Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,6}

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

Overview

Group
SmallGroup(648,297)
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)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)(38,64)(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)(49,80)(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);;
s2 := ( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)(12,50)(13,48)(14,49)(15,56)(16,54)(17,55)(18,53)(19,51)(20,52)(21,40)(22,41)(23,39)(24,46)(25,47)(26,45)(27,43)(28,44)(29,42)(57,60)(58,61)(59,62)(66,80)(67,78)(68,79)(69,77)(70,75)(71,76)(72,83)(73,81)(74,82);;
s3 := ( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)(25,26)(28,29)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)(38,46)(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)(64,74)(65,73)(76,77)(79,80)(82,83);;
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*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*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)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)(38,64)(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)(49,80)(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);
s2 := Sym(83)!( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)(12,50)(13,48)(14,49)(15,56)(16,54)(17,55)(18,53)(19,51)(20,52)(21,40)(22,41)(23,39)(24,46)(25,47)(26,45)(27,43)(28,44)(29,42)(57,60)(58,61)(59,62)(66,80)(67,78)(68,79)(69,77)(70,75)(71,76)(72,83)(73,81)(74,82);
s3 := Sym(83)!( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)(25,26)(28,29)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)(38,46)(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)(64,74)(65,73)(76,77)(79,80)(82,83);
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*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;