Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,9,18}

Atlas Canonical Name {2,9,18}*1944b

Overview

Group
SmallGroup(1944,943)
Rank
4
Schläfli Type
{2,9,18}
Vertices, edges, …
2, 27, 243, 54
Order of s0s1s2s3
6
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

9-fold

27-fold

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