Part of the Atlas of Small Regular Polytopes

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

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

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

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)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34)(36,37)(39,40)(42,43)(45,46)(48,49)(51,52)(54,55)(57,58);;
s3 := ( 5, 6)( 8,12)( 9,11)(10,13)(14,15)(17,21)(18,20)(19,22)(23,24)(26,30)(27,29)(28,31)(32,33)(35,39)(36,38)(37,40)(41,42)(44,48)(45,47)(46,49)(50,51)(53,57)(54,56)(55,58);;
s4 := ( 5, 8)( 6, 9)( 7,10)(14,26)(15,27)(16,28)(17,23)(18,24)(19,25)(20,29)(21,30)(22,31)(32,35)(33,36)(34,37)(41,53)(42,54)(43,55)(44,50)(45,51)(46,52)(47,56)(48,57)(49,58);;
s5 := ( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)(13,49)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58);;
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, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s5*s4*s3*s4*s5*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34)(36,37)(39,40)(42,43)(45,46)(48,49)(51,52)(54,55)(57,58);
s3 := Sym(58)!( 5, 6)( 8,12)( 9,11)(10,13)(14,15)(17,21)(18,20)(19,22)(23,24)(26,30)(27,29)(28,31)(32,33)(35,39)(36,38)(37,40)(41,42)(44,48)(45,47)(46,49)(50,51)(53,57)(54,56)(55,58);
s4 := Sym(58)!( 5, 8)( 6, 9)( 7,10)(14,26)(15,27)(16,28)(17,23)(18,24)(19,25)(20,29)(21,30)(22,31)(32,35)(33,36)(34,37)(41,53)(42,54)(43,55)(44,50)(45,51)(46,52)(47,56)(48,57)(49,58);
s5 := Sym(58)!( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)(13,49)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58);
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, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s5*s4*s3*s4*s5*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;