Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*216
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 9, 54, 27
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 3
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {4,12,2} of size 432
Vertex Figure Of :
   {2,4,12} of size 432
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12}*432a
   3-fold covers : {12,12}*648
   4-fold covers : {4,12}*864a
   6-fold covers : {4,12}*1296, {12,12}*1296c, {12,12}*1296e
   8-fold covers : {4,12}*1728b, {4,24}*1728b, {8,12}*1728b, {4,24}*1728d, {8,12}*1728c
   9-fold covers : {4,12}*1944a, {4,12}*1944b, {4,12}*1944c, {4,12}*1944d
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(11,12)(13,14)(16,18);;
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*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);
s1 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);
s2 := Sym(18)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(11,12)(13,14)(16,18);
poly := sub<Sym(18)|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*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope