Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,6,3}

Atlas Canonical Name {18,6,3}*1944d

Overview

Group
SmallGroup(1944,2340)
Rank
4
Schläfli Type
{18,6,3}
Vertices, edges, …
54, 162, 27, 3
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^9> of order 2

3 facets

27 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^3*s1*s2> of order 3

3 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^3*s2*s1> of order 3

3 facets

18 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3

3 facets

36 vertex figures

Representations

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

References

None.

to this polytope.