Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,18}

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

Overview

Group
SmallGroup(648,301)
Rank
4
Schläfli Type
{2,6,18}
Vertices, edges, …
2, 9, 81, 27
Order of s0s1s2s3
6
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 := ( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)(71,81)(72,80)(73,78)(74,79);;
s2 := ( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)(25,26)(28,29)(30,68)(31,67)(32,66)(33,71)(34,70)(35,69)(36,74)(37,73)(38,72)(39,59)(40,58)(41,57)(42,62)(43,61)(44,60)(45,65)(46,64)(47,63)(48,77)(49,76)(50,75)(51,80)(52,79)(53,78)(54,83)(55,82)(56,81);;
s3 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,42)(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,55)(22,54)(23,56)(24,52)(25,51)(26,53)(27,49)(28,48)(29,50)(57,59)(60,65)(61,64)(62,63)(66,71)(67,70)(68,69)(72,74)(75,81)(76,83)(77,82)(79,80);;
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, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)(71,81)(72,80)(73,78)(74,79);
s2 := 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)(30,68)(31,67)(32,66)(33,71)(34,70)(35,69)(36,74)(37,73)(38,72)(39,59)(40,58)(41,57)(42,62)(43,61)(44,60)(45,65)(46,64)(47,63)(48,77)(49,76)(50,75)(51,80)(52,79)(53,78)(54,83)(55,82)(56,81);
s3 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,42)(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,55)(22,54)(23,56)(24,52)(25,51)(26,53)(27,49)(28,48)(29,50)(57,59)(60,65)(61,64)(62,63)(66,71)(67,70)(68,69)(72,74)(75,81)(76,83)(77,82)(79,80);
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, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;