Part of the Atlas of Small Regular Polytopes

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

Atlas Canonical Name {2,2,6,6,6}*1728g

Overview

Group
SmallGroup(1728,47915)
Rank
6
Schläfli Type
{2,2,6,6,6}
Vertices, edges, …
2, 2, 6, 18, 18, 6
Order of s0s1s2s3s4s5
6
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

12-fold

18-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Representations

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