Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,18,2,2}

Atlas Canonical Name {2,6,18,2,2}*1728b

Overview

Group
SmallGroup(1728,46164)
Rank
6
Schläfli Type
{2,6,18,2,2}
Vertices, edges, …
2, 6, 54, 18, 2, 2
Order of s0s1s2s3s4s5
18
Order of s0s1s2s3s4s5s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

27-fold

Covers minimal covers in bold

None in this atlas.

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)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)(12,30)(13,32)(14,31)(15,36)(16,38)(17,37)(18,33)(19,35)(20,34)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53);;
s4 := (57,58);;
s5 := (59,60);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s4*s5*s4*s5, s1*s2*s3*s1*s2*s1*s2*s3*s1*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(60)!(1,2);
s1 := Sym(60)!( 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(60)!( 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(60)!( 3,39)( 4,41)( 5,40)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)(12,30)(13,32)(14,31)(15,36)(16,38)(17,37)(18,33)(19,35)(20,34)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53);
s4 := Sym(60)!(57,58);
s5 := Sym(60)!(59,60);
poly := sub<Sym(60)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s4*s5*s4*s5, s1*s2*s3*s1*s2*s1*s2*s3*s1*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 >;