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# Polytope of Type {6,6,2}

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

```

to this polytope