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# Polytope of Type {2,68}

Atlas Canonical Name : {2,68}*272
if this polytope has a name.
Group : SmallGroup(272,38)
Rank : 3
Schlafli Type : {2,68}
Number of vertices, edges, etc : 2, 68, 68
Order of s0s1s2 : 68
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,68,2} of size 544
{2,68,4} of size 1088
{2,68,6} of size 1632
{2,68,6} of size 1632
Vertex Figure Of :
{2,2,68} of size 544
{3,2,68} of size 816
{4,2,68} of size 1088
{5,2,68} of size 1360
{6,2,68} of size 1632
{7,2,68} of size 1904
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,34}*136
4-fold quotients : {2,17}*68
17-fold quotients : {2,4}*16
34-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,68}*544, {2,136}*544
3-fold covers : {6,68}*816a, {2,204}*816
4-fold covers : {8,68}*1088a, {4,136}*1088a, {8,68}*1088b, {4,136}*1088b, {4,68}*1088, {2,272}*1088
5-fold covers : {10,68}*1360, {2,340}*1360
6-fold covers : {6,136}*1632, {12,68}*1632, {4,204}*1632a, {2,408}*1632
7-fold covers : {14,68}*1904, {2,476}*1904
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)(21,36)(22,35)
(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,54)(38,70)(39,69)(40,68)(41,67)
(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)
(53,55);;
s2 := ( 3,38)( 4,37)( 5,53)( 6,52)( 7,51)( 8,50)( 9,49)(10,48)(11,47)(12,46)
(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,55)(21,54)(22,70)(23,69)
(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)
(35,57)(36,56);;
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*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(70)!(1,2);
s1 := Sym(70)!( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)(21,36)
(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,54)(38,70)(39,69)(40,68)
(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,60)(49,59)(50,58)(51,57)
(52,56)(53,55);
s2 := Sym(70)!( 3,38)( 4,37)( 5,53)( 6,52)( 7,51)( 8,50)( 9,49)(10,48)(11,47)
(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,55)(21,54)(22,70)
(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)
(34,58)(35,57)(36,56);
poly := sub<Sym(70)|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*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