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# Polytope of Type {6,6,2}

Atlas Canonical Name : {6,6,2}*216
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {6,6,2}
Number of vertices, edges, etc : 9, 27, 9, 2
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,2,2} of size 432
{6,6,2,3} of size 648
{6,6,2,4} of size 864
{6,6,2,5} of size 1080
{6,6,2,6} of size 1296
{6,6,2,7} of size 1512
{6,6,2,8} of size 1728
{6,6,2,9} of size 1944
Vertex Figure Of :
{2,6,6,2} of size 432
{4,6,6,2} of size 864
{6,6,6,2} of size 1296
{4,6,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6,2}*432b
3-fold covers : {6,18,2}*648a, {18,6,2}*648a, {6,6,2}*648a, {6,6,2}*648b, {6,18,2}*648b, {18,6,2}*648b, {6,18,2}*648c, {18,6,2}*648c, {6,6,6}*648b
4-fold covers : {6,12,2}*864b, {12,6,2}*864b, {6,6,4}*864b, {6,6,4}*864d, {6,12,2}*864d, {12,6,2}*864d
5-fold covers : {6,30,2}*1080, {30,6,2}*1080
6-fold covers : {6,18,2}*1296b, {18,6,2}*1296b, {6,6,2}*1296a, {6,6,2}*1296b, {6,18,2}*1296f, {18,6,2}*1296f, {6,18,2}*1296g, {18,6,2}*1296g, {6,6,6}*1296j, {6,6,6}*1296m, {6,6,2}*1296g
7-fold covers : {6,42,2}*1512, {42,6,2}*1512
8-fold covers : {12,6,4}*1728a, {6,12,4}*1728b, {6,24,2}*1728b, {24,6,2}*1728b, {6,6,8}*1728b, {12,12,2}*1728c, {6,6,8}*1728d, {6,6,4}*1728b, {6,12,2}*1728b, {12,6,2}*1728b
9-fold covers : {18,18,2}*1944a, {6,6,2}*1944, {18,18,2}*1944b, {6,18,2}*1944a, {18,6,2}*1944a, {6,18,2}*1944b, {18,6,2}*1944b, {18,18,2}*1944c, {18,18,2}*1944d, {18,18,2}*1944e, {6,54,2}*1944a, {54,6,2}*1944a, {6,18,2}*1944c, {18,6,2}*1944c, {18,18,2}*1944f, {18,18,2}*1944g, {18,18,2}*1944h, {18,18,2}*1944i, {6,18,2}*1944d, {18,6,2}*1944d, {6,54,2}*1944b, {54,6,2}*1944b, {6,54,2}*1944c, {54,6,2}*1944c, {6,18,2}*1944e, {18,6,2}*1944e, {6,18,6}*1944b, {18,6,6}*1944a, {6,6,6}*1944a, {6,6,6}*1944d, {6,6,6}*1944g, {6,6,6}*1944h, {6,18,6}*1944d, {18,6,6}*1944b, {6,18,6}*1944f, {18,6,6}*1944c
Permutation Representation (GAP) :
```s0 := (4,5)(6,7)(8,9);;
s1 := (2,6)(3,4)(5,7);;
s2 := (1,2)(4,9)(5,8)(6,7);;
s3 := (10,11);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!(4,5)(6,7)(8,9);
s1 := Sym(11)!(2,6)(3,4)(5,7);
s2 := Sym(11)!(1,2)(4,9)(5,8)(6,7);
s3 := Sym(11)!(10,11);
poly := sub<Sym(11)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope