Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,2,3,6}

Atlas Canonical Name {2,2,3,6}*1296

Overview

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