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# Polytope of Type {2,21}

Atlas Canonical Name : {2,21}*84
if this polytope has a name.
Group : SmallGroup(84,14)
Rank : 3
Schlafli Type : {2,21}
Number of vertices, edges, etc : 2, 21, 21
Order of s0s1s2 : 42
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,21,2} of size 168
{2,21,4} of size 336
{2,21,6} of size 504
{2,21,6} of size 672
{2,21,4} of size 672
{2,21,14} of size 1176
{2,21,12} of size 1344
{2,21,8} of size 1344
{2,21,6} of size 1512
{2,21,10} of size 1680
Vertex Figure Of :
{2,2,21} of size 168
{3,2,21} of size 252
{4,2,21} of size 336
{5,2,21} of size 420
{6,2,21} of size 504
{7,2,21} of size 588
{8,2,21} of size 672
{9,2,21} of size 756
{10,2,21} of size 840
{11,2,21} of size 924
{12,2,21} of size 1008
{13,2,21} of size 1092
{14,2,21} of size 1176
{15,2,21} of size 1260
{16,2,21} of size 1344
{17,2,21} of size 1428
{18,2,21} of size 1512
{19,2,21} of size 1596
{20,2,21} of size 1680
{21,2,21} of size 1764
{22,2,21} of size 1848
{23,2,21} of size 1932
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,7}*28
7-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,42}*168
3-fold covers : {2,63}*252, {6,21}*252
4-fold covers : {2,84}*336, {4,42}*336a, {4,21}*336
5-fold covers : {2,105}*420
6-fold covers : {2,126}*504, {6,42}*504b, {6,42}*504c
7-fold covers : {2,147}*588, {14,21}*588
8-fold covers : {4,84}*672a, {2,168}*672, {8,42}*672, {8,21}*672, {4,42}*672
9-fold covers : {2,189}*756, {6,63}*756, {6,21}*756
10-fold covers : {10,42}*840, {2,210}*840
11-fold covers : {2,231}*924
12-fold covers : {2,252}*1008, {4,126}*1008a, {4,63}*1008, {12,42}*1008b, {6,84}*1008b, {6,84}*1008c, {12,42}*1008c, {12,21}*1008, {6,21}*1008b
13-fold covers : {2,273}*1092
14-fold covers : {2,294}*1176, {14,42}*1176b, {14,42}*1176c
15-fold covers : {2,315}*1260, {6,105}*1260
16-fold covers : {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b, {2,336}*1344, {16,42}*1344, {8,21}*1344, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
17-fold covers : {2,357}*1428
18-fold covers : {2,378}*1512, {6,126}*1512a, {6,126}*1512b, {18,42}*1512b, {6,42}*1512b, {6,42}*1512c, {6,42}*1512d
19-fold covers : {2,399}*1596
20-fold covers : {20,42}*1680a, {10,84}*1680, {2,420}*1680, {4,210}*1680a, {4,105}*1680
21-fold covers : {2,441}*1764, {6,147}*1764, {14,63}*1764, {42,21}*1764
22-fold covers : {22,42}*1848, {2,462}*1848
23-fold covers : {2,483}*1932
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23);;
s2 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22);;
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*s1*s0*s1, s0*s2*s0*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*s1*s2 ];;
poly := F / rels;;

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

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

```

to this polytope