Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,6}

Atlas Canonical Name {18,6}*1944s

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,2345)
Rank
3
Schläfli Type
{18,6}
Vertices, edges, …
162, 486, 54
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=<(s0*s1)^4*s2*s1*s0*(s2*s1)^2> of order 2

27 facets

81 vertex figures

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

30 facets

81 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2, (s0*s1)^4*s2*s1*s0*(s2*s1)^2*s2> of order 6

12 facets

27 vertex figures

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

12 facets

27 vertex figures

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

6 facets

18 vertex figures

Representations

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

References

None.

to this polytope.

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