Polytope of Type {4,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*432b
if this polytope has a name.
Group : SmallGroup(432,741)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 36, 108, 54
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,6,2} of size 864
   {4,6,3} of size 1296
   {4,6,4} of size 1728
   {4,6,4} of size 1728
Vertex Figure Of :
   {2,4,6} of size 864
   {3,4,6} of size 1296
   {4,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,6}*144
   6-fold quotients : {4,6}*72
   9-fold quotients : {4,6}*48a
   18-fold quotients : {2,6}*24
   27-fold quotients : {4,2}*16
   36-fold quotients : {2,3}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12}*864c, {8,6}*864b
   3-fold covers : {4,18}*1296b, {4,6}*1296a, {12,6}*1296j, {12,6}*1296k, {12,6}*1296s, {12,6}*1296t
   4-fold covers : {4,24}*1728e, {4,24}*1728g, {16,6}*1728b, {8,12}*1728g, {8,12}*1728h, {4,12}*1728c, {4,6}*1728
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      27 facets:
         27 of {4}*8
      21 vertex figures:
         15 of {6}*12
         6 of {3}*6
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {2}*4
         24 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         9 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      15 facets:
         3 of {2}*4
         12 of {4}*8
      12 vertex figures:
         6 of {6}*12
         6 of {3}*6
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 6.
      9 facets:
         9 of {4}*8
      9 vertex figures:
         6 of {3}*6
         3 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 6.
      12 facets:
         6 of {2}*4
         6 of {4}*8
      6 vertex figures:
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 6.
      12 facets:
         6 of {2}*4
         6 of {4}*8
      6 vertex figures:
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 9.
      6 facets:
         6 of {4}*8
      6 vertex figures:
         3 of {6}*12
         3 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s1*s2> of order 9.
      6 facets:
         6 of {4}*8
      8 vertex figures:
         2 of {6}*12
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 12.
      6 facets:
         3 of {2}*4
         3 of {4}*8
      5 vertex figures:
         4 of {3}*6
         1 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,6}

Twisty Puzzle