Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1296,2984)
Rank
5
Schläfli Type
{2,6,6,9}
Vertices, edges, …
2, 6, 18, 27, 9
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

6-fold

9-fold

18-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Representations

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