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}*768a
if this polytope has a name.
Group : SmallGroup(768,90280)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 96, 192, 32
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*384a
   3-fold quotients : {4,4}*256
   4-fold quotients : {12,4}*192a
   6-fold quotients : {4,4}*128
   8-fold quotients : {12,4}*96a
   12-fold quotients : {4,4}*64
   16-fold quotients : {12,2}*48, {6,4}*48a
   24-fold quotients : {4,4}*32
   32-fold quotients : {6,2}*24
   48-fold quotients : {2,4}*16, {4,2}*16
   64-fold quotients : {3,2}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 2.
      16 facets:
         16 of {12}*24
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      18 facets:
         4 of {6}*12
         14 of {12}*24
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {12}*24
      54 vertex figures:
         12 of {2}*4
         42 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 2.
      16 facets:
         16 of {12}*24
      48 vertex figures:
         48 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 4.
      8 facets:
         8 of {12}*24
      27 vertex figures:
         6 of {2}*4
         21 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
      8 facets:
         8 of {12}*24
      30 vertex figures:
         12 of {2}*4
         18 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      10 facets:
         4 of {6}*12
         6 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 4.
      9 facets:
         7 of {12}*24
         2 of {6}*12
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 8.
      5 facets:
         3 of {12}*24
         2 of {6}*12
      15 vertex figures:
         6 of {2}*4
         9 of {4}*8

Permutation Representation (GAP) :

s0 := ( 1,49)( 2,51)( 3,50)( 4,52)( 5,54)( 6,53)( 7,58)( 8,60)( 9,59)(10,55)(11,57)(12,56)(13,64)(14,66)(15,65)(16,61)(17,63)(18,62)(19,67)(20,69)(21,68)(22,70)(23,72)(24,71)(25,73)(26,75)(27,74)(28,76)(29,78)(30,77)(31,82)(32,84)(33,83)(34,79)(35,81)(36,80)(37,88)(38,90)(39,89)(40,85)(41,87)(42,86)(43,91)(44,93)(45,92)(46,94)(47,96)(48,95);;
s1 := ( 1, 3)( 4, 6)( 7, 9)(10,12)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,27)(28,30)(31,33)(34,36)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)(49,75)(50,74)(51,73)(52,78)(53,77)(54,76)(55,81)(56,80)(57,79)(58,84)(59,83)(60,82)(61,93)(62,92)(63,91)(64,96)(65,95)(66,94)(67,87)(68,86)(69,85)(70,90)(71,89)(72,88);;
s2 := ( 7,10)( 8,11)( 9,12)(19,22)(20,23)(21,24)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72)(73,94)(74,95)(75,96)(76,91)(77,92)(78,93)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90);;
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, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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, 
s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(96)!( 1,49)( 2,51)( 3,50)( 4,52)( 5,54)( 6,53)( 7,58)( 8,60)( 9,59)(10,55)(11,57)(12,56)(13,64)(14,66)(15,65)(16,61)(17,63)(18,62)(19,67)(20,69)(21,68)(22,70)(23,72)(24,71)(25,73)(26,75)(27,74)(28,76)(29,78)(30,77)(31,82)(32,84)(33,83)(34,79)(35,81)(36,80)(37,88)(38,90)(39,89)(40,85)(41,87)(42,86)(43,91)(44,93)(45,92)(46,94)(47,96)(48,95);
s1 := Sym(96)!( 1, 3)( 4, 6)( 7, 9)(10,12)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,27)(28,30)(31,33)(34,36)(37,45)(38,44)(39,43)(40,48)(41,47)(42,46)(49,75)(50,74)(51,73)(52,78)(53,77)(54,76)(55,81)(56,80)(57,79)(58,84)(59,83)(60,82)(61,93)(62,92)(63,91)(64,96)(65,95)(66,94)(67,87)(68,86)(69,85)(70,90)(71,89)(72,88);
s2 := Sym(96)!( 7,10)( 8,11)( 9,12)(19,22)(20,23)(21,24)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(55,58)(56,59)(57,60)(67,70)(68,71)(69,72)(73,94)(74,95)(75,96)(76,91)(77,92)(78,93)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90);
poly := sub<Sym(96)|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, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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, 
s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {12,4}

Twisty Puzzle