# Polytope of Type {5,10,5}

Atlas Canonical Name : {5,10,5}*500
if this polytope has a name.
Group : SmallGroup(500,27)
Rank : 4
Schlafli Type : {5,10,5}
Number of vertices, edges, etc : 5, 25, 25, 5
Order of s0s1s2s3 : 5
Order of s0s1s2s3s2s1 : 10
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{5,10,5,2} of size 1000
Vertex Figure Of :
{2,5,10,5} of size 1000
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {5,2,5}*100
Covers (Minimal Covers in Boldface) :
2-fold covers : {5,10,10}*1000a, {10,10,5}*1000a
3-fold covers : {5,10,15}*1500, {15,10,5}*1500
4-fold covers : {5,10,20}*2000a, {20,10,5}*2000a, {10,10,10}*2000a
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)
(14,18)(15,17);;
s1 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)(15,22)
(17,20)(18,19);;
s2 := ( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24);;
s3 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := 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);
s1 := Sym(25)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)
(15,22)(17,20)(18,19);
s2 := Sym(25)!( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)
(22,24);
s3 := Sym(25)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)
(23,24);
poly := sub<Sym(25)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope