Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,18}

Atlas Canonical Name {6,18}*1944n

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Overview

Group
SmallGroup(1944,2340)
Rank
3
Schläfli Type
{6,18}
Vertices, edges, …
54, 486, 162
Order of s0s1s2
18
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

162-fold

243-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=<(s1*s2)^2*s1*(s0*(s2*s1)^3)^2*s2> of order 2

81 facets

30 vertex figures

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

81 facets

27 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^4*s2> of order 3

54 facets

18 vertex figures

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

72 facets

18 vertex figures

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

54 facets

18 vertex figures

P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, (s1*s2)^2*s1*(s0*(s2*s1)^3)^2*s2> of order 6

27 facets

12 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, (s0*s1*s2*s1)^2> of order 9

30 facets

6 vertex figures

P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2, (s0*(s2*s1)^3)^2> of order 9

18 facets

6 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s2> of order 9

18 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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