Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*432d
Also Known As : {12,6}₃. if this polytope has another name.
Group : SmallGroup(432,523)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 36, 108, 18
Order of s₀s₁s₂ : 3
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 864
   {12,6,4} of size 1728
Vertex Figure Of :
   {2,12,6} of size 864
   {4,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,6}*144d
   4-fold quotients : {6,6}*108
   9-fold quotients : {4,6}*48b
   18-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*864b
   3-fold covers : {36,6}*1296i, {36,6}*1296j, {36,6}*1296k, {12,18}*1296i, {12,18}*1296j, {12,6}*1296e, {12,18}*1296k, {12,6}*1296f
   4-fold covers : {24,6}*1728a, {12,12}*1728l, {12,6}*1728b, {24,6}*1728c, {24,6}*1728e, {12,12}*1728p, {12,12}*1728u
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
      12 facets:
         6 of {12}*24
         6 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      10 facets:
         6 of {4}*8
         4 of {12}*24
      12 vertex figures:
         12 of {6}*12

Permutation Representation (GAP) :

s0 := ( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,27)(14,28)(15,25)(16,26)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,16)( 5,17)( 6,19)( 7,18)( 8,20)( 9,21)(10,23)(11,22)(12,24)(26,27)(30,31)(34,35);;
s2 := ( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,21)(14,24)(15,23)(16,22)(18,20)(25,29)(26,32)(27,31)(28,30)(34,36);;
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*s2*s0*s1*s2*s0*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(36)!( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,27)(14,28)(15,25)(16,26)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30);
s1 := Sym(36)!( 1,13)( 2,15)( 3,14)( 4,16)( 5,17)( 6,19)( 7,18)( 8,20)( 9,21)(10,23)(11,22)(12,24)(26,27)(30,31)(34,35);
s2 := Sym(36)!( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,21)(14,24)(15,23)(16,22)(18,20)(25,29)(26,32)(27,31)(28,30)(34,36);
poly := sub<Sym(36)|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*s2*s0*s1*s2*s0*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >;

References : None.
to this polytope

Polytope of Type {12,6}

Twisty Puzzle