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Polytope of Type {10,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,36}*720
Also Known As : {10,36|2}. if this polytope has another name.
Group : SmallGroup(720,136)
Rank : 3
Schlafli Type : {10,36}
Number of vertices, edges, etc : 10, 180, 36
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 :
   {10,36,2} of size 1440
Vertex Figure Of :
   {2,10,36} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,18}*360
   3-fold quotients : {10,12}*240
   5-fold quotients : {2,36}*144
   6-fold quotients : {10,6}*120
   9-fold quotients : {10,4}*80
   10-fold quotients : {2,18}*72
   15-fold quotients : {2,12}*48
   18-fold quotients : {10,2}*40
   20-fold quotients : {2,9}*36
   30-fold quotients : {2,6}*24
   36-fold quotients : {5,2}*20
   45-fold quotients : {2,4}*16
   60-fold quotients : {2,3}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,72}*1440, {20,36}*1440
Permutation Representation (GAP) :
s0 := (  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 19, 28)( 20, 29)
( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)( 37, 40)
( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)
( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)( 80, 89)
( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)( 97,100)
( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)
(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)(140,149)
(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)(157,160)
(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)( 16, 36)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)( 47, 51)
( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 86)
( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91,139)( 92,141)( 93,140)( 94,136)
( 95,138)( 96,137)( 97,148)( 98,150)( 99,149)(100,145)(101,147)(102,146)
(103,142)(104,144)(105,143)(106,171)(107,170)(108,169)(109,168)(110,167)
(111,166)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)
(119,173)(120,172)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)
(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)
(135,157);;
s2 := (  1,106)(  2,108)(  3,107)(  4,109)(  5,111)(  6,110)(  7,112)(  8,114)
(  9,113)( 10,115)( 11,117)( 12,116)( 13,118)( 14,120)( 15,119)( 16, 91)
( 17, 93)( 18, 92)( 19, 94)( 20, 96)( 21, 95)( 22, 97)( 23, 99)( 24, 98)
( 25,100)( 26,102)( 27,101)( 28,103)( 29,105)( 30,104)( 31,123)( 32,122)
( 33,121)( 34,126)( 35,125)( 36,124)( 37,129)( 38,128)( 39,127)( 40,132)
( 41,131)( 42,130)( 43,135)( 44,134)( 45,133)( 46,151)( 47,153)( 48,152)
( 49,154)( 50,156)( 51,155)( 52,157)( 53,159)( 54,158)( 55,160)( 56,162)
( 57,161)( 58,163)( 59,165)( 60,164)( 61,136)( 62,138)( 63,137)( 64,139)
( 65,141)( 66,140)( 67,142)( 68,144)( 69,143)( 70,145)( 71,147)( 72,146)
( 73,148)( 74,150)( 75,149)( 76,168)( 77,167)( 78,166)( 79,171)( 80,170)
( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)( 88,180)
( 89,179)( 90,178);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(180)!(  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 19, 28)
( 20, 29)( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)
( 37, 40)( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)
( 54, 57)( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)
(114,117)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)
(157,160)(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)
(174,177);
s1 := Sym(180)!(  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)
( 16, 36)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)
( 47, 51)( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)
( 63, 79)( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)
( 71, 86)( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91,139)( 92,141)( 93,140)
( 94,136)( 95,138)( 96,137)( 97,148)( 98,150)( 99,149)(100,145)(101,147)
(102,146)(103,142)(104,144)(105,143)(106,171)(107,170)(108,169)(109,168)
(110,167)(111,166)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)
(118,174)(119,173)(120,172)(121,156)(122,155)(123,154)(124,153)(125,152)
(126,151)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)
(134,158)(135,157);
s2 := Sym(180)!(  1,106)(  2,108)(  3,107)(  4,109)(  5,111)(  6,110)(  7,112)
(  8,114)(  9,113)( 10,115)( 11,117)( 12,116)( 13,118)( 14,120)( 15,119)
( 16, 91)( 17, 93)( 18, 92)( 19, 94)( 20, 96)( 21, 95)( 22, 97)( 23, 99)
( 24, 98)( 25,100)( 26,102)( 27,101)( 28,103)( 29,105)( 30,104)( 31,123)
( 32,122)( 33,121)( 34,126)( 35,125)( 36,124)( 37,129)( 38,128)( 39,127)
( 40,132)( 41,131)( 42,130)( 43,135)( 44,134)( 45,133)( 46,151)( 47,153)
( 48,152)( 49,154)( 50,156)( 51,155)( 52,157)( 53,159)( 54,158)( 55,160)
( 56,162)( 57,161)( 58,163)( 59,165)( 60,164)( 61,136)( 62,138)( 63,137)
( 64,139)( 65,141)( 66,140)( 67,142)( 68,144)( 69,143)( 70,145)( 71,147)
( 72,146)( 73,148)( 74,150)( 75,149)( 76,168)( 77,167)( 78,166)( 79,171)
( 80,170)( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)
( 88,180)( 89,179)( 90,178);
poly := sub<Sym(180)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >; 
 
References : None.
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