Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,27,6}

Atlas Canonical Name {3,2,27,6}*1944

Overview

Group
SmallGroup(1944,2343)
Rank
5
Schläfli Type
{3,2,27,6}
Vertices, edges, …
3, 3, 27, 81, 6
Order of s0s1s2s3s4
54
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 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,11)( 8,10)( 9,12)(13,22)(14,24)(15,23)(16,29)(17,28)(18,30)(19,26)(20,25)(21,27)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,65)(38,64)(39,66)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,83)(47,82)(48,84)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68)(55,74)(56,73)(57,75);;
s3 := ( 4,40)( 5,42)( 6,41)( 7,47)( 8,46)( 9,48)(10,44)(11,43)(12,45)(13,31)(14,33)(15,32)(16,38)(17,37)(18,39)(19,35)(20,34)(21,36)(22,49)(23,51)(24,50)(25,56)(26,55)(27,57)(28,53)(29,52)(30,54)(58,70)(59,72)(60,71)(61,67)(62,69)(63,68)(64,74)(65,73)(66,75)(76,79)(77,81)(78,80)(82,83);;
s4 := (13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84);;
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*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, 
s0*s1*s0*s1*s0*s1, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 5, 6)( 7,11)( 8,10)( 9,12)(13,22)(14,24)(15,23)(16,29)(17,28)(18,30)(19,26)(20,25)(21,27)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,65)(38,64)(39,66)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,83)(47,82)(48,84)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68)(55,74)(56,73)(57,75);
s3 := Sym(84)!( 4,40)( 5,42)( 6,41)( 7,47)( 8,46)( 9,48)(10,44)(11,43)(12,45)(13,31)(14,33)(15,32)(16,38)(17,37)(18,39)(19,35)(20,34)(21,36)(22,49)(23,51)(24,50)(25,56)(26,55)(27,57)(28,53)(29,52)(30,54)(58,70)(59,72)(60,71)(61,67)(62,69)(63,68)(64,74)(65,73)(66,75)(76,79)(77,81)(78,80)(82,83);
s4 := Sym(84)!(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81)(73,82)(74,83)(75,84);
poly := sub<Sym(84)|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*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, s0*s1*s0*s1*s0*s1, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s2*s3*s4*s3*s2*s3*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;