Polytope of Type {12,21}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,21}*1008
if this polytope has a name.
Group : SmallGroup(1008,903)
Rank : 3
Schlafli Type : {12,21}
Number of vertices, edges, etc : 24, 252, 42
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 12
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) :
   3-fold quotients : {4,21}*336
   4-fold quotients : {6,21}*252
   6-fold quotients : {4,21}*168
   7-fold quotients : {12,3}*144
   12-fold quotients : {2,21}*84
   21-fold quotients : {4,3}*48
   28-fold quotients : {6,3}*36
   36-fold quotients : {2,7}*28
   42-fold quotients : {4,3}*24
   84-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      28 facets:
         14 of {6}*12
         14 of {12}*24
      12 vertex figures:
         12 of {21}*42
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      21 facets:
         21 of {12}*24
      12 vertex figures:
         12 of {21}*42

Permutation Representation (GAP) :

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

Twisty Puzzle