Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,3,2}

Atlas Canonical Name {18,3,2}*648

Overview

Group
SmallGroup(648,301)
Rank
4
Schläfli Type
{18,3,2}
Vertices, edges, …
54, 81, 9, 2
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
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

2-fold

3-fold

Representations

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