Questions?
See the FAQ
or other info.

# Polytope of Type {4,5}

Atlas Canonical Name : {4,5}*720
Also Known As : {4,5|5}. if this polytope has another name.
Group : SmallGroup(720,764)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 72, 180, 90
Order of s0s1s2 : 8
Order of s0s1s2s1 : 5
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,5,2} of size 1440
Vertex Figure Of :
{2,4,5} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,5}*1440, {4,10}*1440e, {4,10}*1440f
Permutation Representation (GAP) :
```s0 := ( 1, 3)( 2, 8)( 4,10)( 5, 6)( 7, 9);;
s1 := ( 1, 3)( 2, 4)( 5, 8)( 6, 9)( 7,10);;
s2 := ( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10);;
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope