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Polytope of Type {120}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {120}*240
Also Known As : 120-gon, {120}. if this polytope has another name.
Group : SmallGroup(240,68)
Rank : 2
Schlafli Type : {120}
Number of vertices, edges, etc : 120, 120
Order of s0s1 : 120
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {120,2} of size 480
   {120,4} of size 960
   {120,4} of size 960
   {120,4} of size 960
   {120,4} of size 960
   {120,6} of size 1440
   {120,6} of size 1440
   {120,6} of size 1440
   {120,4} of size 1920
   {120,8} of size 1920
   {120,8} of size 1920
   {120,8} of size 1920
   {120,8} of size 1920
   {120,4} of size 1920
   {120,6} of size 1920
   {120,6} of size 1920
   {120,4} of size 1920
   {120,4} of size 1920
   {120,4} of size 1920
   {120,4} of size 1920
Vertex Figure Of :
   {2,120} of size 480
   {4,120} of size 960
   {4,120} of size 960
   {4,120} of size 960
   {4,120} of size 960
   {6,120} of size 1440
   {6,120} of size 1440
   {6,120} of size 1440
   {4,120} of size 1920
   {8,120} of size 1920
   {8,120} of size 1920
   {8,120} of size 1920
   {8,120} of size 1920
   {4,120} of size 1920
   {6,120} of size 1920
   {6,120} of size 1920
   {4,120} of size 1920
   {4,120} of size 1920
   {4,120} of size 1920
   {4,120} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {60}*120
   3-fold quotients : {40}*80
   4-fold quotients : {30}*60
   5-fold quotients : {24}*48
   6-fold quotients : {20}*40
   8-fold quotients : {15}*30
   10-fold quotients : {12}*24
   12-fold quotients : {10}*20
   15-fold quotients : {8}*16
   20-fold quotients : {6}*12
   24-fold quotients : {5}*10
   30-fold quotients : {4}*8
   40-fold quotients : {3}*6
   60-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {240}*480
   3-fold covers : {360}*720
   4-fold covers : {480}*960
   5-fold covers : {600}*1200
   6-fold covers : {720}*1440
   7-fold covers : {840}*1680
   8-fold covers : {960}*1920
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 31, 46)( 32, 50)
( 33, 49)( 34, 48)( 35, 47)( 36, 56)( 37, 60)( 38, 59)( 39, 58)( 40, 57)
( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 61, 91)( 62, 95)( 63, 94)
( 64, 93)( 65, 92)( 66,101)( 67,105)( 68,104)( 69,103)( 70,102)( 71, 96)
( 72,100)( 73, 99)( 74, 98)( 75, 97)( 76,106)( 77,110)( 78,109)( 79,108)
( 80,107)( 81,116)( 82,120)( 83,119)( 84,118)( 85,117)( 86,111)( 87,115)
( 88,114)( 89,113)( 90,112);;
s1 := (  1, 67)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 62)(  7, 61)(  8, 65)
(  9, 64)( 10, 63)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)( 16, 82)
( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 77)( 22, 76)( 23, 80)( 24, 79)
( 25, 78)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31,112)( 32,111)
( 33,115)( 34,114)( 35,113)( 36,107)( 37,106)( 38,110)( 39,109)( 40,108)
( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46, 97)( 47, 96)( 48,100)
( 49, 99)( 50, 98)( 51, 92)( 52, 91)( 53, 95)( 54, 94)( 55, 93)( 56,102)
( 57,101)( 58,105)( 59,104)( 60,103);;
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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(120)!(  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 31, 46)
( 32, 50)( 33, 49)( 34, 48)( 35, 47)( 36, 56)( 37, 60)( 38, 59)( 39, 58)
( 40, 57)( 41, 51)( 42, 55)( 43, 54)( 44, 53)( 45, 52)( 61, 91)( 62, 95)
( 63, 94)( 64, 93)( 65, 92)( 66,101)( 67,105)( 68,104)( 69,103)( 70,102)
( 71, 96)( 72,100)( 73, 99)( 74, 98)( 75, 97)( 76,106)( 77,110)( 78,109)
( 79,108)( 80,107)( 81,116)( 82,120)( 83,119)( 84,118)( 85,117)( 86,111)
( 87,115)( 88,114)( 89,113)( 90,112);
s1 := Sym(120)!(  1, 67)(  2, 66)(  3, 70)(  4, 69)(  5, 68)(  6, 62)(  7, 61)
(  8, 65)(  9, 64)( 10, 63)( 11, 72)( 12, 71)( 13, 75)( 14, 74)( 15, 73)
( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 77)( 22, 76)( 23, 80)
( 24, 79)( 25, 78)( 26, 87)( 27, 86)( 28, 90)( 29, 89)( 30, 88)( 31,112)
( 32,111)( 33,115)( 34,114)( 35,113)( 36,107)( 37,106)( 38,110)( 39,109)
( 40,108)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46, 97)( 47, 96)
( 48,100)( 49, 99)( 50, 98)( 51, 92)( 52, 91)( 53, 95)( 54, 94)( 55, 93)
( 56,102)( 57,101)( 58,105)( 59,104)( 60,103);
poly := sub<Sym(120)|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 >; 
 
References : None.
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