Part of the Atlas of Small Regular Polytopes

Polytope of Type {21,6}

Atlas Canonical Name {21,6}*1008b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1008,904)
Rank
3
Schläfli Type
{21,6}
Vertices, edges, …
84, 252, 24
Order of s0s1s2
84
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

7-fold

12-fold

21-fold

28-fold

36-fold

42-fold

84-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 2

12 facets

42 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3

8 facets

42 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2> of order 4

6 facets

21 vertex figures

Representations

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

References

None.

to this polytope.

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