# Polytope of Type {15,6}

Atlas Canonical Name : {15,6}*540
if this polytope has a name.
Group : SmallGroup(540,54)
Rank : 3
Schlafli Type : {15,6}
Number of vertices, edges, etc : 45, 135, 18
Order of s0s1s2 : 30
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 1080
{15,6,3} of size 1620
Vertex Figure Of :
{2,15,6} of size 1080
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {15,6}*180
5-fold quotients : {3,6}*108
9-fold quotients : {15,2}*60
15-fold quotients : {3,6}*36
27-fold quotients : {5,2}*20
45-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {30,6}*1080b
3-fold covers : {45,6}*1620a, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
Permutation Representation (GAP) :
```s0 := ( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(16,31)(17,32)(18,33)(19,43)
(20,44)(21,45)(22,40)(23,41)(24,42)(25,37)(26,38)(27,39)(28,34)(29,35)
(30,36);;
s1 := ( 1,20)( 2,21)( 3,19)( 4,17)( 5,18)( 6,16)( 7,29)( 8,30)( 9,28)(10,26)
(11,27)(12,25)(13,23)(14,24)(15,22)(31,34)(32,35)(33,36)(37,43)(38,44)
(39,45);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(45)!( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(16,31)(17,32)(18,33)
(19,43)(20,44)(21,45)(22,40)(23,41)(24,42)(25,37)(26,38)(27,39)(28,34)(29,35)
(30,36);
s1 := Sym(45)!( 1,20)( 2,21)( 3,19)( 4,17)( 5,18)( 6,16)( 7,29)( 8,30)( 9,28)
(10,26)(11,27)(12,25)(13,23)(14,24)(15,22)(31,34)(32,35)(33,36)(37,43)(38,44)
(39,45);
s2 := Sym(45)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45);
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, 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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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 >;

```
References : None.
to this polytope