Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,2,18,6}

Atlas Canonical Name {2,2,18,6}*1296c

Overview

Group
SmallGroup(1296,1862)
Rank
5
Schläfli Type
{2,2,18,6}
Vertices, edges, …
2, 2, 27, 81, 9
Order of s0s1s2s3s4
6
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

Covers minimal covers in bold

None in this atlas.

Representations

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