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# Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*216c
if this polytope has a name.
Group : SmallGroup(216,159)
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
{6,12,4} of size 864
{6,12,6} of size 1296
{6,12,6} of size 1296
{6,12,6} of size 1296
{6,12,8} of size 1728
{6,12,6} of size 1728
{6,12,4} 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}*432i
3-fold covers : {6,36}*648a, {6,12}*648, {6,36}*648b, {6,36}*648c
4-fold covers : {6,24}*864h, {12,12}*864l, {12,12}*864o
5-fold covers : {6,60}*1080c
6-fold covers : {6,36}*1296m, {6,12}*1296o, {6,36}*1296n, {6,36}*1296o, {6,12}*1296t, {6,12}*1296u
7-fold covers : {6,84}*1512c
8-fold covers : {6,48}*1728h, {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x, {12,12}*1728ab
9-fold covers : {18,12}*1944g, {6,36}*1944, {6,12}*1944c, {18,12}*1944h, {18,12}*1944i, {6,108}*1944a, {6,108}*1944b, {6,108}*1944c
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18);;
s1 := ( 2, 3)( 5, 6)( 8, 9)(10,13)(11,15)(12,14)(17,18);;
s2 := ( 1,11)( 2,10)( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18);;
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*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*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)( 5, 6)( 8, 9)(10,13)(11,15)(12,14)(17,18);
s2 := Sym(18)!( 1,11)( 2,10)( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18);
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*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2 >;

```
References : None.
to this polytope