Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,6}

Atlas Canonical Name {2,18,6}*648b

Overview

Group
SmallGroup(648,300)
Rank
4
Schläfli Type
{2,18,6}
Vertices, edges, …
2, 27, 81, 9
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
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

2-fold

3-fold

Representations

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