Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*432i
if this polytope has a name.
Group : SmallGroup(432,756)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 18, 108, 36
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,12,2} of size 864
   {6,12,4} of size 1728
   {6,12,4} of size 1728
Vertex Figure Of :
   {2,6,12} of size 864
   {3,6,12} of size 1296
   {4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*216c
   3-fold quotients : {6,4}*144
   6-fold quotients : {6,4}*72
   9-fold quotients : {2,12}*48
   18-fold quotients : {2,6}*24
   27-fold quotients : {2,4}*16
   36-fold quotients : {2,3}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,24}*864h, {12,12}*864l
   3-fold covers : {6,36}*1296m, {6,12}*1296o, {6,36}*1296n, {6,36}*1296o, {6,12}*1296t, {6,12}*1296u
   4-fold covers : {6,48}*1728h, {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x, {12,12}*1728ab
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 3.
      24 facets:
         18 of {2}*4
         6 of {6}*12
      6 vertex figures:
         6 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      12 facets:
         12 of {6}*12
      6 vertex figures:
         6 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      12 facets:
         12 of {6}*12
      6 vertex figures:
         6 of {12}*24

Permutation Representation (GAP) :

s0 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)(46,49)(47,51)(48,50)(53,54);;
s2 := ( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);;
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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(54)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);
s1 := Sym(54)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)(46,49)(47,51)(48,50)(53,54);
s2 := Sym(54)!( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);
poly := sub<Sym(54)|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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope

Polytope of Type {6,12}

Twisty Puzzle