Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1944,2346)
Rank
5
Schläfli Type
{3,18,2,3}
Vertices, edges, …
9, 81, 54, 3, 3
Order of s0s1s2s3s4
6
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)( 4, 6)( 7, 8)(10,19)(11,21)(12,20)(13,24)(14,23)(15,22)(16,26)(17,25)(18,27)(28,57)(29,56)(30,55)(31,59)(32,58)(33,60)(34,61)(35,63)(36,62)(37,75)(38,74)(39,73)(40,77)(41,76)(42,78)(43,79)(44,81)(45,80)(46,66)(47,65)(48,64)(49,68)(50,67)(51,69)(52,70)(53,72)(54,71);;
s1 := ( 1,43)( 2,45)( 3,44)( 4,37)( 5,39)( 6,38)( 7,40)( 8,42)( 9,41)(10,31)(11,33)(12,32)(13,34)(14,36)(15,35)(16,28)(17,30)(18,29)(19,47)(20,46)(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,72)(56,71)(57,70)(58,66)(59,65)(60,64)(61,69)(62,68)(63,67)(74,75)(77,78)(80,81);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)(47,74)(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);;
s3 := (83,84);;
s4 := (82,83);;
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, 
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*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(84)!( 2, 3)( 4, 6)( 7, 8)(10,19)(11,21)(12,20)(13,24)(14,23)(15,22)(16,26)(17,25)(18,27)(28,57)(29,56)(30,55)(31,59)(32,58)(33,60)(34,61)(35,63)(36,62)(37,75)(38,74)(39,73)(40,77)(41,76)(42,78)(43,79)(44,81)(45,80)(46,66)(47,65)(48,64)(49,68)(50,67)(51,69)(52,70)(53,72)(54,71);
s1 := Sym(84)!( 1,43)( 2,45)( 3,44)( 4,37)( 5,39)( 6,38)( 7,40)( 8,42)( 9,41)(10,31)(11,33)(12,32)(13,34)(14,36)(15,35)(16,28)(17,30)(18,29)(19,47)(20,46)(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,72)(56,71)(57,70)(58,66)(59,65)(60,64)(61,69)(62,68)(63,67)(74,75)(77,78)(80,81);
s2 := Sym(84)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,57)(29,56)(30,55)(31,63)(32,62)(33,61)(34,60)(35,59)(36,58)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,75)(47,74)(48,73)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76);
s3 := Sym(84)!(83,84);
s4 := Sym(84)!(82,83);
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, 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*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;