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# Polytope of Type {2,78}

Atlas Canonical Name : {2,78}*312
if this polytope has a name.
Group : SmallGroup(312,60)
Rank : 3
Schlafli Type : {2,78}
Number of vertices, edges, etc : 2, 78, 78
Order of s0s1s2 : 78
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,78,2} of size 624
{2,78,4} of size 1248
{2,78,4} of size 1248
{2,78,4} of size 1248
{2,78,6} of size 1872
{2,78,6} of size 1872
{2,78,6} of size 1872
Vertex Figure Of :
{2,2,78} of size 624
{3,2,78} of size 936
{4,2,78} of size 1248
{5,2,78} of size 1560
{6,2,78} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,39}*156
3-fold quotients : {2,26}*104
6-fold quotients : {2,13}*52
13-fold quotients : {2,6}*24
26-fold quotients : {2,3}*12
39-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,156}*624, {4,78}*624a
3-fold covers : {2,234}*936, {6,78}*936b, {6,78}*936c
4-fold covers : {4,156}*1248a, {2,312}*1248, {8,78}*1248, {4,78}*1248
5-fold covers : {10,78}*1560, {2,390}*1560
6-fold covers : {2,468}*1872, {4,234}*1872a, {12,78}*1872b, {6,156}*1872b, {6,156}*1872c, {12,78}*1872c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,29)(17,41)(18,40)(19,39)
(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,54)(44,53)
(45,52)(46,51)(47,50)(48,49)(55,68)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)
(62,74)(63,73)(64,72)(65,71)(66,70)(67,69);;
s2 := ( 3,56)( 4,55)( 5,67)( 6,66)( 7,65)( 8,64)( 9,63)(10,62)(11,61)(12,60)
(13,59)(14,58)(15,57)(16,43)(17,42)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)
(24,48)(25,47)(26,46)(27,45)(28,44)(29,69)(30,68)(31,80)(32,79)(33,78)(34,77)
(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70);;
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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(80)!(1,2);
s1 := Sym(80)!( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,29)(17,41)(18,40)
(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,54)
(44,53)(45,52)(46,51)(47,50)(48,49)(55,68)(56,80)(57,79)(58,78)(59,77)(60,76)
(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69);
s2 := Sym(80)!( 3,56)( 4,55)( 5,67)( 6,66)( 7,65)( 8,64)( 9,63)(10,62)(11,61)
(12,60)(13,59)(14,58)(15,57)(16,43)(17,42)(18,54)(19,53)(20,52)(21,51)(22,50)
(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,69)(30,68)(31,80)(32,79)(33,78)
(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70);
poly := sub<Sym(80)|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*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope