Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,4,18}

Atlas Canonical Name {2,4,18}*1296

Overview

Group
SmallGroup(1296,1813)
Rank
4
Schläfli Type
{2,4,18}
Vertices, edges, …
2, 18, 162, 81
Order of s0s1s2s3
4
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

9-fold

Covers minimal covers in bold

None in this atlas.

Representations

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