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# Polytope of Type {21,2}

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

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

```

to this polytope