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# Polytope of Type {2,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {2,6,6}
Number of vertices, edges, etc : 2, 72, 216, 72
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,6}*576a, {2,6,6}*576b
4-fold quotients : {2,6,6}*432d
6-fold quotients : {2,3,6}*288, {2,6,3}*288
9-fold quotients : {2,6,6}*192
12-fold quotients : {2,6,6}*144a, {2,6,6}*144b, {2,6,6}*144c
18-fold quotients : {2,3,6}*96, {2,6,3}*96
24-fold quotients : {2,3,6}*72, {2,6,3}*72
36-fold quotients : {2,3,3}*48, {2,2,6}*48, {2,6,2}*48
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 7,11)( 8,13)( 9,12)(10,14)(16,17)(19,23)(20,25)(21,24)(22,26)
(28,29)(31,35)(32,37)(33,36)(34,38);;
s2 := ( 3, 7)( 4, 8)( 5,10)( 6, 9)(13,14)(15,31)(16,32)(17,34)(18,33)(19,27)
(20,28)(21,30)(22,29)(23,35)(24,36)(25,38)(26,37);;
s3 := ( 3,18)( 4,16)( 5,17)( 6,15)( 7,22)( 8,20)( 9,21)(10,19)(11,26)(12,24)
(13,25)(14,23)(27,30)(31,34)(35,38);;
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*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(38)!(1,2);
s1 := Sym(38)!( 4, 5)( 7,11)( 8,13)( 9,12)(10,14)(16,17)(19,23)(20,25)(21,24)
(22,26)(28,29)(31,35)(32,37)(33,36)(34,38);
s2 := Sym(38)!( 3, 7)( 4, 8)( 5,10)( 6, 9)(13,14)(15,31)(16,32)(17,34)(18,33)
(19,27)(20,28)(21,30)(22,29)(23,35)(24,36)(25,38)(26,37);
s3 := Sym(38)!( 3,18)( 4,16)( 5,17)( 6,15)( 7,22)( 8,20)( 9,21)(10,19)(11,26)
(12,24)(13,25)(14,23)(27,30)(31,34)(35,38);
poly := sub<Sym(38)|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*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2 >;

```

to this polytope