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# Polytope of Type {52}

Atlas Canonical Name : {52}*104
Also Known As : 52-gon, {52}. if this polytope has another name.
Group : SmallGroup(104,6)
Rank : 2
Schlafli Type : {52}
Number of vertices, edges, etc : 52, 52
Order of s0s1 : 52
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{52,2} of size 208
{52,4} of size 416
{52,6} of size 624
{52,6} of size 624
{52,4} of size 832
{52,8} of size 832
{52,8} of size 832
{52,6} of size 936
{52,10} of size 1040
{52,12} of size 1248
{52,6} of size 1248
{52,14} of size 1456
{52,8} of size 1664
{52,16} of size 1664
{52,16} of size 1664
{52,4} of size 1664
{52,8} of size 1664
{52,18} of size 1872
{52,18} of size 1872
{52,4} of size 1872
{52,6} of size 1872
Vertex Figure Of :
{2,52} of size 208
{4,52} of size 416
{6,52} of size 624
{6,52} of size 624
{4,52} of size 832
{8,52} of size 832
{8,52} of size 832
{6,52} of size 936
{10,52} of size 1040
{12,52} of size 1248
{6,52} of size 1248
{14,52} of size 1456
{8,52} of size 1664
{16,52} of size 1664
{16,52} of size 1664
{4,52} of size 1664
{8,52} of size 1664
{18,52} of size 1872
{18,52} of size 1872
{4,52} of size 1872
{6,52} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {26}*52
4-fold quotients : {13}*26
13-fold quotients : {4}*8
26-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {104}*208
3-fold covers : {156}*312
4-fold covers : {208}*416
5-fold covers : {260}*520
6-fold covers : {312}*624
7-fold covers : {364}*728
8-fold covers : {416}*832
9-fold covers : {468}*936
10-fold covers : {520}*1040
11-fold covers : {572}*1144
12-fold covers : {624}*1248
13-fold covers : {676}*1352
14-fold covers : {728}*1456
15-fold covers : {780}*1560
16-fold covers : {832}*1664
17-fold covers : {884}*1768
18-fold covers : {936}*1872
19-fold covers : {988}*1976
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 5)( 7,10)( 8, 9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)
(23,26)(24,25)(27,28)(29,30)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41)(43,44)
(45,46)(47,50)(48,49)(51,52);;
s1 := ( 1, 7)( 2, 4)( 3,13)( 5,15)( 6, 9)( 8,11)(10,21)(12,23)(14,17)(16,19)
(18,29)(20,31)(22,25)(24,27)(26,37)(28,39)(30,33)(32,35)(34,45)(36,47)(38,41)
(40,43)(42,51)(44,48)(46,49)(50,52);;
poly := Group([s0,s1]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;
rels := [ s0*s0, s1*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*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*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(52)!( 2, 3)( 4, 5)( 7,10)( 8, 9)(11,12)(13,14)(15,18)(16,17)(19,20)
(21,22)(23,26)(24,25)(27,28)(29,30)(31,34)(32,33)(35,36)(37,38)(39,42)(40,41)
(43,44)(45,46)(47,50)(48,49)(51,52);
s1 := Sym(52)!( 1, 7)( 2, 4)( 3,13)( 5,15)( 6, 9)( 8,11)(10,21)(12,23)(14,17)
(16,19)(18,29)(20,31)(22,25)(24,27)(26,37)(28,39)(30,33)(32,35)(34,45)(36,47)
(38,41)(40,43)(42,51)(44,48)(46,49)(50,52);
poly := sub<Sym(52)|s0,s1>;

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

```
References : None.
to this polytope