Polytope of Type {6,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*648e
if this polytope has a name.
Group : SmallGroup(648,555)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 54, 162, 54
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 1296
   {6,6,3} of size 1944
Vertex Figure Of :
   {2,6,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*216c, {6,6}*216d
   6-fold quotients : {3,6}*108
   9-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {3,6}*36, {6,3}*36
   27-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,3}*12, {3,2}*12
   81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*1296h, {6,12}*1296i
   3-fold covers : {18,6}*1944m, {6,18}*1944o, {6,6}*1944d, {6,6}*1944e, {18,6}*1944p, {18,6}*1944q, {18,6}*1944r, {6,6}*1944j, {6,18}*1944u
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 2.
      27 facets:
         27 of {6}*12
      30 vertex figures:
         24 of {6}*12
         6 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      36 facets:
         18 of {3}*6
         18 of {6}*12
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 3.
      24 facets:
         15 of {6}*12
         9 of {2}*4
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      18 facets:
         18 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 3.
      18 facets:
         18 of {6}*12
      24 vertex figures:
         15 of {6}*12
         9 of {2}*4
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 6.
      9 facets:
         9 of {6}*12
      12 vertex figures:
         6 of {3}*6
         6 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
      6 facets:
         6 of {6}*12
      10 vertex figures:
         4 of {6}*12
         6 of {2}*4

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,6}

Twisty Puzzle