Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,14}

Atlas Canonical Name {6,14}*168

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Overview

Group
SmallGroup(168,50)
Rank
3
Schläfli Type
{6,14}
Vertices, edges, …
6, 42, 14
Order of s0s1s2
42
Order of s0s1s2s1
2
Also known as
{6,14|2}. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

6-fold

7-fold

14-fold

21-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

10-fold

11-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

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