Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,9}

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

Overview

Group
SmallGroup(1944,943)
Rank
4
Schläfli Type
{2,18,9}
Vertices, edges, …
2, 54, 243, 27
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)( 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);;
s2 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,41)(13,40)(14,39)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53)(57,59)(60,65)(61,64)(62,63)(66,67)(69,73)(70,72)(71,74)(76,77)(78,81)(79,83)(80,82);;
s3 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(12,18)(13,20)(14,19)(16,17)(21,24)(22,26)(23,25)(28,29)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,83)(37,82)(38,81)(39,59)(40,58)(41,57)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66);;
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*s2*s3*s2*s3*s1*s2*s1*s2*s3*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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(83)!(1,2);
s1 := 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);
s2 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,41)(13,40)(14,39)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,49)(22,48)(23,50)(24,55)(25,54)(26,56)(27,52)(28,51)(29,53)(57,59)(60,65)(61,64)(62,63)(66,67)(69,73)(70,72)(71,74)(76,77)(78,81)(79,83)(80,82);
s3 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(12,18)(13,20)(14,19)(16,17)(21,24)(22,26)(23,25)(28,29)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,83)(37,82)(38,81)(39,59)(40,58)(41,57)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66);
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*s2*s3*s2*s3*s1*s2*s1*s2*s3*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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2 >;