Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,14}

Atlas Canonical Name {3,6,14}*1512

Overview

Group
SmallGroup(1512,486)
Rank
4
Schläfli Type
{3,6,14}
Vertices, edges, …
9, 27, 126, 14
Order of s0s1s2s3
42
Order of s0s1s2s3s2s1
2
Also known as
{{3,6}6,{6,14|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

7-fold

9-fold

18-fold

21-fold

63-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=<s0*(s1*s0*s2)^2*s1> of order 3

14 facets

  • 14 of 3-fold non-regular quotient of {3,6}*108

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);;
s1 := ( 1,23)( 2,24)( 3,22)( 4,26)( 5,27)( 6,25)( 7,29)( 8,30)( 9,28)(10,32)(11,33)(12,31)(13,35)(14,36)(15,34)(16,38)(17,39)(18,37)(19,41)(20,42)(21,40);;
s2 := ( 2, 3)( 4,19)( 5,21)( 6,20)( 7,16)( 8,18)( 9,17)(10,13)(11,15)(12,14)(23,24)(25,40)(26,42)(27,41)(28,37)(29,39)(30,38)(31,34)(32,36)(33,35)(44,45)(46,61)(47,63)(48,62)(49,58)(50,60)(51,59)(52,55)(53,57)(54,56);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 7,19)( 8,20)( 9,21)(10,16)(11,17)(12,18)(22,25)(23,26)(24,27)(28,40)(29,41)(30,42)(31,37)(32,38)(33,39)(43,46)(44,47)(45,48)(49,61)(50,62)(51,63)(52,58)(53,59)(54,60);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(63)!(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63);
s1 := Sym(63)!( 1,23)( 2,24)( 3,22)( 4,26)( 5,27)( 6,25)( 7,29)( 8,30)( 9,28)(10,32)(11,33)(12,31)(13,35)(14,36)(15,34)(16,38)(17,39)(18,37)(19,41)(20,42)(21,40);
s2 := Sym(63)!( 2, 3)( 4,19)( 5,21)( 6,20)( 7,16)( 8,18)( 9,17)(10,13)(11,15)(12,14)(23,24)(25,40)(26,42)(27,41)(28,37)(29,39)(30,38)(31,34)(32,36)(33,35)(44,45)(46,61)(47,63)(48,62)(49,58)(50,60)(51,59)(52,55)(53,57)(54,56);
s3 := Sym(63)!( 1, 4)( 2, 5)( 3, 6)( 7,19)( 8,20)( 9,21)(10,16)(11,17)(12,18)(22,25)(23,26)(24,27)(28,40)(29,41)(30,42)(31,37)(32,38)(33,39)(43,46)(44,47)(45,48)(49,61)(50,62)(51,63)(52,58)(53,59)(54,60);
poly := sub<Sym(63)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.