Polytope of Type {78,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,2}*312
if this polytope has a name.
Group : SmallGroup(312,60)
Rank : 3
Schlafli Type : {78,2}
Number of vertices, edges, etc : 78, 78, 2
Order of s₀s₁s₂ : 78
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {78,2,2} of size 624
   {78,2,3} of size 936
   {78,2,4} of size 1248
   {78,2,5} of size 1560
   {78,2,6} of size 1872
Vertex Figure Of :
   {2,78,2} of size 624
   {4,78,2} of size 1248
   {4,78,2} of size 1248
   {4,78,2} of size 1248
   {6,78,2} of size 1872
   {6,78,2} of size 1872
   {6,78,2} of size 1872
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {39,2}*156
   3-fold quotients : {26,2}*104
   6-fold quotients : {13,2}*52
   13-fold quotients : {6,2}*24
   26-fold quotients : {3,2}*12
   39-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {156,2}*624, {78,4}*624a
   3-fold covers : {234,2}*936, {78,6}*936b, {78,6}*936c
   4-fold covers : {156,4}*1248a, {312,2}*1248, {78,8}*1248, {78,4}*1248
   5-fold covers : {78,10}*1560, {390,2}*1560
   6-fold covers : {468,2}*1872, {234,4}*1872a, {78,12}*1872b, {156,6}*1872b, {156,6}*1872c, {78,12}*1872c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope