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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,26}*1872
if this polytope has a name.
Group : SmallGroup(1872,1061)
Rank : 5
Schlafli Type : {3,2,6,26}
Number of vertices, edges, etc : 3, 3, 6, 78, 26
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 : {3,2,2,26}*624
   6-fold quotients : {3,2,2,13}*312
   13-fold quotients : {3,2,6,2}*144
   26-fold quotients : {3,2,3,2}*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 := (17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)
(27,40)(28,41)(29,42)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)
(64,77)(65,78)(66,79)(67,80)(68,81);;
s3 := ( 4,17)( 5,29)( 6,28)( 7,27)( 8,26)( 9,25)(10,24)(11,23)(12,22)(13,21)
(14,20)(15,19)(16,18)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,56)(44,68)
(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)
(70,81)(71,80)(72,79)(73,78)(74,77)(75,76);;
s4 := ( 4,44)( 5,43)( 6,55)( 7,54)( 8,53)( 9,52)(10,51)(11,50)(12,49)(13,48)
(14,47)(15,46)(16,45)(17,57)(18,56)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)
(25,62)(26,61)(27,60)(28,59)(29,58)(30,70)(31,69)(32,81)(33,80)(34,79)(35,78)
(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope