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# Polytope of Type {2,6,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,22}*528
if this polytope has a name.
Group : SmallGroup(528,164)
Rank : 4
Schlafli Type : {2,6,22}
Number of vertices, edges, etc : 2, 6, 66, 22
Order of s0s1s2s3 : 66
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,22,2} of size 1056
Vertex Figure Of :
{2,2,6,22} of size 1056
{3,2,6,22} of size 1584
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,22}*176
6-fold quotients : {2,2,11}*88
11-fold quotients : {2,6,2}*48
22-fold quotients : {2,3,2}*24
33-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,12,22}*1056, {2,6,44}*1056a, {4,6,22}*1056a
3-fold covers : {2,18,22}*1584, {6,6,22}*1584a, {6,6,22}*1584b, {2,6,66}*1584a, {2,6,66}*1584b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)
(24,35)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)
(57,68);;
s2 := ( 3,14)( 4,24)( 5,23)( 6,22)( 7,21)( 8,20)( 9,19)(10,18)(11,17)(12,16)
(13,15)(26,35)(27,34)(28,33)(29,32)(30,31)(36,47)(37,57)(38,56)(39,55)(40,54)
(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(59,68)(60,67)(61,66)(62,65)
(63,64);;
s3 := ( 3,37)( 4,36)( 5,46)( 6,45)( 7,44)( 8,43)( 9,42)(10,41)(11,40)(12,39)
(13,38)(14,48)(15,47)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)
(24,49)(25,59)(26,58)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)
(35,60);;
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*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(68)!(1,2);
s1 := Sym(68)!(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)
(23,34)(24,35)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)
(56,67)(57,68);
s2 := Sym(68)!( 3,14)( 4,24)( 5,23)( 6,22)( 7,21)( 8,20)( 9,19)(10,18)(11,17)
(12,16)(13,15)(26,35)(27,34)(28,33)(29,32)(30,31)(36,47)(37,57)(38,56)(39,55)
(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(59,68)(60,67)(61,66)(62,65)
(63,64);
s3 := Sym(68)!( 3,37)( 4,36)( 5,46)( 6,45)( 7,44)( 8,43)( 9,42)(10,41)(11,40)
(12,39)(13,38)(14,48)(15,47)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)
(23,50)(24,49)(25,59)(26,58)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)
(34,61)(35,60);
poly := sub<Sym(68)|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*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope