Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,3,18}

Atlas Canonical Name {2,3,18}*648

Overview

Group
SmallGroup(648,301)
Rank
4
Schläfli Type
{2,3,18}
Vertices, edges, …
2, 9, 81, 54
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

Covers minimal covers in bold

2-fold

3-fold

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := ( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)(18,28)(19,27)(20,29)(30,59)(31,58)(32,57)(33,61)(34,60)(35,62)(36,63)(37,65)(38,64)(39,77)(40,76)(41,75)(42,79)(43,78)(44,80)(45,81)(46,83)(47,82)(48,68)(49,67)(50,66)(51,70)(52,69)(53,71)(54,72)(55,74)(56,73);;
s2 := ( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)(12,33)(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,74)(58,73)(59,72)(60,68)(61,67)(62,66)(63,71)(64,70)(65,69)(76,77)(79,80)(82,83);;
s3 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)(18,28)(19,27)(20,29)(30,59)(31,58)(32,57)(33,61)(34,60)(35,62)(36,63)(37,65)(38,64)(39,77)(40,76)(41,75)(42,79)(43,78)(44,80)(45,81)(46,83)(47,82)(48,68)(49,67)(50,66)(51,70)(52,69)(53,71)(54,72)(55,74)(56,73);
s2 := Sym(83)!( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)(12,33)(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,74)(58,73)(59,72)(60,68)(61,67)(62,66)(63,71)(64,70)(65,69)(76,77)(79,80)(82,83);
s3 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);
poly := sub<Sym(83)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 >;