Polytope of Type {9,4}

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

s0 := ( 2, 3)( 4,19)( 5,21)( 6,20)( 7,10)( 8,12)( 9,11)(13,25)(14,27)(15,26)
(17,18)(23,24);;
s1 := ( 1, 4)( 2,22)( 3,13)( 5,19)( 6,10)( 8,25)( 9,16)(11,24)(12,15)(14,21)
(17,27)(20,23);;
s2 := ( 1,22)( 2,23)( 3,24)( 4,19)( 5,20)( 6,21)( 7,25)( 8,26)( 9,27)(10,13)
(11,14)(12,15);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope