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# Polytope of Type {2,6,6}

Atlas Canonical Name : {2,6,6}*432c
if this polytope has a name.
Group : SmallGroup(432,545)
Rank : 4
Schlafli Type : {2,6,6}
Number of vertices, edges, etc : 2, 18, 54, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,6,2} of size 864
{2,6,6,3} of size 1296
{2,6,6,4} of size 1728
Vertex Figure Of :
{2,2,6,6} of size 864
{3,2,6,6} of size 1296
{4,2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,3,6}*216
3-fold quotients : {2,6,6}*144c
6-fold quotients : {2,3,6}*72
9-fold quotients : {2,6,2}*48
18-fold quotients : {2,3,2}*24
27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,12,6}*864a, {4,6,6}*864a, {2,6,12}*864c
3-fold covers : {2,18,6}*1296a, {2,18,6}*1296c, {2,18,6}*1296d, {2,18,6}*1296e, {2,6,6}*1296d, {2,6,18}*1296h, {6,6,6}*1296d, {6,6,6}*1296e, {2,6,6}*1296e
4-fold covers : {4,12,6}*1728a, {2,24,6}*1728a, {8,6,6}*1728a, {2,12,12}*1728a, {2,6,24}*1728c, {4,6,12}*1728c, {4,6,6}*1728a, {2,6,6}*1728a, {2,6,12}*1728a
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 3, 9)( 4,10)( 5,11)(12,18)(13,19)(14,20);;
s2 := ( 3,12)( 4,13)( 5,14)( 6,19)( 7,20)( 8,18)( 9,17)(10,15)(11,16);;
s3 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(20)!(1,2);
s1 := Sym(20)!( 3, 9)( 4,10)( 5,11)(12,18)(13,19)(14,20);
s2 := Sym(20)!( 3,12)( 4,13)( 5,14)( 6,19)( 7,20)( 8,18)( 9,17)(10,15)(11,16);
s3 := Sym(20)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20);
poly := sub<Sym(20)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

```

to this polytope