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# Polytope of Type {3,5}

Atlas Canonical Name : {3,5}*60
Also Known As : hemiicosahedron, {3,5}5if this polytope has another name.
Group : SmallGroup(60,5)
Rank : 3
Schlafli Type : {3,5}
Number of vertices, edges, etc : 6, 15, 10
Order of s0s1s2 : 5
Order of s0s1s2s1 : 5
Special Properties :
Projective
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,5,2} of size 120
{3,5,3} of size 660
{3,5,4} of size 960
{3,5,6} of size 1320
{3,5,4} of size 1920
{3,5,4} of size 1920
{3,5,4} of size 1920
Vertex Figure Of :
{2,3,5} of size 120
{4,3,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,5}*120, {3,10}*120a, {3,10}*120b, {6,5}*120b, {6,5}*120c
4-fold covers : {3,10}*240, {6,5}*240b, {6,10}*240c, {6,10}*240d, {6,10}*240e, {6,10}*240f
6-fold covers : {3,10}*360, {3,15}*360, {6,15}*360
8-fold covers : {6,20}*480a, {6,20}*480b, {12,10}*480c, {12,10}*480d, {3,20}*480, {12,5}*480, {6,10}*480c
10-fold covers : {6,5}*600, {15,5}*600, {15,10}*600
12-fold covers : {3,10}*720b, {3,30}*720, {6,10}*720b, {6,10}*720c, {6,15}*720c, {6,15}*720d, {6,30}*720a, {6,30}*720b
14-fold covers : {6,35}*840, {21,10}*840
16-fold covers : {6,40}*960a, {6,40}*960b, {24,10}*960c, {24,10}*960d, {6,20}*960c, {12,10}*960c, {6,20}*960d, {12,10}*960d, {6,10}*960b, {6,5}*960
18-fold covers : {6,45}*1080, {9,10}*1080
20-fold covers : {6,5}*1200b, {6,10}*1200a, {6,10}*1200b, {15,10}*1200a, {15,10}*1200b, {30,5}*1200b, {30,10}*1200b, {30,10}*1200c
22-fold covers : {6,55}*1320, {33,10}*1320
24-fold covers : {6,60}*1440a, {6,60}*1440b, {12,10}*1440e, {12,10}*1440f, {3,20}*1440a, {3,60}*1440, {12,15}*1440a, {12,15}*1440b, {3,15}*1440, {3,20}*1440b, {12,15}*1440d, {6,10}*1440f, {6,30}*1440e, {6,30}*1440f
26-fold covers : {6,65}*1560, {39,10}*1560
28-fold covers : {6,35}*1680c, {6,70}*1680a, {6,70}*1680b, {21,10}*1680, {42,10}*1680b, {42,10}*1680c
30-fold covers : {6,15}*1800, {15,10}*1800a, {15,15}*1800a
32-fold covers : {6,80}*1920a, {6,80}*1920b, {48,10}*1920c, {48,10}*1920d, {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, {6,10}*1920a, {6,5}*1920b, {6,5}*1920c, {6,5}*1920d, {6,10}*1920b, {6,10}*1920c, {6,10}*1920d, {6,10}*1920e, {12,5}*1920c, {12,5}*1920d, {12,5}*1920e, {12,5}*1920f
Permutation Representation (GAP) :
```s0 := (2,3)(4,5);;
s1 := (1,2)(4,5);;
s2 := (2,4)(3,5);;
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, s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(5)!(2,3)(4,5);
s1 := Sym(5)!(1,2)(4,5);
s2 := Sym(5)!(2,4)(3,5);
poly := sub<Sym(5)|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,
s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 >;

```
References : None.
to this polytope