Polytope of Type {2,2,6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,27}*1296
if this polytope has a name.
Group : SmallGroup(1296,1859)
Rank : 5
Schlafli Type : {2,2,6,27}
Number of vertices, edges, etc : 2, 2, 6, 81, 27
Order of s₀s₁s₂s₃s₄ : 54
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 : {2,2,2,27}*432, {2,2,6,9}*432
   9-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
   27-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope