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# Polytope of Type {6,21}

Atlas Canonical Name : {6,21}*336
if this polytope has a name.
Group : SmallGroup(336,212)
Rank : 3
Schlafli Type : {6,21}
Number of vertices, edges, etc : 8, 84, 28
Order of s0s1s2 : 28
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,21,2} of size 672
Vertex Figure Of :
{2,6,21} of size 672
{4,6,21} of size 1344
{3,6,21} of size 1680
Quotients (Maximal Quotients in Boldface) :
7-fold quotients : {6,3}*48
12-fold quotients : {2,7}*28
14-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,21}*672, {6,42}*672
3-fold covers : {6,21}*1008b
4-fold covers : {6,21}*1344, {6,84}*1344a, {12,42}*1344a, {6,42}*1344, {6,84}*1344b, {12,42}*1344b
5-fold covers : {6,105}*1680
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27);;
s1 := ( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)
(14,18)(15,20)(16,19);;
s2 := ( 1, 8)( 2, 6)( 3, 7)( 4, 5)( 9,28)(10,26)(11,27)(12,25)(13,24)(14,22)
(15,23)(16,21)(17,20);;
poly := Group([s0,s1,s2]);;

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

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

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

```
References : None.
to this polytope