Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,18}

Atlas Canonical Name {2,6,18}*432a

Overview

Group
SmallGroup(432,544)
Rank
4
Schläfli Type
{2,6,18}
Vertices, edges, …
2, 6, 54, 18
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

Representations

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