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Polytope of Type {4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9}*648
if this polytope has a name.
Group : SmallGroup(648,703)
Rank : 3
Schlafli Type : {4,9}
Number of vertices, edges, etc : 36, 162, 81
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,9,2} of size 1296
Vertex Figure Of :
   {2,4,9} of size 1296
Quotients (Maximal Quotients in Boldface) :
   27-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,9}*1296b, {4,18}*1296c, {4,18}*1296d
   3-fold covers : {4,9}*1944
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)
(17,27)(18,26);;
s1 := ( 2, 3)( 4,19)( 5,21)( 6,20)( 7,10)( 8,12)( 9,11)(13,25)(14,27)(15,26)
(17,18)(23,24);;
s2 := ( 1, 4)( 2,22)( 3,13)( 5,19)( 6,10)( 8,25)( 9,16)(11,24)(12,15)(14,21)
(17,27)(20,23);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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(27)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)
(16,25)(17,27)(18,26);
s1 := Sym(27)!( 2, 3)( 4,19)( 5,21)( 6,20)( 7,10)( 8,12)( 9,11)(13,25)(14,27)
(15,26)(17,18)(23,24);
s2 := Sym(27)!( 1, 4)( 2,22)( 3,13)( 5,19)( 6,10)( 8,25)( 9,16)(11,24)(12,15)
(14,21)(17,27)(20,23);
poly := sub<Sym(27)|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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope