Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,18}

Atlas Canonical Name {2,6,18}*1296i

Overview

Group
SmallGroup(1296,2984)
Rank
4
Schläfli Type
{2,6,18}
Vertices, edges, …
2, 18, 162, 54
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

Covers minimal covers in bold

None in this atlas.

Representations

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