# Polytope of Type {4,5}

Atlas Canonical Name : {4,5}*120
if this polytope has a name.
Group : SmallGroup(120,34)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 12, 30, 15
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,5,2} of size 240
{4,5,4} of size 1440
{4,5,3} of size 1920
{4,5,4} of size 1920
{4,5,4} of size 1920
Vertex Figure Of :
{2,4,5} of size 240
{4,4,5} of size 480
{6,4,5} of size 720
{8,4,5} of size 960
{10,4,5} of size 1200
{12,4,5} of size 1440
{14,4,5} of size 1680
{16,4,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,5}*240, {4,10}*240a, {4,10}*240b
4-fold covers : {8,5}*480, {4,10}*480a, {4,10}*480b, {4,10}*480c
6-fold covers : {4,15}*720, {12,10}*720
8-fold covers : {8,10}*960a, {8,10}*960b, {4,20}*960a, {4,20}*960b, {8,10}*960c, {8,10}*960d, {4,20}*960c, {4,20}*960d, {4,10}*960
10-fold covers : {4,5}*1200, {20,5}*1200, {20,10}*1200
12-fold covers : {12,10}*1440a, {12,10}*1440b, {8,15}*1440, {4,30}*1440, {12,10}*1440g
14-fold covers : {4,35}*1680, {28,10}*1680
16-fold covers : {16,10}*1920a, {16,10}*1920b, {4,20}*1920a, {8,10}*1920a, {4,40}*1920a, {4,40}*1920b, {8,10}*1920b, {4,20}*1920b, {4,10}*1920, {8,20}*1920a, {8,20}*1920b, {8,20}*1920c, {8,20}*1920d, {4,40}*1920c, {4,40}*1920d, {4,5}*1920, {8,5}*1920a, {8,5}*1920b
Permutation Representation (GAP) :
```s0 := (4,5);;
s1 := (2,4)(3,5);;
s2 := (1,2)(4,5);;
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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope