Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*216a
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 18, 54, 18
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 432
   {6,6,4} of size 864
   {6,6,4} of size 864
   {6,6,4} of size 864
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {6,6,8} of size 1728
   {6,6,4} of size 1728
   {6,6,6} of size 1944
Vertex Figure Of :
   {2,6,6} of size 432
   {3,6,6} of size 648
   {4,6,6} of size 864
   {6,6,6} of size 1296
   {6,6,6} of size 1296
   {8,6,6} of size 1728
   {3,6,6} of size 1944
   {9,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,3}*108
   3-fold quotients : {6,6}*72b
   6-fold quotients : {6,3}*36
   9-fold quotients : {2,6}*24
   18-fold quotients : {2,3}*12
   27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12}*432a, {12,6}*432c
   3-fold covers : {6,18}*648a, {6,18}*648c, {6,18}*648d, {6,18}*648e, {6,6}*648c, {18,6}*648h, {6,6}*648f
   4-fold covers : {6,24}*864a, {12,12}*864b, {24,6}*864c, {6,6}*864b, {12,6}*864a
   5-fold covers : {30,6}*1080a, {6,30}*1080b
   6-fold covers : {6,36}*1296a, {6,36}*1296c, {6,36}*1296d, {6,36}*1296e, {18,12}*1296d, {6,12}*1296c, {12,18}*1296e, {12,18}*1296f, {12,18}*1296g, {12,18}*1296h, {12,6}*1296d, {36,6}*1296h, {6,12}*1296h, {12,6}*1296i
   7-fold covers : {42,6}*1512a, {6,42}*1512b
   8-fold covers : {6,48}*1728a, {24,12}*1728a, {12,12}*1728b, {24,12}*1728b, {12,24}*1728c, {12,24}*1728e, {48,6}*1728c, {12,12}*1728k, {6,12}*1728a, {12,12}*1728n, {12,6}*1728c, {24,6}*1728b, {6,6}*1728a, {24,6}*1728d, {6,12}*1728d, {12,6}*1728e, {12,6}*1728f
   9-fold covers : {18,18}*1944b, {6,18}*1944a, {18,6}*1944b, {6,18}*1944d, {18,18}*1944e, {6,18}*1944f, {18,18}*1944g, {18,18}*1944j, {18,18}*1944n, {6,54}*1944a, {6,18}*1944h, {18,18}*1944p, {18,18}*1944r, {18,18}*1944w, {18,18}*1944aa, {6,18}*1944i, {18,18}*1944ac, {6,54}*1944c, {6,54}*1944e, {6,6}*1944c, {18,6}*1944k, {6,18}*1944m, {18,6}*1944o, {6,6}*1944d, {6,6}*1944f, {6,18}*1944p, {6,18}*1944q, {6,18}*1944r, {6,6}*1944i, {18,6}*1944u
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope