Polytope of Type {12,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3}*648
if this polytope has a name.
Group : SmallGroup(648,703)
Rank : 3
Schlafli Type : {12,3}
Number of vertices, edges, etc : 108, 162, 27
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,3,2} of size 1296
Vertex Figure Of :
   {2,12,3} of size 1296
Quotients (Maximal Quotients in Boldface) :
   27-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,3}*1296b, {12,6}*1296q, {12,6}*1296r
   3-fold covers : {12,9}*1944b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      15 facets:
         9 of {4}*8
         6 of {12}*24
      36 vertex figures:
         36 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 3.
      9 facets:
         9 of {12}*24
      36 vertex figures:
         36 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> of order 3.
      9 facets:
         9 of {12}*24
      36 vertex figures:
         36 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 9.
      3 facets:
         3 of {12}*24
      12 vertex figures:
         12 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 9.
      5 facets:
         3 of {4}*8
         2 of {12}*24
      12 vertex figures:
         12 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 9.
      7 facets:
         6 of {4}*8
         1 of {12}*24
      12 vertex figures:
         12 of {3}*6

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {12,3}

Twisty Puzzle