Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1728,46164)
Rank
6
Schläfli Type
{2,2,2,18,6}
Vertices, edges, …
2, 2, 2, 18, 54, 6
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

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