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# Polytope of Type {2,70}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,70}*280
if this polytope has a name.
Group : SmallGroup(280,39)
Rank : 3
Schlafli Type : {2,70}
Number of vertices, edges, etc : 2, 70, 70
Order of s0s1s2 : 70
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,70,2} of size 560
{2,70,4} of size 1120
{2,70,6} of size 1680
Vertex Figure Of :
{2,2,70} of size 560
{3,2,70} of size 840
{4,2,70} of size 1120
{5,2,70} of size 1400
{6,2,70} of size 1680
{7,2,70} of size 1960
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,35}*140
5-fold quotients : {2,14}*56
7-fold quotients : {2,10}*40
10-fold quotients : {2,7}*28
14-fold quotients : {2,5}*20
35-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,140}*560, {4,70}*560
3-fold covers : {6,70}*840, {2,210}*840
4-fold covers : {4,140}*1120, {2,280}*1120, {8,70}*1120
5-fold covers : {2,350}*1400, {10,70}*1400b, {10,70}*1400c
6-fold covers : {12,70}*1680, {6,140}*1680a, {2,420}*1680, {4,210}*1680a
7-fold covers : {2,490}*1960, {14,70}*1960b, {14,70}*1960c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(10,31)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)
(17,24)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(39,44)(40,43)(41,42)(45,66)
(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,59)(53,65)(54,64)(55,63)(56,62)
(57,61)(58,60);;
s2 := ( 3,46)( 4,45)( 5,51)( 6,50)( 7,49)( 8,48)( 9,47)(10,39)(11,38)(12,44)
(13,43)(14,42)(15,41)(16,40)(17,67)(18,66)(19,72)(20,71)(21,70)(22,69)(23,68)
(24,60)(25,59)(26,65)(27,64)(28,63)(29,62)(30,61)(31,53)(32,52)(33,58)(34,57)
(35,56)(36,55)(37,54);;
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*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(72)!(1,2);
s1 := Sym(72)!( 4, 9)( 5, 8)( 6, 7)(10,31)(11,37)(12,36)(13,35)(14,34)(15,33)
(16,32)(17,24)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(39,44)(40,43)(41,42)
(45,66)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,59)(53,65)(54,64)(55,63)
(56,62)(57,61)(58,60);
s2 := Sym(72)!( 3,46)( 4,45)( 5,51)( 6,50)( 7,49)( 8,48)( 9,47)(10,39)(11,38)
(12,44)(13,43)(14,42)(15,41)(16,40)(17,67)(18,66)(19,72)(20,71)(21,70)(22,69)
(23,68)(24,60)(25,59)(26,65)(27,64)(28,63)(29,62)(30,61)(31,53)(32,52)(33,58)
(34,57)(35,56)(36,55)(37,54);
poly := sub<Sym(72)|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*s1*s2*s1*s2 >;

```

to this polytope