Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,6}

Atlas Canonical Name {18,6}*1944n

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,2340)
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)^9> of order 2

30 facets

81 vertex figures

P/N, where N=<(s0*s1)^5*(s0*s2*s1)^3*s0*s1> of order 2

27 facets

81 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

72 vertex figures

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

18 facets

54 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, (s0*s1)^9> of order 6

12 facets

27 vertex figures

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

6 facets

30 vertex figures

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

6 facets

18 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*s2*(s1*s0)^4*s2*s1> of order 9

6 facets

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

Twisty Puzzle