Polytope of Type {3,2,26,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,26,6}*1872
if this polytope has a name.
Group : SmallGroup(1872,1061)
Rank : 5
Schlafli Type : {3,2,26,6}
Number of vertices, edges, etc : 3, 3, 26, 78, 6
Order of s₀s₁s₂s₃s₄ : 78
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,26,2}*624
   6-fold quotients : {3,2,13,2}*312
   13-fold quotients : {3,2,2,6}*144
   26-fold quotients : {3,2,2,3}*72
   39-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(18,29)(19,28)(20,27)(21,26)
(22,25)(23,24)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(44,55)(45,54)(46,53)
(47,52)(48,51)(49,50)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(70,81)(71,80)
(72,79)(73,78)(74,77)(75,76);;
s3 := ( 4, 5)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)(17,31)(18,30)(19,42)(20,41)
(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(43,44)(45,55)
(46,54)(47,53)(48,52)(49,51)(56,70)(57,69)(58,81)(59,80)(60,79)(61,78)(62,77)
(63,76)(64,75)(65,74)(66,73)(67,72)(68,71);;
s4 := ( 4,56)( 5,57)( 6,58)( 7,59)( 8,60)( 9,61)(10,62)(11,63)(12,64)(13,65)
(14,66)(15,67)(16,68)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)
(25,51)(26,52)(27,53)(28,54)(29,55)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)
(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!(2,3);
s1 := Sym(81)!(1,2);
s2 := Sym(81)!( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(18,29)(19,28)(20,27)
(21,26)(22,25)(23,24)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(44,55)(45,54)
(46,53)(47,52)(48,51)(49,50)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(70,81)
(71,80)(72,79)(73,78)(74,77)(75,76);
s3 := Sym(81)!( 4, 5)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)(17,31)(18,30)(19,42)
(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(43,44)
(45,55)(46,54)(47,53)(48,52)(49,51)(56,70)(57,69)(58,81)(59,80)(60,79)(61,78)
(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71);
s4 := Sym(81)!( 4,56)( 5,57)( 6,58)( 7,59)( 8,60)( 9,61)(10,62)(11,63)(12,64)
(13,65)(14,66)(15,67)(16,68)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)
(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,69)(31,70)(32,71)(33,72)(34,73)
(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81);
poly := sub<Sym(81)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope