# Polytope of Type {6,42}

Atlas Canonical Name : {6,42}*504a
if this polytope has a name.
Group : SmallGroup(504,172)
Rank : 3
Schlafli Type : {6,42}
Number of vertices, edges, etc : 6, 126, 42
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,42,2} of size 1008
Vertex Figure Of :
{2,6,42} of size 1008
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,14}*168
7-fold quotients : {6,6}*72c
9-fold quotients : {2,14}*56
14-fold quotients : {3,6}*36
18-fold quotients : {2,7}*28
21-fold quotients : {6,2}*24
42-fold quotients : {3,2}*12
63-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,84}*1008a, {12,42}*1008a
3-fold covers : {18,42}*1512a, {6,42}*1512a, {6,42}*1512d
Permutation Representation (GAP) :
```s0 := ( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)(24,45)
(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)
(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56);;
s1 := ( 1,29)( 2,35)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,22)( 9,28)(10,27)
(11,26)(12,25)(13,24)(14,23)(15,36)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)
(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,63)(59,62)(60,61);;
s2 := ( 1, 2)( 3, 7)( 4, 6)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)(14,17)
(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)(43,44)
(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59);;
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*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(63)!( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,43)(23,44)
(24,45)(25,46)(26,47)(27,48)(28,49)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)
(35,63)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56);
s1 := Sym(63)!( 1,29)( 2,35)( 3,34)( 4,33)( 5,32)( 6,31)( 7,30)( 8,22)( 9,28)
(10,27)(11,26)(12,25)(13,24)(14,23)(15,36)(16,42)(17,41)(18,40)(19,39)(20,38)
(21,37)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,63)(59,62)(60,61);
s2 := Sym(63)!( 1, 2)( 3, 7)( 4, 6)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)
(14,17)(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)
(43,44)(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59);
poly := sub<Sym(63)|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*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 >;

```
References : None.
to this polytope