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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,6}*864b
if this polytope has a name.
Group : SmallGroup(864,4704)
Rank : 5
Schlafli Type : {2,6,6,6}
Number of vertices, edges, etc : 2, 6, 18, 18, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,6,6,2} of size 1728
Vertex Figure Of :
{2,2,6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,6,6}*288a, {2,6,2,6}*288, {2,6,6,2}*288a
6-fold quotients : {2,3,2,6}*144, {2,6,2,3}*144
9-fold quotients : {2,2,2,6}*96, {2,2,6,2}*96, {2,6,2,2}*96
12-fold quotients : {2,3,2,3}*72
18-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48, {2,3,2,2}*48
27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,6,12}*1728b, {2,6,12,6}*1728b, {2,12,6,6}*1728b, {4,6,6,6}*1728d
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)(31,32)
(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56);;
s2 := ( 3, 4)( 6,10)( 7, 9)( 8,11)(12,13)(15,19)(16,18)(17,20)(21,22)(24,28)
(25,27)(26,29)(30,31)(33,37)(34,36)(35,38)(39,40)(42,46)(43,45)(44,47)(48,49)
(51,55)(52,54)(53,56);;
s3 := ( 3, 6)( 4, 7)( 5, 8)(12,24)(13,25)(14,26)(15,21)(16,22)(17,23)(18,27)
(19,28)(20,29)(30,33)(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)(44,50)
(45,54)(46,55)(47,56);;
s4 := ( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,30)
(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,48)(22,49)(23,50)
(24,51)(25,52)(26,53)(27,54)(28,55)(29,56);;
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, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s3*s2*s3*s4*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*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(56)!(1,2);
s1 := Sym(56)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)
(31,32)(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56);
s2 := Sym(56)!( 3, 4)( 6,10)( 7, 9)( 8,11)(12,13)(15,19)(16,18)(17,20)(21,22)
(24,28)(25,27)(26,29)(30,31)(33,37)(34,36)(35,38)(39,40)(42,46)(43,45)(44,47)
(48,49)(51,55)(52,54)(53,56);
s3 := Sym(56)!( 3, 6)( 4, 7)( 5, 8)(12,24)(13,25)(14,26)(15,21)(16,22)(17,23)
(18,27)(19,28)(20,29)(30,33)(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)
(44,50)(45,54)(46,55)(47,56);
s4 := Sym(56)!( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)
(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,48)(22,49)
(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56);
poly := sub<Sym(56)|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,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s4*s3*s2*s3*s4*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*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

```

to this polytope