Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,18,6}

Atlas Canonical Name {3,2,18,6}*1944c

Overview

Group
SmallGroup(1944,2346)
Rank
5
Schläfli Type
{3,2,18,6}
Vertices, edges, …
3, 3, 27, 81, 9
Order of s0s1s2s3s4
3
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

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