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# Polytope of Type {4,12}

Atlas Canonical Name : {4,12}*216
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 9, 54, 27
Order of s0s1s2 : 6
Order of s0s1s2s1 : 3
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{4,12,2} of size 432
Vertex Figure Of :
{2,4,12} of size 432
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12}*432a
3-fold covers : {12,12}*648
4-fold covers : {4,12}*864a
6-fold covers : {4,12}*1296, {12,12}*1296c, {12,12}*1296e
8-fold covers : {4,12}*1728b, {4,24}*1728b, {8,12}*1728b, {4,24}*1728d, {8,12}*1728c
9-fold covers : {4,12}*1944a, {4,12}*1944b, {4,12}*1944c, {4,12}*1944d
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(11,12)(13,14)(16,18);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);
s1 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);
s2 := Sym(18)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(11,12)(13,14)(16,18);
poly := sub<Sym(18)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope