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# Polytope of Type {42,6}

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

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

```
References : None.
to this polytope