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# Polytope of Type {2,6,6}

Atlas Canonical Name : {2,6,6}*432d
if this polytope has a name.
Group : SmallGroup(432,759)
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
{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) :
3-fold quotients : {2,6,6}*144a, {2,6,6}*144b, {2,6,6}*144c
6-fold quotients : {2,3,6}*72, {2,6,3}*72
9-fold quotients : {2,2,6}*48, {2,6,2}*48
18-fold quotients : {2,2,3}*24, {2,3,2}*24
27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,12}*864g, {2,12,6}*864g, {4,6,6}*864h
3-fold covers : {2,6,18}*1296i, {2,18,6}*1296i, {2,6,6}*1296e, {2,6,6}*1296f, {2,6,6}*1296g, {6,6,6}*1296q, {6,6,6}*1296r
4-fold covers : {2,6,24}*1728f, {2,24,6}*1728f, {8,6,6}*1728e, {2,12,12}*1728h, {4,12,6}*1728j, {4,6,12}*1728h, {4,6,6}*1728c, {2,6,6}*1728c, {2,6,12}*1728c, {2,12,6}*1728c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)(18,24)
(19,25)(20,26);;
s2 := ( 3,15)( 4,17)( 5,16)( 6,12)( 7,14)( 8,13)( 9,18)(10,20)(11,19)(21,24)
(22,26)(23,25)(28,29);;
s3 := ( 3, 4)( 6, 7)( 9,10)(12,22)(13,21)(14,23)(15,25)(16,24)(17,26)(18,28)
(19,27)(20,29);;
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(29)!(1,2);
s1 := Sym(29)!( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)
(18,24)(19,25)(20,26);
s2 := Sym(29)!( 3,15)( 4,17)( 5,16)( 6,12)( 7,14)( 8,13)( 9,18)(10,20)(11,19)
(21,24)(22,26)(23,25)(28,29);
s3 := Sym(29)!( 3, 4)( 6, 7)( 9,10)(12,22)(13,21)(14,23)(15,25)(16,24)(17,26)
(18,28)(19,27)(20,29);
poly := sub<Sym(29)|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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2 >;

```

to this polytope