Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,9,18}

Atlas Canonical Name {3,2,9,18}*1944

Overview

Group
SmallGroup(1944,2339)
Rank
5
Schläfli Type
{3,2,9,18}
Vertices, edges, …
3, 3, 9, 81, 18
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
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

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,10)( 8,12)( 9,11)(13,23)(14,22)(15,24)(16,29)(17,28)(18,30)(19,26)(20,25)(21,27)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,83)(47,82)(48,84)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(55,74)(56,73)(57,75);;
s3 := ( 4,40)( 5,42)( 6,41)( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)(13,31)(14,33)(15,32)(16,37)(17,39)(18,38)(19,34)(20,36)(21,35)(22,50)(23,49)(24,51)(25,56)(26,55)(27,57)(28,53)(29,52)(30,54)(58,70)(59,72)(60,71)(61,67)(62,69)(63,68)(64,73)(65,75)(66,74)(76,80)(77,79)(78,81)(82,83);;
s4 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78)(55,82)(56,83)(57,84);;
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, s4*s2*s3*s4*s3*s4*s2*s3*s4*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(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 5, 6)( 7,10)( 8,12)( 9,11)(13,23)(14,22)(15,24)(16,29)(17,28)(18,30)(19,26)(20,25)(21,27)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,83)(47,82)(48,84)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69)(55,74)(56,73)(57,75);
s3 := Sym(84)!( 4,40)( 5,42)( 6,41)( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)(13,31)(14,33)(15,32)(16,37)(17,39)(18,38)(19,34)(20,36)(21,35)(22,50)(23,49)(24,51)(25,56)(26,55)(27,57)(28,53)(29,52)(30,54)(58,70)(59,72)(60,71)(61,67)(62,69)(63,68)(64,73)(65,75)(66,74)(76,80)(77,79)(78,81)(82,83);
s4 := Sym(84)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78)(55,82)(56,83)(57,84);
poly := sub<Sym(84)|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, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;