# Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1080a
if this polytope has a name.
Group : SmallGroup(1080,287)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 18, 270, 90
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,30}*360a
5-fold quotients : {6,6}*216c
9-fold quotients : {6,10}*120
10-fold quotients : {3,6}*108
15-fold quotients : {6,6}*72c
27-fold quotients : {2,10}*40
30-fold quotients : {3,6}*36
45-fold quotients : {6,2}*24
54-fold quotients : {2,5}*20
90-fold quotients : {3,2}*12
135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
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 := ( 1,17)( 2,18)( 3,16)( 4,29)( 5,30)( 6,28)( 7,26)( 8,27)( 9,25)(10,23)
(11,24)(12,22)(13,20)(14,21)(15,19)(34,43)(35,44)(36,45)(37,40)(38,41)
(39,42);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,19)(17,21)(18,20)
(22,28)(23,30)(24,29)(26,27)(31,34)(32,36)(33,35)(37,43)(38,45)(39,44)
(41,42);;
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*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 ];;
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)!( 1,17)( 2,18)( 3,16)( 4,29)( 5,30)( 6,28)( 7,26)( 8,27)( 9,25)
(10,23)(11,24)(12,22)(13,20)(14,21)(15,19)(34,43)(35,44)(36,45)(37,40)(38,41)
(39,42);
s2 := Sym(45)!( 1, 4)( 2, 6)( 3, 5)( 7,13)( 8,15)( 9,14)(11,12)(16,19)(17,21)
(18,20)(22,28)(23,30)(24,29)(26,27)(31,34)(32,36)(33,35)(37,43)(38,45)(39,44)
(41,42);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 >;

```
References : None.
to this polytope