Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,2,18,6}

Atlas Canonical Name {4,2,18,6}*1728a

Overview

Group
SmallGroup(1728,30790)
Rank
5
Schläfli Type
{4,2,18,6}
Vertices, edges, …
4, 4, 18, 54, 6
Order of s0s1s2s3s4
36
Order of s0s1s2s3s4s3s2s1
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

36-fold

54-fold

Covers minimal covers in bold

None in this atlas.

Representations

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