Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1296,1861)
Rank
5
Schläfli Type
{2,9,6,2}
Vertices, edges, …
2, 27, 81, 18, 2
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 := ( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)(18,28)(19,27)(20,29)(30,58)(31,57)(32,59)(33,60)(34,62)(35,61)(36,65)(37,64)(38,63)(39,76)(40,75)(41,77)(42,78)(43,80)(44,79)(45,83)(46,82)(47,81)(48,67)(49,66)(50,68)(51,69)(52,71)(53,70)(54,74)(55,73)(56,72);;
s2 := ( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)(12,33)(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,73)(58,72)(59,74)(60,67)(61,66)(62,68)(63,70)(64,69)(65,71)(75,77)(78,80)(81,83);;
s3 := ( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)(71,81)(72,80)(73,78)(74,79);;
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*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(85)!(1,2);
s1 := Sym(85)!( 4, 5)( 6, 8)( 9,10)(12,21)(13,23)(14,22)(15,26)(16,25)(17,24)(18,28)(19,27)(20,29)(30,58)(31,57)(32,59)(33,60)(34,62)(35,61)(36,65)(37,64)(38,63)(39,76)(40,75)(41,77)(42,78)(43,80)(44,79)(45,83)(46,82)(47,81)(48,67)(49,66)(50,68)(51,69)(52,71)(53,70)(54,74)(55,73)(56,72);
s2 := Sym(85)!( 3,45)( 4,47)( 5,46)( 6,39)( 7,41)( 8,40)( 9,42)(10,44)(11,43)(12,33)(13,35)(14,34)(15,36)(16,38)(17,37)(18,30)(19,32)(20,31)(21,49)(22,48)(23,50)(24,52)(25,51)(26,53)(27,55)(28,54)(29,56)(57,73)(58,72)(59,74)(60,67)(61,66)(62,68)(63,70)(64,69)(65,71)(75,77)(78,80)(81,83);
s3 := Sym(85)!( 6,10)( 7,11)( 8, 9)(12,21)(13,22)(14,23)(15,28)(16,29)(17,27)(18,26)(19,24)(20,25)(33,37)(34,38)(35,36)(39,48)(40,49)(41,50)(42,55)(43,56)(44,54)(45,53)(46,51)(47,52)(60,64)(61,65)(62,63)(66,75)(67,76)(68,77)(69,82)(70,83)(71,81)(72,80)(73,78)(74,79);
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*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;