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# Polytope of Type {2,52}

Atlas Canonical Name : {2,52}*208
if this polytope has a name.
Group : SmallGroup(208,37)
Rank : 3
Schlafli Type : {2,52}
Number of vertices, edges, etc : 2, 52, 52
Order of s0s1s2 : 52
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,52,2} of size 416
{2,52,4} of size 832
{2,52,6} of size 1248
{2,52,6} of size 1248
{2,52,8} of size 1664
{2,52,8} of size 1664
{2,52,4} of size 1664
{2,52,6} of size 1872
Vertex Figure Of :
{2,2,52} of size 416
{3,2,52} of size 624
{4,2,52} of size 832
{5,2,52} of size 1040
{6,2,52} of size 1248
{7,2,52} of size 1456
{8,2,52} of size 1664
{9,2,52} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,26}*104
4-fold quotients : {2,13}*52
13-fold quotients : {2,4}*16
26-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,52}*416, {2,104}*416
3-fold covers : {6,52}*624a, {2,156}*624
4-fold covers : {4,104}*832a, {4,52}*832, {4,104}*832b, {8,52}*832a, {8,52}*832b, {2,208}*832
5-fold covers : {10,52}*1040, {2,260}*1040
6-fold covers : {6,104}*1248, {12,52}*1248, {4,156}*1248a, {2,312}*1248
7-fold covers : {14,52}*1456, {2,364}*1456
8-fold covers : {8,52}*1664a, {4,104}*1664a, {8,104}*1664a, {8,104}*1664b, {8,104}*1664c, {8,104}*1664d, {16,52}*1664a, {4,208}*1664a, {16,52}*1664b, {4,208}*1664b, {4,52}*1664, {4,104}*1664b, {8,52}*1664b, {2,416}*1664
9-fold covers : {18,52}*1872a, {2,468}*1872, {6,156}*1872a, {6,156}*1872b, {6,156}*1872c, {6,52}*1872
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)
(25,28)(26,27)(29,30)(31,32)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)(45,46)
(47,48)(49,52)(50,51)(53,54);;
s2 := ( 3, 9)( 4, 6)( 5,15)( 7,17)( 8,11)(10,13)(12,23)(14,25)(16,19)(18,21)
(20,31)(22,33)(24,27)(26,29)(28,39)(30,41)(32,35)(34,37)(36,47)(38,49)(40,43)
(42,45)(44,53)(46,50)(48,51)(52,54);;
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*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*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(54)!(1,2);
s1 := Sym(54)!( 4, 5)( 6, 7)( 9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)
(23,24)(25,28)(26,27)(29,30)(31,32)(33,36)(34,35)(37,38)(39,40)(41,44)(42,43)
(45,46)(47,48)(49,52)(50,51)(53,54);
s2 := Sym(54)!( 3, 9)( 4, 6)( 5,15)( 7,17)( 8,11)(10,13)(12,23)(14,25)(16,19)
(18,21)(20,31)(22,33)(24,27)(26,29)(28,39)(30,41)(32,35)(34,37)(36,47)(38,49)
(40,43)(42,45)(44,53)(46,50)(48,51)(52,54);
poly := sub<Sym(54)|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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope