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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,3,6}*1296
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 5
Schlafli Type : {2,2,3,6}
Number of vertices, edges, etc : 2, 2, 27, 81, 54
Order of s₀s₁s₂s₃s₄ : 18
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 : {2,2,3,6}*432
   9-fold quotients : {2,2,3,6}*144
   27-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8,10)(11,12)(14,23)(15,25)(16,24)(17,28)(18,27)(19,26)(20,30)
(21,29)(22,31)(32,61)(33,60)(34,59)(35,63)(36,62)(37,64)(38,65)(39,67)(40,66)
(41,79)(42,78)(43,77)(44,81)(45,80)(46,82)(47,83)(48,85)(49,84)(50,70)(51,69)
(52,68)(53,72)(54,71)(55,73)(56,74)(57,76)(58,75);;
s3 := ( 5,47)( 6,49)( 7,48)( 8,41)( 9,43)(10,42)(11,44)(12,46)(13,45)(14,35)
(15,37)(16,36)(17,38)(18,40)(19,39)(20,32)(21,34)(22,33)(23,51)(24,50)(25,52)
(26,54)(27,53)(28,55)(29,57)(30,56)(31,58)(59,76)(60,75)(61,74)(62,70)(63,69)
(64,68)(65,73)(66,72)(67,71)(78,79)(81,82)(84,85);;
s4 := ( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)(20,28)
(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)(46,56)
(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)(72,85)
(73,83)(74,82)(75,80)(76,81);;
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*s1*s0*s1, 
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, s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 6, 7)( 8,10)(11,12)(14,23)(15,25)(16,24)(17,28)(18,27)(19,26)
(20,30)(21,29)(22,31)(32,61)(33,60)(34,59)(35,63)(36,62)(37,64)(38,65)(39,67)
(40,66)(41,79)(42,78)(43,77)(44,81)(45,80)(46,82)(47,83)(48,85)(49,84)(50,70)
(51,69)(52,68)(53,72)(54,71)(55,73)(56,74)(57,76)(58,75);
s3 := Sym(85)!( 5,47)( 6,49)( 7,48)( 8,41)( 9,43)(10,42)(11,44)(12,46)(13,45)
(14,35)(15,37)(16,36)(17,38)(18,40)(19,39)(20,32)(21,34)(22,33)(23,51)(24,50)
(25,52)(26,54)(27,53)(28,55)(29,57)(30,56)(31,58)(59,76)(60,75)(61,74)(62,70)
(63,69)(64,68)(65,73)(66,72)(67,71)(78,79)(81,82)(84,85);
s4 := Sym(85)!( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)
(20,28)(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)
(46,56)(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)
(72,85)(73,83)(74,82)(75,80)(76,81);
poly := sub<Sym(85)|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*s1*s0*s1, 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, 
s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3 >;

to this polytope