# Polytope of Type {4,15}

Atlas Canonical Name : {4,15}*120
if this polytope has a name.
Group : SmallGroup(120,38)
Rank : 3
Schlafli Type : {4,15}
Number of vertices, edges, etc : 4, 30, 15
Order of s0s1s2 : 15
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,15,2} of size 240
{4,15,4} of size 480
{4,15,6} of size 720
{4,15,4} of size 960
{4,15,10} of size 1200
{4,15,6} of size 1440
{4,15,10} of size 1440
{4,15,8} of size 1920
{4,15,4} of size 1920
Vertex Figure Of :
{2,4,15} of size 240
{4,4,15} of size 960
{4,4,15} of size 1920
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,15}*240, {4,30}*240b, {4,30}*240c
3-fold covers : {4,45}*360
4-fold covers : {4,60}*480b, {4,60}*480c, {8,15}*480, {4,30}*480
5-fold covers : {4,75}*600
6-fold covers : {4,45}*720, {4,90}*720b, {4,90}*720c, {12,15}*720, {12,30}*720d
7-fold covers : {4,105}*840
8-fold covers : {4,30}*960a, {8,15}*960a, {8,30}*960a, {4,120}*960c, {4,120}*960d, {4,60}*960b, {4,30}*960b, {4,60}*960c, {8,30}*960b, {8,30}*960c
9-fold covers : {4,135}*1080
10-fold covers : {4,75}*1200, {4,150}*1200b, {4,150}*1200c, {20,15}*1200, {20,30}*1200d
11-fold covers : {4,165}*1320
12-fold covers : {4,180}*1440b, {4,180}*1440c, {8,45}*1440, {4,90}*1440, {24,15}*1440, {12,30}*1440a, {12,30}*1440b
13-fold covers : {4,195}*1560
14-fold covers : {28,30}*1680b, {4,105}*1680, {4,210}*1680b, {4,210}*1680c
15-fold covers : {4,225}*1800
16-fold covers : {4,60}*1920b, {4,60}*1920c, {8,15}*1920a, {8,30}*1920a, {8,60}*1920c, {8,60}*1920d, {8,30}*1920b, {8,30}*1920c, {4,240}*1920c, {4,240}*1920d, {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,30}*1920a, {8,30}*1920d, {8,30}*1920e, {8,30}*1920f, {8,60}*1920g, {8,60}*1920h, {4,120}*1920c, {4,120}*1920d, {8,30}*1920g, {4,60}*1920e, {4,120}*1920e, {4,30}*1920b, {4,120}*1920f, {4,15}*1920b
Permutation Representation (GAP) :
```s0 := ( 1, 4)( 2, 6)( 3, 8)( 5,11)( 7,15)( 9,10)(12,16)(13,14)(17,20)(18,19);;
s1 := ( 2, 3)( 4, 9)( 5, 7)( 6,12)( 8,13)(11,17)(14,16)(15,18)(19,20);;
s2 := ( 1, 2)( 3, 5)( 4, 6)( 8,11)( 9,14)(10,13)(12,19)(16,18)(17,20);;
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*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope