Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*432b
if this polytope has a name.
Group : SmallGroup(432,741)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 54, 108, 18
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {12,4,2} of size 864
   {12,4,4} of size 1728
Vertex Figure Of :
   {2,12,4} of size 864
   {4,12,4} of size 1728
   {4,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,4}*144
   6-fold quotients : {4,4}*72
   18-fold quotients : {6,2}*24
   36-fold quotients : {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,4}*864c
   3-fold covers : {36,4}*1296, {12,4}*1296, {12,12}*1296c, {12,12}*1296d, {12,12}*1296f, {12,12}*1296h
   4-fold covers : {24,4}*1728e, {12,8}*1728e, {24,4}*1728h, {12,8}*1728f, {12,4}*1728d, {12,4}*1728e
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> of order 2.
      9 facets:
         9 of {12}*24
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      9 facets:
         9 of {12}*24
      27 vertex figures:
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      10 facets:
         2 of {6}*12
         8 of {12}*24
      30 vertex figures:
         24 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {12}*24
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {12}*24
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      5 facets:
         1 of {6}*12
         4 of {12}*24
      15 vertex figures:
         12 of {4}*8
         3 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      4 facets:
         2 of {6}*12
         2 of {12}*24
      12 vertex figures:
         6 of {4}*8
         6 of {2}*4

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {12,4}

Twisty Puzzle