Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,78}

Atlas Canonical Name {2,78}*312

Overview

Group
SmallGroup(312,60)
Rank
3
Schläfli Type
{2,78}
Vertices, edges, …
2, 78, 78
Order of s0s1s2
78
Order of s0s1s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

13-fold

26-fold

39-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Representations

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 >;