Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*324b
if this polytope has a name.
Group : SmallGroup(324,39)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 27, 81, 27
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 648
   {6,6,4} of size 1296
   {6,6,6} of size 1944
Vertex Figure Of :
   {2,6,6} of size 648
   {4,6,6} of size 1296
   {3,6,6} of size 1296
   {4,6,6} of size 1296
   {6,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6}*648a
   3-fold covers : {6,6}*972, {18,6}*972a, {18,6}*972b, {18,6}*972c, {6,18}*972d
   4-fold covers : {12,6}*1296a, {6,12}*1296b, {12,6}*1296e, {6,12}*1296f
   5-fold covers : {6,30}*1620a, {30,6}*1620b
   6-fold covers : {6,6}*1944a, {18,6}*1944c, {18,6}*1944e, {18,6}*1944g, {6,18}*1944j, {6,6}*1944g
Permutation Representation (GAP) :

s0 := (1,7)(2,8)(3,9);;
s1 := (2,3)(4,7)(5,9)(6,8);;
s2 := (2,3)(4,5)(8,9);;
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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(9)!(1,7)(2,8)(3,9);
s1 := Sym(9)!(2,3)(4,7)(5,9)(6,8);
s2 := Sym(9)!(2,3)(4,5)(8,9);
poly := sub<Sym(9)|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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 >;

References : None.
to this polytope