Polytope of Type {6,21}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,21}*1008b
if this polytope has a name.
Group : SmallGroup(1008,904)
Rank : 3
Schlafli Type : {6,21}
Number of vertices, edges, etc : 24, 252, 84
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 6
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 : {6,21}*336
   4-fold quotients : {6,21}*252
   7-fold quotients : {6,3}*144
   12-fold quotients : {2,21}*84
   21-fold quotients : {6,3}*48
   28-fold quotients : {6,3}*36
   36-fold quotients : {2,7}*28
   42-fold quotients : {3,3}*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*s2*s1*s0*s1*s0*s2*s1> of order 2.
      42 facets:
         42 of {6}*12
      12 vertex figures:
         12 of {21}*42
   P/N, where N=<s0*s1*s0*s1> of order 3.
      42 facets:
         21 of {2}*4
         21 of {6}*12
      8 vertex figures:
         8 of {21}*42
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      21 facets:
         21 of {6}*12
      6 vertex figures:
         6 of {21}*42

Permutation Representation (GAP) :

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

Twisty Puzzle