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# Polytope of Type {3,5}

Atlas Canonical Name : {3,5}*120
Also Known As : icosahedron, {3,5}. if this polytope has another name.
Group : SmallGroup(120,35)
Rank : 3
Schlafli Type : {3,5}
Number of vertices, edges, etc : 12, 30, 20
Order of s0s1s2 : 10
Order of s0s1s2s1 : 5
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,5,2} of size 240
{3,5,4} of size 1920
Vertex Figure Of :
{2,3,5} of size 240
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,5}*60
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,10}*240, {6,5}*240b
3-fold covers : {3,15}*360
4-fold covers : {3,20}*480, {12,5}*480, {6,10}*480c
5-fold covers : {15,5}*600
6-fold covers : {3,10}*720b, {3,30}*720, {6,15}*720c, {6,15}*720d
8-fold covers : {6,20}*960c, {12,10}*960c, {6,20}*960d, {12,10}*960d, {6,10}*960b
10-fold covers : {6,5}*1200b, {15,10}*1200a, {15,10}*1200b, {30,5}*1200b
12-fold covers : {3,20}*1440a, {3,60}*1440, {12,15}*1440a, {12,15}*1440b, {3,15}*1440, {6,10}*1440f, {6,30}*1440e, {6,30}*1440f
14-fold covers : {6,35}*1680c, {21,10}*1680
15-fold covers : {15,15}*1800a
16-fold covers : {12,20}*1920g, {6,40}*1920f, {24,10}*1920d, {6,20}*1920d, {12,10}*1920c, {12,20}*1920k, {12,20}*1920l, {12,20}*1920m, {6,40}*1920h, {24,10}*1920f, {6,5}*1920d
Permutation Representation (GAP) :
```s0 := ( 2, 9)( 4,12)( 5, 7)( 6, 8);;
s1 := ( 1, 2)( 3, 4)( 5,11)( 8,10);;
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, s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 2, 9)( 4,12)( 5, 7)( 6, 8);
s1 := Sym(12)!( 1, 2)( 3, 4)( 5,11)( 8,10);
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, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope