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# Polytope of Type {15,4}

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

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

```
References : None.
to this polytope