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}*576
Also Known As : {12,4}₄. if this polytope has another name.
Group : SmallGroup(576,5296)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 72, 144, 24
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {12,4,2} of size 1152
Vertex Figure Of :
   {2,12,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*288
   4-fold quotients : {6,4}*144
   8-fold quotients : {6,4}*72
   9-fold quotients : {4,4}*64
   18-fold quotients : {4,4}*32
   36-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,4}*1152a, {12,8}*1152a, {12,4}*1152a, {12,8}*1152b, {24,4}*1152b
   3-fold covers : {12,4}*1728b, {12,12}*1728f, {12,12}*1728g, {12,4}*1728c, {12,12}*1728q, {12,12}*1728t
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.
      18 facets:
         6 of {12}*24
         12 of {6}*12
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      16 facets:
         12 of {4}*8
         4 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 3.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2, s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
      10 facets:
         3 of {12}*24
         5 of {6}*12
         2 of {3}*6
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 6.
      6 facets:
         2 of {12}*24
         4 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 6.
      14 facets:
         12 of {2}*4
         2 of {12}*24
      12 vertex figures:
         12 of {4}*8
   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.
      10 facets:
         6 of {4}*8
         4 of {6}*12
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 12.
      6 facets:
         3 of {4}*8
         1 of {6}*12
         2 of {3}*6
      6 vertex figures:
         6 of {4}*8

Permutation Representation (GAP) :

s0 := ( 1,46)( 2,48)( 3,47)( 4,52)( 5,54)( 6,53)( 7,49)( 8,51)( 9,50)(10,37)(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,64)(20,66)(21,65)(22,70)(23,72)(24,71)(25,67)(26,69)(27,68)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59);;
s1 := ( 1, 5)( 3, 8)( 6, 7)(10,14)(12,17)(15,16)(19,23)(21,26)(24,25)(28,32)(30,35)(33,34)(37,68)(38,65)(39,71)(40,67)(41,64)(42,70)(43,69)(44,66)(45,72)(46,59)(47,56)(48,62)(49,58)(50,55)(51,61)(52,60)(53,57)(54,63);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*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*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(72)!( 1,46)( 2,48)( 3,47)( 4,52)( 5,54)( 6,53)( 7,49)( 8,51)( 9,50)(10,37)(11,39)(12,38)(13,43)(14,45)(15,44)(16,40)(17,42)(18,41)(19,64)(20,66)(21,65)(22,70)(23,72)(24,71)(25,67)(26,69)(27,68)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59);
s1 := Sym(72)!( 1, 5)( 3, 8)( 6, 7)(10,14)(12,17)(15,16)(19,23)(21,26)(24,25)(28,32)(30,35)(33,34)(37,68)(38,65)(39,71)(40,67)(41,64)(42,70)(43,69)(44,66)(45,72)(46,59)(47,56)(48,62)(49,58)(50,55)(51,61)(52,60)(53,57)(54,63);
s2 := Sym(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
poly := sub<Sym(72)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*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*s0*s1 >;

References : None.
to this polytope

Polytope of Type {12,4}

Twisty Puzzle