# Polytope of Type {3,6,15}

Atlas Canonical Name : {3,6,15}*540
if this polytope has a name.
Group : SmallGroup(540,54)
Rank : 4
Schlafli Type : {3,6,15}
Number of vertices, edges, etc : 3, 9, 45, 15
Order of s0s1s2s3 : 15
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,15,2} of size 1080
Vertex Figure Of :
{2,3,6,15} of size 1080
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,2,15}*180
5-fold quotients : {3,6,3}*108
9-fold quotients : {3,2,5}*60
15-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,30}*1080a, {6,6,15}*1080a
3-fold covers : {3,6,45}*1620, {9,6,15}*1620, {3,6,15}*1620a, {3,6,15}*1620b
Permutation Representation (GAP) :
```s0 := ( 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);;
s1 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,17)(19,20)(22,23)(25,26)(28,29)
(31,33)(34,36)(37,39)(40,42)(43,45);;
s2 := ( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)(10,22)
(11,24)(12,23)(13,19)(14,21)(15,20)(32,33)(34,43)(35,45)(36,44)(37,40)(38,42)
(39,41);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,34)(17,36)(18,35)
(19,31)(20,33)(21,32)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(28,37)(29,39)
(30,38);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := 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);
s1 := Sym(45)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,17)(19,20)(22,23)(25,26)
(28,29)(31,33)(34,36)(37,39)(40,42)(43,45);
s2 := Sym(45)!( 1,16)( 2,18)( 3,17)( 4,28)( 5,30)( 6,29)( 7,25)( 8,27)( 9,26)
(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(32,33)(34,43)(35,45)(36,44)(37,40)
(38,42)(39,41);
s3 := Sym(45)!( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,34)(17,36)
(18,35)(19,31)(20,33)(21,32)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(28,37)
(29,39)(30,38);
poly := sub<Sym(45)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope