Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,2,6,9}

Atlas Canonical Name {2,2,6,9}*1296d

Overview

Group
SmallGroup(1296,1861)
Rank
5
Schläfli Type
{2,2,6,9}
Vertices, edges, …
2, 2, 18, 81, 27
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
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

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)(20,28)(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)(46,56)(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)(72,85)(73,83)(74,82)(75,80)(76,81);;
s3 := ( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)(24,25)(27,28)(30,31)(32,69)(33,68)(34,70)(35,72)(36,71)(37,73)(38,75)(39,74)(40,76)(41,60)(42,59)(43,61)(44,63)(45,62)(46,64)(47,66)(48,65)(49,67)(50,78)(51,77)(52,79)(53,81)(54,80)(55,82)(56,84)(57,83)(58,85);;
s4 := ( 5,32)( 6,34)( 7,33)( 8,37)( 9,36)(10,35)(11,39)(12,38)(13,40)(14,57)(15,56)(16,58)(17,50)(18,52)(19,51)(20,55)(21,54)(22,53)(23,44)(24,46)(25,45)(26,49)(27,48)(28,47)(29,42)(30,41)(31,43)(59,60)(63,64)(65,67)(68,85)(69,84)(70,83)(71,78)(72,77)(73,79)(74,80)(75,82)(76,81);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3*s2*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)(20,28)(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)(46,56)(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)(72,85)(73,83)(74,82)(75,80)(76,81);
s3 := Sym(85)!( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)(24,25)(27,28)(30,31)(32,69)(33,68)(34,70)(35,72)(36,71)(37,73)(38,75)(39,74)(40,76)(41,60)(42,59)(43,61)(44,63)(45,62)(46,64)(47,66)(48,65)(49,67)(50,78)(51,77)(52,79)(53,81)(54,80)(55,82)(56,84)(57,83)(58,85);
s4 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,37)( 9,36)(10,35)(11,39)(12,38)(13,40)(14,57)(15,56)(16,58)(17,50)(18,52)(19,51)(20,55)(21,54)(22,53)(23,44)(24,46)(25,45)(26,49)(27,48)(28,47)(29,42)(30,41)(31,43)(59,60)(63,64)(65,67)(68,85)(69,84)(70,83)(71,78)(72,77)(73,79)(74,80)(75,82)(76,81);
poly := sub<Sym(85)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3*s2*s4*s3*s4 >;