Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1296,1862)
Rank
5
Schläfli Type
{2,18,6,2}
Vertices, edges, …
2, 27, 81, 9, 2
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 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);;
s2 := ( 3,30)( 4,32)( 5,31)( 6,35)( 7,34)( 8,33)( 9,37)(10,36)(11,38)(12,55)(13,54)(14,56)(15,48)(16,50)(17,49)(18,53)(19,52)(20,51)(21,42)(22,44)(23,43)(24,47)(25,46)(26,45)(27,40)(28,39)(29,41)(57,59)(60,61)(64,65)(66,81)(67,83)(68,82)(69,77)(70,76)(71,75)(72,79)(73,78)(74,80);;
s3 := ( 3,12)( 4,13)( 5,14)( 6,18)( 7,19)( 8,20)( 9,15)(10,16)(11,17)(24,27)(25,28)(26,29)(30,39)(31,40)(32,41)(33,45)(34,46)(35,47)(36,42)(37,43)(38,44)(51,54)(52,55)(53,56)(57,66)(58,67)(59,68)(60,72)(61,73)(62,74)(63,69)(64,70)(65,71)(78,81)(79,82)(80,83);;
s4 := (84,85);;
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, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(85)!(1,2);
s1 := Sym(85)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);
s2 := Sym(85)!( 3,30)( 4,32)( 5,31)( 6,35)( 7,34)( 8,33)( 9,37)(10,36)(11,38)(12,55)(13,54)(14,56)(15,48)(16,50)(17,49)(18,53)(19,52)(20,51)(21,42)(22,44)(23,43)(24,47)(25,46)(26,45)(27,40)(28,39)(29,41)(57,59)(60,61)(64,65)(66,81)(67,83)(68,82)(69,77)(70,76)(71,75)(72,79)(73,78)(74,80);
s3 := Sym(85)!( 3,12)( 4,13)( 5,14)( 6,18)( 7,19)( 8,20)( 9,15)(10,16)(11,17)(24,27)(25,28)(26,29)(30,39)(31,40)(32,41)(33,45)(34,46)(35,47)(36,42)(37,43)(38,44)(51,54)(52,55)(53,56)(57,66)(58,67)(59,68)(60,72)(61,73)(62,74)(63,69)(64,70)(65,71)(78,81)(79,82)(80,83);
s4 := Sym(85)!(84,85);
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 >;