# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*216a
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 9, 54, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,2} of size 432
{6,12,4} of size 864
{6,12,6} of size 1296
{6,12,8} of size 1728
Vertex Figure Of :
{2,6,12} of size 432
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*432e
3-fold covers : {6,12}*648
4-fold covers : {6,24}*864d, {12,12}*864g
5-fold covers : {6,60}*1080a
6-fold covers : {6,12}*1296j, {6,12}*1296k, {6,12}*1296o
7-fold covers : {6,84}*1512a
8-fold covers : {6,48}*1728d, {12,12}*1728g, {12,24}*1728h, {24,12}*1728j, {24,12}*1728l, {12,24}*1728n
9-fold covers : {18,12}*1944a, {6,12}*1944a, {18,12}*1944d, {18,12}*1944e, {6,36}*1944
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18);;
s1 := ( 2, 3)( 4, 5)( 7, 9)(10,16)(11,18)(12,17)(14,15);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope