# Polytope of Type {2,2,6,6}

Atlas Canonical Name : {2,2,6,6}*864d
if this polytope has a name.
Group : SmallGroup(864,4704)
Rank : 5
Schlafli Type : {2,2,6,6}
Number of vertices, edges, etc : 2, 2, 18, 54, 18
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,2,6,6,2} of size 1728
Vertex Figure Of :
{2,2,2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,6,6}*288a, {2,2,6,6}*288b, {2,2,6,6}*288c
6-fold quotients : {2,2,3,6}*144, {2,2,6,3}*144
9-fold quotients : {2,2,2,6}*96, {2,2,6,2}*96
18-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48
27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,2,6,6}*1728d, {2,2,6,12}*1728g, {2,2,12,6}*1728g, {2,4,6,6}*1728h
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)(20,26)
(21,27)(22,28);;
s3 := ( 5,17)( 6,19)( 7,18)( 8,14)( 9,16)(10,15)(11,20)(12,22)(13,21)(23,26)
(24,28)(25,27)(30,31);;
s4 := ( 5, 6)( 8, 9)(11,12)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)(20,30)
(21,29)(22,31);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(31)!(1,2);
s1 := Sym(31)!(3,4);
s2 := Sym(31)!( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)
(20,26)(21,27)(22,28);
s3 := Sym(31)!( 5,17)( 6,19)( 7,18)( 8,14)( 9,16)(10,15)(11,20)(12,22)(13,21)
(23,26)(24,28)(25,27)(30,31);
s4 := Sym(31)!( 5, 6)( 8, 9)(11,12)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)
(20,30)(21,29)(22,31);
poly := sub<Sym(31)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3 >;

```

to this polytope