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# Polytope of Type {15,6}

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

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

```
References : None.
to this polytope