Part of the Atlas of Small Regular Polytopes

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

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

Overview

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

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