Polytope of Type {6,26}

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

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

References : None.
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