( 48,110)( 49,109)( 50,108)( 51,107)( 52,106)( 53,105)( 54,104)( 55,103)
( 56,102)( 57,101)( 58,100)( 59, 99)( 60, 98)( 61, 97)( 62, 96)( 63,157)
( 64,156)( 65,186)( 66,185)( 67,184)( 68,183)( 69,182)( 70,181)( 71,180)
( 72,179)( 73,178)( 74,177)( 75,176)( 76,175)( 77,174)( 78,173)( 79,172)
( 80,171)( 81,170)( 82,169)( 83,168)( 84,167)( 85,166)( 86,165)( 87,164)
( 88,163)( 89,162)( 90,161)( 91,160)( 92,159)( 93,158);
poly := sub<Sym(186)|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 >; 
 
References : None.
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