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# Polytope of Type {30,12}

Atlas Canonical Name : {30,12}*720d
if this polytope has a name.
Group : SmallGroup(720,793)
Rank : 3
Schlafli Type : {30,12}
Number of vertices, edges, etc : 30, 180, 12
Order of s0s1s2 : 15
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{30,12,2} of size 1440
Vertex Figure Of :
{2,30,12} of size 1440
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {30,4}*240c
5-fold quotients : {6,12}*144d
6-fold quotients : {15,4}*120
15-fold quotients : {6,4}*48b
30-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {30,12}*1440a
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5,17)( 6,19)( 7,18)( 8,20)( 9,13)(10,15)(11,14)(12,16)(22,23)
(25,37)(26,39)(27,38)(28,40)(29,33)(30,35)(31,34)(32,36)(42,43)(45,57)(46,59)
(47,58)(48,60)(49,53)(50,55)(51,54)(52,56);;
s1 := ( 1, 5)( 2, 8)( 3, 7)( 4, 6)( 9,17)(10,20)(11,19)(12,18)(14,16)(21,45)
(22,48)(23,47)(24,46)(25,41)(26,44)(27,43)(28,42)(29,57)(30,60)(31,59)(32,58)
(33,53)(34,56)(35,55)(36,54)(37,49)(38,52)(39,51)(40,50);;
s2 := ( 1,24)( 2,23)( 3,22)( 4,21)( 5,28)( 6,27)( 7,26)( 8,25)( 9,32)(10,31)
(11,30)(12,29)(13,36)(14,35)(15,34)(16,33)(17,40)(18,39)(19,38)(20,37)(41,44)
(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59);;
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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*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(60)!( 2, 3)( 5,17)( 6,19)( 7,18)( 8,20)( 9,13)(10,15)(11,14)(12,16)
(22,23)(25,37)(26,39)(27,38)(28,40)(29,33)(30,35)(31,34)(32,36)(42,43)(45,57)
(46,59)(47,58)(48,60)(49,53)(50,55)(51,54)(52,56);
s1 := Sym(60)!( 1, 5)( 2, 8)( 3, 7)( 4, 6)( 9,17)(10,20)(11,19)(12,18)(14,16)
(21,45)(22,48)(23,47)(24,46)(25,41)(26,44)(27,43)(28,42)(29,57)(30,60)(31,59)
(32,58)(33,53)(34,56)(35,55)(36,54)(37,49)(38,52)(39,51)(40,50);
s2 := Sym(60)!( 1,24)( 2,23)( 3,22)( 4,21)( 5,28)( 6,27)( 7,26)( 8,25)( 9,32)
(10,31)(11,30)(12,29)(13,36)(14,35)(15,34)(16,33)(17,40)(18,39)(19,38)(20,37)
(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59);
poly := sub<Sym(60)|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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope