# Polytope of Type {5,10}

Atlas Canonical Name : {5,10}*500
if this polytope has a name.
Group : SmallGroup(500,27)
Rank : 3
Schlafli Type : {5,10}
Number of vertices, edges, etc : 25, 125, 50
Order of s0s1s2 : 10
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{5,10,2} of size 1000
{5,10,4} of size 2000
Vertex Figure Of :
{2,5,10} of size 1000
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {5,10}*100
25-fold quotients : {5,2}*20
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,10}*1000a
3-fold covers : {15,10}*1500e
4-fold covers : {20,10}*2000a, {10,20}*2000c
Permutation Representation (GAP) :
```s0 := ( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20);;
s1 := ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5, 6)(11,23)(12,24)(13,25)(14,21)(15,22);;
s2 := ( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)
(14,18)(15,17);;
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*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(25)!( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)
(15,20);
s1 := Sym(25)!( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5, 6)(11,23)(12,24)(13,25)(14,21)
(15,22);
s2 := Sym(25)!( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)
(13,19)(14,18)(15,17);
poly := sub<Sym(25)|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*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope