# Polytope of Type {4,39}

Atlas Canonical Name : {4,39}*312
if this polytope has a name.
Group : SmallGroup(312,48)
Rank : 3
Schlafli Type : {4,39}
Number of vertices, edges, etc : 4, 78, 39
Order of s0s1s2 : 39
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,39,2} of size 624
{4,39,4} of size 1248
{4,39,6} of size 1872
Vertex Figure Of :
{2,4,39} of size 624
Quotients (Maximal Quotients in Boldface) :
13-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,39}*624, {4,78}*624b, {4,78}*624c
3-fold covers : {4,117}*936
4-fold covers : {4,156}*1248b, {4,156}*1248c, {8,39}*1248, {4,78}*1248
5-fold covers : {4,195}*1560
6-fold covers : {4,117}*1872, {4,234}*1872b, {4,234}*1872c, {12,39}*1872, {12,78}*1872d
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)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52);;
s1 := ( 2, 3)( 5,49)( 6,51)( 7,50)( 8,52)( 9,45)(10,47)(11,46)(12,48)(13,41)
(14,43)(15,42)(16,44)(17,37)(18,39)(19,38)(20,40)(21,33)(22,35)(23,34)(24,36)
(25,29)(26,31)(27,30)(28,32);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,49)(10,50)(11,52)(12,51)(13,45)(14,46)
(15,48)(16,47)(17,41)(18,42)(19,44)(20,43)(21,37)(22,38)(23,40)(24,39)(25,33)
(26,34)(27,36)(28,35)(31,32);;
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*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(52)!( 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)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)
(41,42)(43,44)(45,46)(47,48)(49,50)(51,52);
s1 := Sym(52)!( 2, 3)( 5,49)( 6,51)( 7,50)( 8,52)( 9,45)(10,47)(11,46)(12,48)
(13,41)(14,43)(15,42)(16,44)(17,37)(18,39)(19,38)(20,40)(21,33)(22,35)(23,34)
(24,36)(25,29)(26,31)(27,30)(28,32);
s2 := Sym(52)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,49)(10,50)(11,52)(12,51)(13,45)
(14,46)(15,48)(16,47)(17,41)(18,42)(19,44)(20,43)(21,37)(22,38)(23,40)(24,39)
(25,33)(26,34)(27,36)(28,35)(31,32);
poly := sub<Sym(52)|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*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