Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,6,18}

Atlas Canonical Name {3,2,6,18}*1944b

Overview

Group
SmallGroup(1944,2345)
Rank
5
Schläfli Type
{3,2,6,18}
Vertices, edges, …
3, 3, 9, 81, 27
Order of s0s1s2s3s4
9
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

Covers minimal covers in bold

None in this atlas.

Representations

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