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

Atlas Canonical Name : {2,6,6,6}*864g
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) :
2-fold quotients : {2,3,6,6}*432b
3-fold quotients : {2,6,2,6}*288, {2,6,6,2}*288c
6-fold quotients : {2,3,2,6}*144, {2,3,6,2}*144, {2,6,2,3}*144
9-fold quotients : {2,2,2,6}*96, {2,6,2,2}*96
12-fold quotients : {2,3,2,3}*72
18-fold quotients : {2,2,2,3}*48, {2,3,2,2}*48
27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,12,6,6}*1728d, {4,6,6,6}*1728f, {2,6,6,12}*1728e, {2,6,12,6}*1728f
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)
(25,29)(26,28)(31,32)(33,36)(34,38)(35,37)(40,41)(42,45)(43,47)(44,46)(49,50)
(51,54)(52,56)(53,55);;
s2 := ( 3,34)( 4,33)( 5,35)( 6,31)( 7,30)( 8,32)( 9,37)(10,36)(11,38)(12,43)
(13,42)(14,44)(15,40)(16,39)(17,41)(18,46)(19,45)(20,47)(21,52)(22,51)(23,53)
(24,49)(25,48)(26,50)(27,55)(28,54)(29,56);;
s3 := ( 4, 5)( 7, 8)(10,11)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)
(19,29)(20,28)(31,32)(34,35)(37,38)(39,48)(40,50)(41,49)(42,51)(43,53)(44,52)
(45,54)(46,56)(47,55);;
s4 := ( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)(30,39)
(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
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)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)
(24,27)(25,29)(26,28)(31,32)(33,36)(34,38)(35,37)(40,41)(42,45)(43,47)(44,46)
(49,50)(51,54)(52,56)(53,55);
s2 := Sym(56)!( 3,34)( 4,33)( 5,35)( 6,31)( 7,30)( 8,32)( 9,37)(10,36)(11,38)
(12,43)(13,42)(14,44)(15,40)(16,39)(17,41)(18,46)(19,45)(20,47)(21,52)(22,51)
(23,53)(24,49)(25,48)(26,50)(27,55)(28,54)(29,56);
s3 := Sym(56)!( 4, 5)( 7, 8)(10,11)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)
(18,27)(19,29)(20,28)(31,32)(34,35)(37,38)(39,48)(40,50)(41,49)(42,51)(43,53)
(44,52)(45,54)(46,56)(47,55);
s4 := Sym(56)!( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)
(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47);
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

```

to this polytope