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# Polytope of Type {6,15}

Atlas Canonical Name : {6,15}*240
if this polytope has a name.
Group : SmallGroup(240,194)
Rank : 3
Schlafli Type : {6,15}
Number of vertices, edges, etc : 8, 60, 20
Order of s0s1s2 : 20
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,15,2} of size 480
{6,15,4} of size 1920
Vertex Figure Of :
{2,6,15} of size 480
{4,6,15} of size 960
{3,6,15} of size 1200
{6,6,15} of size 1440
{4,6,15} of size 1920
{8,6,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {6,3}*48
10-fold quotients : {3,3}*24
12-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,15}*480, {6,30}*480
3-fold covers : {6,15}*720e
4-fold covers : {6,15}*960, {6,60}*960a, {12,30}*960a, {6,30}*960, {6,60}*960b, {12,30}*960b
5-fold covers : {6,75}*1200, {30,15}*1200
6-fold covers : {12,15}*1440c, {6,30}*1440g, {6,30}*1440h
7-fold covers : {6,105}*1680
8-fold covers : {12,15}*1920, {6,30}*1920a, {12,60}*1920a, {12,60}*1920b, {6,60}*1920, {6,30}*1920b, {6,30}*1920c, {6,120}*1920a, {6,120}*1920b, {12,60}*1920c, {24,30}*1920a, {12,30}*1920, {12,60}*1920d, {24,30}*1920b
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20);;
s1 := ( 2, 3)( 5,17)( 6,19)( 7,18)( 8,20)( 9,13)(10,15)(11,14)(12,16);;
s2 := ( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,18)(10,17)(11,19)(12,20)(13,14);;
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*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope