Polytope of Type {6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*216
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 27, 54, 18
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {6,4,2} of size 432
   {6,4,4} of size 864
   {6,4,6} of size 1296
   {6,4,8} of size 1728
Vertex Figure Of :
   {2,6,4} of size 432
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,4}*432a
   3-fold covers : {6,12}*648
   4-fold covers : {6,8}*864a, {12,4}*864b
   5-fold covers : {6,20}*1080
   6-fold covers : {6,4}*1296a, {6,12}*1296m, {6,12}*1296o
   7-fold covers : {6,28}*1512
   8-fold covers : {6,16}*1728a, {12,4}*1728b, {12,8}*1728a, {24,4}*1728a, {24,4}*1728c, {12,8}*1728d
   9-fold covers : {18,4}*1944a, {6,4}*1944, {18,4}*1944b, {18,4}*1944c, {6,36}*1944
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope