# Polytope of Type {2,74}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,74}*296
if this polytope has a name.
Group : SmallGroup(296,13)
Rank : 3
Schlafli Type : {2,74}
Number of vertices, edges, etc : 2, 74, 74
Order of s0s1s2 : 74
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,74,2} of size 592
{2,74,4} of size 1184
{2,74,6} of size 1776
Vertex Figure Of :
{2,2,74} of size 592
{3,2,74} of size 888
{4,2,74} of size 1184
{5,2,74} of size 1480
{6,2,74} of size 1776
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,37}*148
37-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,148}*592, {4,74}*592
3-fold covers : {6,74}*888, {2,222}*888
4-fold covers : {4,148}*1184, {8,74}*1184, {2,296}*1184
5-fold covers : {10,74}*1480, {2,370}*1480
6-fold covers : {12,74}*1776, {6,148}*1776a, {2,444}*1776, {4,222}*1776a
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)(13,30)
(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(41,76)(42,75)(43,74)
(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)
(55,62)(56,61)(57,60)(58,59);;
s2 := ( 3,41)( 4,40)( 5,76)( 6,75)( 7,74)( 8,73)( 9,72)(10,71)(11,70)(12,69)
(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)
(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)
(35,46)(36,45)(37,44)(38,43)(39,42);;
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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(76)!(1,2);
s1 := Sym(76)!( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)
(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(41,76)(42,75)
(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)
(54,63)(55,62)(56,61)(57,60)(58,59);
s2 := Sym(76)!( 3,41)( 4,40)( 5,76)( 6,75)( 7,74)( 8,73)( 9,72)(10,71)(11,70)
(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)
(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)
(34,47)(35,46)(36,45)(37,44)(38,43)(39,42);
poly := sub<Sym(76)|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*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope