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# Polytope of Type {6,6}

Atlas Canonical Name : {6,6}*216c
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 18, 54, 18
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 432
{6,6,3} of size 648
{6,6,4} of size 864
{6,6,6} of size 1296
{6,6,6} of size 1296
{6,6,8} of size 1728
{6,6,3} of size 1944
{6,6,9} of size 1944
Vertex Figure Of :
{2,6,6} of size 432
{4,6,6} of size 864
{4,6,6} of size 864
{4,6,6} of size 864
{6,6,6} of size 1296
{6,6,6} of size 1296
{8,6,6} of size 1728
{4,6,6} of size 1728
{6,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6}*108
3-fold quotients : {6,6}*72c
6-fold quotients : {3,6}*36
9-fold quotients : {6,2}*24
18-fold quotients : {3,2}*12
27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,6}*432a, {6,12}*432c
3-fold covers : {18,6}*648a, {18,6}*648c, {18,6}*648d, {18,6}*648e, {6,6}*648d, {6,18}*648h, {6,6}*648e
4-fold covers : {24,6}*864a, {12,12}*864a, {6,24}*864c, {6,6}*864a, {6,12}*864a
5-fold covers : {6,30}*1080a, {30,6}*1080b
6-fold covers : {36,6}*1296a, {36,6}*1296c, {36,6}*1296d, {36,6}*1296e, {12,18}*1296d, {12,6}*1296c, {18,12}*1296e, {18,12}*1296f, {18,12}*1296g, {18,12}*1296h, {6,12}*1296d, {6,36}*1296h, {12,6}*1296h, {6,12}*1296i
7-fold covers : {6,42}*1512a, {42,6}*1512b
8-fold covers : {48,6}*1728a, {12,24}*1728a, {12,12}*1728a, {12,24}*1728b, {24,12}*1728c, {24,12}*1728e, {6,48}*1728c, {12,12}*1728i, {12,6}*1728a, {12,12}*1728m, {6,12}*1728c, {6,24}*1728b, {6,6}*1728b, {6,24}*1728d, {12,6}*1728d, {6,12}*1728e, {6,12}*1728f
9-fold covers : {18,18}*1944a, {18,6}*1944a, {6,18}*1944b, {18,6}*1944d, {18,18}*1944f, {18,6}*1944f, {18,18}*1944h, {18,18}*1944l, {18,18}*1944o, {54,6}*1944a, {18,6}*1944h, {18,18}*1944q, {18,18}*1944t, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {18,18}*1944ab, {54,6}*1944c, {54,6}*1944e, {6,6}*1944b, {6,18}*1944k, {18,6}*1944m, {6,18}*1944o, {6,6}*1944d, {6,6}*1944e, {18,6}*1944p, {18,6}*1944q, {18,6}*1944r, {6,6}*1944j, {6,18}*1944u
Permutation Representation (GAP) :
```s0 := ( 1, 7)( 2, 8)( 3, 9)(10,16)(11,17)(12,18);;
s1 := ( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,15)( 8,13)( 9,14);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!( 1, 7)( 2, 8)( 3, 9)(10,16)(11,17)(12,18);
s1 := Sym(18)!( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,15)( 8,13)( 9,14);
s2 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope