Polytope of Type {4,12}

Play with this polytope as a twisty puzzle

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

Permutation Representation (GAP) : 
s0 := ( 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);;
s1 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,19)(14,21)(15,20)(16,22)(17,24)(18,23)(26,27)(29,30)(32,33)(35,36)(37,43)(38,45)(39,44)(40,46)(41,48)(42,47)(49,73)(50,75)(51,74)(52,76)(53,78)(54,77)(55,79)(56,81)(57,80)(58,82)(59,84)(60,83)(61,91)(62,93)(63,92)(64,94)(65,96)(66,95)(67,85)(68,87)(69,86)(70,88)(71,90)(72,89);;
s2 := ( 1,51)( 2,50)( 3,49)( 4,54)( 5,53)( 6,52)( 7,60)( 8,59)( 9,58)(10,57)(11,56)(12,55)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,69)(20,68)(21,67)(22,72)(23,71)(24,70)(25,75)(26,74)(27,73)(28,78)(29,77)(30,76)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,93)(44,92)(45,91)(46,96)(47,95)(48,94);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*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, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := 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);
s1 := Sym(96)!( 2, 3)( 5, 6)( 8, 9)(11,12)(13,19)(14,21)(15,20)(16,22)(17,24)(18,23)(26,27)(29,30)(32,33)(35,36)(37,43)(38,45)(39,44)(40,46)(41,48)(42,47)(49,73)(50,75)(51,74)(52,76)(53,78)(54,77)(55,79)(56,81)(57,80)(58,82)(59,84)(60,83)(61,91)(62,93)(63,92)(64,94)(65,96)(66,95)(67,85)(68,87)(69,86)(70,88)(71,90)(72,89);
s2 := Sym(96)!( 1,51)( 2,50)( 3,49)( 4,54)( 5,53)( 6,52)( 7,60)( 8,59)( 9,58)(10,57)(11,56)(12,55)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,69)(20,68)(21,67)(22,72)(23,71)(24,70)(25,75)(26,74)(27,73)(28,78)(29,77)(30,76)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,93)(44,92)(45,91)(46,96)(47,95)(48,94);
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*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, 
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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