Polytope of Type {26,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {26,6}*312
Also Known As : {26,6|2}. if this polytope has another name.
Group : SmallGroup(312,54)
Rank : 3
Schlafli Type : {26,6}
Number of vertices, edges, etc : 26, 78, 6
Order of s₀s₁s₂ : 78
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {26,6,2} of size 624
   {26,6,3} of size 936
   {26,6,4} of size 1248
   {26,6,3} of size 1248
   {26,6,4} of size 1248
   {26,6,6} of size 1872
   {26,6,6} of size 1872
   {26,6,6} of size 1872
Vertex Figure Of :
   {2,26,6} of size 624
   {4,26,6} of size 1248
   {6,26,6} of size 1872
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {26,2}*104
   6-fold quotients : {13,2}*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 : {26,12}*624, {52,6}*624a
   3-fold covers : {26,18}*936, {78,6}*936a, {78,6}*936b
   4-fold covers : {26,24}*1248, {104,6}*1248, {52,12}*1248, {52,6}*1248
   5-fold covers : {26,30}*1560, {130,6}*1560
   6-fold covers : {26,36}*1872, {52,18}*1872a, {156,6}*1872a, {78,12}*1872a, {78,12}*1872b, {156,6}*1872b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope