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# Polytope of Type {6,15}

Atlas Canonical Name : {6,15}*180
if this polytope has a name.
Group : SmallGroup(180,29)
Rank : 3
Schlafli Type : {6,15}
Number of vertices, edges, etc : 6, 45, 15
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,15,2} of size 360
{6,15,4} of size 720
{6,15,6} of size 1080
{6,15,4} of size 1440
{6,15,10} of size 1800
Vertex Figure Of :
{2,6,15} of size 360
{3,6,15} of size 540
{4,6,15} of size 720
{6,6,15} of size 1080
{6,6,15} of size 1080
{8,6,15} of size 1440
{9,6,15} of size 1620
{3,6,15} of size 1620
{10,6,15} of size 1800
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,15}*60
5-fold quotients : {6,3}*36
9-fold quotients : {2,5}*20
15-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,30}*360c
3-fold covers : {6,45}*540, {6,15}*540
4-fold covers : {6,60}*720c, {12,30}*720c, {12,15}*720, {6,15}*720e
5-fold covers : {6,75}*900, {30,15}*900
6-fold covers : {6,90}*1080b, {6,30}*1080b, {6,30}*1080d
7-fold covers : {6,105}*1260
8-fold covers : {6,120}*1440c, {12,60}*1440c, {24,30}*1440c, {24,15}*1440, {12,15}*1440c, {12,30}*1440b, {6,30}*1440h
9-fold covers : {18,45}*1620, {6,45}*1620a, {6,135}*1620, {6,45}*1620b, {6,45}*1620c, {6,45}*1620d, {6,15}*1620, {18,15}*1620
10-fold covers : {6,150}*1800c, {30,30}*1800c, {30,30}*1800h
11-fold covers : {6,165}*1980
Permutation Representation (GAP) :
```s0 := (16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)
(26,41)(27,42)(28,43)(29,44)(30,45);;
s1 := ( 1,16)( 2,20)( 3,19)( 4,18)( 5,17)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)
(11,21)(12,25)(13,24)(14,23)(15,22)(32,35)(33,34)(36,41)(37,45)(38,44)(39,43)
(40,42);;
s2 := ( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)(11,12)(13,15)(16,37)(17,36)(18,40)
(19,39)(20,38)(21,32)(22,31)(23,35)(24,34)(25,33)(26,42)(27,41)(28,45)(29,44)
(30,43);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(45)!(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)
(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);
s1 := Sym(45)!( 1,16)( 2,20)( 3,19)( 4,18)( 5,17)( 6,26)( 7,30)( 8,29)( 9,28)
(10,27)(11,21)(12,25)(13,24)(14,23)(15,22)(32,35)(33,34)(36,41)(37,45)(38,44)
(39,43)(40,42);
s2 := Sym(45)!( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)(11,12)(13,15)(16,37)(17,36)
(18,40)(19,39)(20,38)(21,32)(22,31)(23,35)(24,34)(25,33)(26,42)(27,41)(28,45)
(29,44)(30,43);
poly := sub<Sym(45)|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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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