Part of the Atlas of Small Regular Polytopes

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

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

Overview

Group
SmallGroup(1944,2346)
Rank
5
Schläfli Type
{3,2,3,6}
Vertices, edges, …
3, 3, 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 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 9)(10,11)(13,22)(14,24)(15,23)(16,27)(17,26)(18,25)(19,29)(20,28)(21,30)(31,60)(32,59)(33,58)(34,62)(35,61)(36,63)(37,64)(38,66)(39,65)(40,78)(41,77)(42,76)(43,80)(44,79)(45,81)(46,82)(47,84)(48,83)(49,69)(50,68)(51,67)(52,71)(53,70)(54,72)(55,73)(56,75)(57,74);;
s3 := ( 4,46)( 5,48)( 6,47)( 7,40)( 8,42)( 9,41)(10,43)(11,45)(12,44)(13,34)(14,36)(15,35)(16,37)(17,39)(18,38)(19,31)(20,33)(21,32)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(28,56)(29,55)(30,57)(58,75)(59,74)(60,73)(61,69)(62,68)(63,67)(64,72)(65,71)(66,70)(77,78)(80,81)(83,84);;
s4 := ( 7,11)( 8,12)( 9,10)(13,22)(14,23)(15,24)(16,29)(17,30)(18,28)(19,27)(20,25)(21,26)(34,38)(35,39)(36,37)(40,49)(41,50)(42,51)(43,56)(44,57)(45,55)(46,54)(47,52)(48,53)(61,65)(62,66)(63,64)(67,76)(68,77)(69,78)(70,83)(71,84)(72,82)(73,81)(74,79)(75,80);;
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, 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(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 5, 6)( 7, 9)(10,11)(13,22)(14,24)(15,23)(16,27)(17,26)(18,25)(19,29)(20,28)(21,30)(31,60)(32,59)(33,58)(34,62)(35,61)(36,63)(37,64)(38,66)(39,65)(40,78)(41,77)(42,76)(43,80)(44,79)(45,81)(46,82)(47,84)(48,83)(49,69)(50,68)(51,67)(52,71)(53,70)(54,72)(55,73)(56,75)(57,74);
s3 := Sym(84)!( 4,46)( 5,48)( 6,47)( 7,40)( 8,42)( 9,41)(10,43)(11,45)(12,44)(13,34)(14,36)(15,35)(16,37)(17,39)(18,38)(19,31)(20,33)(21,32)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(28,56)(29,55)(30,57)(58,75)(59,74)(60,73)(61,69)(62,68)(63,67)(64,72)(65,71)(66,70)(77,78)(80,81)(83,84);
s4 := Sym(84)!( 7,11)( 8,12)( 9,10)(13,22)(14,23)(15,24)(16,29)(17,30)(18,28)(19,27)(20,25)(21,26)(34,38)(35,39)(36,37)(40,49)(41,50)(42,51)(43,56)(44,57)(45,55)(46,54)(47,52)(48,53)(61,65)(62,66)(63,64)(67,76)(68,77)(69,78)(70,83)(71,84)(72,82)(73,81)(74,79)(75,80);
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, 
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 >;