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# Polytope of Type {4,21}

Atlas Canonical Name : {4,21}*168
if this polytope has a name.
Group : SmallGroup(168,46)
Rank : 3
Schlafli Type : {4,21}
Number of vertices, edges, etc : 4, 42, 21
Order of s0s1s2 : 21
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,21,2} of size 336
{4,21,4} of size 672
{4,21,6} of size 1008
{4,21,4} of size 1344
Vertex Figure Of :
{2,4,21} of size 336
{4,4,21} of size 1344
Quotients (Maximal Quotients in Boldface) :
7-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,21}*336, {4,42}*336b, {4,42}*336c
3-fold covers : {4,63}*504
4-fold covers : {4,84}*672b, {4,84}*672c, {8,21}*672, {4,42}*672
5-fold covers : {4,105}*840
6-fold covers : {4,63}*1008, {4,126}*1008b, {4,126}*1008c, {12,21}*1008, {12,42}*1008d
7-fold covers : {4,147}*1176
8-fold covers : {4,42}*1344a, {8,21}*1344, {8,42}*1344a, {4,168}*1344c, {4,168}*1344d, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
9-fold covers : {4,189}*1512
10-fold covers : {20,42}*1680b, {4,105}*1680, {4,210}*1680b, {4,210}*1680c
11-fold covers : {4,231}*1848
Permutation Representation (GAP) :
```s0 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28);;
s1 := ( 2, 3)( 5,25)( 6,27)( 7,26)( 8,28)( 9,21)(10,23)(11,22)(12,24)(13,17)
(14,19)(15,18)(16,20);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,25)(10,26)(11,28)(12,27)(13,21)(14,22)
(15,24)(16,23)(19,20);;
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*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope