Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,18}

Atlas Canonical Name {6,18}*1944u

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Overview

Group
SmallGroup(1944,2346)
Rank
3
Schläfli Type
{6,18}
Vertices, edges, …
54, 486, 162
Order of s0s1s2
6
Order of s0s1s2s1
18
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=<s0*s1*s0*s2*(s1*s0)^2*(s1*s2)^2> of order 2

81 facets

30 vertex figures

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

108 facets

27 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^4*s0*s2*s1> of order 3

54 facets

18 vertex figures

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

27 facets

12 vertex figures

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

27 facets

12 vertex figures

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

36 facets

9 vertex figures

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

18 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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