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}*432
Also Known As : {12,3}₆. if this polytope has another name.
Group : SmallGroup(432,523)
Rank : 3
Schlafli Type : {12,3}
Number of vertices, edges, etc : 72, 108, 18
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,3,2} of size 864
   {12,3,4} of size 1728
Vertex Figure Of :
   {2,12,3} of size 864
   {4,12,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,3}*144
   4-fold quotients : {6,3}*108
   9-fold quotients : {4,3}*48
   12-fold quotients : {6,3}*36
   18-fold quotients : {4,3}*24
   36-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,3}*864, {12,6}*864a
   3-fold covers : {36,3}*1296, {12,3}*1296a, {12,9}*1296a, {12,9}*1296b, {12,9}*1296c, {12,9}*1296d
   4-fold covers : {24,3}*1728, {12,12}*1728k, {12,12}*1728n, {24,6}*1728b, {24,6}*1728d, {12,6}*1728f, {12,3}*1728
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
      36 vertex figures:
         36 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 2.
      9 facets:
         9 of {12}*24
      36 vertex figures:
         36 of {3}*6
   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
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      6 facets:
         3 of {4}*8
         1 of {12}*24
         2 of {6}*12
      12 vertex figures:
         12 of {3}*6
   P/N, where N=<s0*s2*s1*s0*s1*s2> of order 6.
      8 facets:
         2 of {12}*24
         6 of {2}*4
      12 vertex figures:
         12 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 6.
      5 facets:
         3 of {4}*8
         2 of {12}*24
      12 vertex figures:
         12 of {3}*6

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)( 6, 8)(10,12)(13,29)(14,32)(15,31)(16,30)(17,33)(18,36)(19,35)(20,34)(21,25)(22,28)(23,27)(24,26);;
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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*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 ];;
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)( 6, 8)(10,12)(13,29)(14,32)(15,31)(16,30)(17,33)(18,36)(19,35)(20,34)(21,25)(22,28)(23,27)(24,26);
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, s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*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 >;

References : None.
to this polytope

Polytope of Type {12,3}

Twisty Puzzle