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# Polytope of Type {6,30}

Atlas Canonical Name : {6,30}*360a
if this polytope has a name.
Group : SmallGroup(360,137)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 6, 90, 30
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 :
{6,30,2} of size 720
{6,30,4} of size 1440
Vertex Figure Of :
{2,6,30} of size 720
{4,6,30} of size 1440
{4,6,30} of size 1440
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,10}*120
5-fold quotients : {6,6}*72c
9-fold quotients : {2,10}*40
10-fold quotients : {3,6}*36
15-fold quotients : {6,2}*24
18-fold quotients : {2,5}*20
30-fold quotients : {3,2}*12
45-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,60}*720a, {12,30}*720a
3-fold covers : {18,30}*1080a, {6,30}*1080a, {6,30}*1080d
4-fold covers : {6,120}*1440a, {24,30}*1440a, {12,60}*1440a, {6,30}*1440g, {6,60}*1440c
5-fold covers : {6,150}*1800a, {30,30}*1800a, {30,30}*1800b
Permutation Representation (GAP) :
```s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)
(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40);;
s1 := ( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,16)( 7,20)( 8,19)( 9,18)(10,17)
(11,26)(12,30)(13,29)(14,28)(15,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)
(43,44);;
s2 := ( 1, 2)( 3, 5)( 6,12)( 7,11)( 8,15)( 9,14)(10,13)(16,17)(18,20)(21,27)
(22,26)(23,30)(24,29)(25,28)(31,32)(33,35)(36,42)(37,41)(38,45)(39,44)
(40,43);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope