Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,4}

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

Overview

Group
SmallGroup(1296,1813)
Rank
4
Schläfli Type
{2,18,4}
Vertices, edges, …
2, 81, 162, 18
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, 5)( 6, 9)( 7,11)( 8,10)(12,28)(13,27)(14,29)(15,25)(16,24)(17,26)(18,22)(19,21)(20,23)(30,63)(31,65)(32,64)(33,60)(34,62)(35,61)(36,57)(37,59)(38,58)(39,79)(40,78)(41,80)(42,76)(43,75)(44,77)(45,82)(46,81)(47,83)(48,70)(49,69)(50,71)(51,67)(52,66)(53,68)(54,73)(55,72)(56,74);;
s2 := ( 3,30)( 4,34)( 5,38)( 6,36)( 7,31)( 8,35)( 9,33)(10,37)(11,32)(12,42)(13,46)(14,41)(15,39)(16,43)(17,47)(18,45)(19,40)(20,44)(21,54)(22,49)(23,53)(24,51)(25,55)(26,50)(27,48)(28,52)(29,56)(57,63)(59,62)(61,64)(67,70)(68,74)(69,72)(75,78)(76,82)(80,83);;
s3 := ( 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);;
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, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)(12,28)(13,27)(14,29)(15,25)(16,24)(17,26)(18,22)(19,21)(20,23)(30,63)(31,65)(32,64)(33,60)(34,62)(35,61)(36,57)(37,59)(38,58)(39,79)(40,78)(41,80)(42,76)(43,75)(44,77)(45,82)(46,81)(47,83)(48,70)(49,69)(50,71)(51,67)(52,66)(53,68)(54,73)(55,72)(56,74);
s2 := Sym(83)!( 3,30)( 4,34)( 5,38)( 6,36)( 7,31)( 8,35)( 9,33)(10,37)(11,32)(12,42)(13,46)(14,41)(15,39)(16,43)(17,47)(18,45)(19,40)(20,44)(21,54)(22,49)(23,53)(24,51)(25,55)(26,50)(27,48)(28,52)(29,56)(57,63)(59,62)(61,64)(67,70)(68,74)(69,72)(75,78)(76,82)(80,83);
s3 := 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);
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, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;