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Polytope of Type {78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78}*156
Also Known As : 78-gon, {78}. if this polytope has another name.
Group : SmallGroup(156,17)
Rank : 2
Schlafli Type : {78}
Number of vertices, edges, etc : 78, 78
Order of s0s1 : 78
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {78,2} of size 312
   {78,4} of size 624
   {78,4} of size 624
   {78,4} of size 624
   {78,6} of size 936
   {78,6} of size 936
   {78,6} of size 936
   {78,8} of size 1248
   {78,6} of size 1248
   {78,4} of size 1248
   {78,6} of size 1404
   {78,10} of size 1560
   {78,12} of size 1872
   {78,12} of size 1872
   {78,12} of size 1872
   {78,4} of size 1872
   {78,12} of size 1872
Vertex Figure Of :
   {2,78} of size 312
   {4,78} of size 624
   {4,78} of size 624
   {4,78} of size 624
   {6,78} of size 936
   {6,78} of size 936
   {6,78} of size 936
   {8,78} of size 1248
   {6,78} of size 1248
   {4,78} of size 1248
   {6,78} of size 1404
   {10,78} of size 1560
   {12,78} of size 1872
   {12,78} of size 1872
   {12,78} of size 1872
   {4,78} of size 1872
   {12,78} of size 1872
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {39}*78
   3-fold quotients : {26}*52
   6-fold quotients : {13}*26
   13-fold quotients : {6}*12
   26-fold quotients : {3}*6
   39-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {156}*312
   3-fold covers : {234}*468
   4-fold covers : {312}*624
   5-fold covers : {390}*780
   6-fold covers : {468}*936
   7-fold covers : {546}*1092
   8-fold covers : {624}*1248
   9-fold covers : {702}*1404
   10-fold covers : {780}*1560
   11-fold covers : {858}*1716
   12-fold covers : {936}*1872
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);;
poly := Group([s0,s1]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*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*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(78)!( 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(78)!( 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);
poly := sub<Sym(78)|s0,s1>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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*s0*s1 >; 
 
References : None.
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