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Polytope of Type {5,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,3}*120
Also Known As : dodecahedron, {5,3}. if this polytope has another name.
Group : SmallGroup(120,35)
Rank : 3
Schlafli Type : {5,3}
Number of vertices, edges, etc : 20, 30, 12
Order of s0s1s2 : 10
Order of s0s1s2s1 : 5
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {5,3,2} of size 240
Vertex Figure Of :
   {2,5,3} of size 240
   {4,5,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {5,3}*60
Covers (Minimal Covers in Boldface) :
   2-fold covers : {5,6}*240b, {10,3}*240
   3-fold covers : {15,3}*360
   4-fold covers : {5,12}*480, {20,3}*480, {10,6}*480c
   5-fold covers : {5,15}*600
   6-fold covers : {10,3}*720b, {15,6}*720c, {15,6}*720d, {30,3}*720
   8-fold covers : {10,12}*960c, {20,6}*960c, {10,12}*960d, {20,6}*960d, {10,6}*960b
   10-fold covers : {5,6}*1200b, {5,30}*1200b, {10,15}*1200a, {10,15}*1200b
   12-fold covers : {15,12}*1440a, {15,12}*1440b, {20,3}*1440a, {60,3}*1440, {15,3}*1440, {10,6}*1440f, {30,6}*1440e, {30,6}*1440f
   14-fold covers : {10,21}*1680, {35,6}*1680c
   15-fold covers : {15,15}*1800b
   16-fold covers : {20,12}*1920g, {10,24}*1920d, {40,6}*1920f, {10,12}*1920c, {20,6}*1920d, {20,12}*1920k, {20,12}*1920l, {20,12}*1920m, {10,24}*1920f, {40,6}*1920h, {5,6}*1920d
Permutation Representation (GAP) :
s0 := ( 2, 9)( 4,12)( 5, 7)( 6, 8);;
s1 := ( 3, 5)( 4,11)( 6,12)( 7, 9);;
s2 := ( 1, 3)( 2, 6)( 8, 9)(10,11);;
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, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(12)!( 2, 9)( 4,12)( 5, 7)( 6, 8);
s1 := Sym(12)!( 3, 5)( 4,11)( 6,12)( 7, 9);
s2 := Sym(12)!( 1, 3)( 2, 6)( 8, 9)(10,11);
poly := sub<Sym(12)|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, s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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