Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,6}

Atlas Canonical Name {18,6}*1944m

▶ 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

36 facets

81 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

72 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*s2*s1)^2*s0*(s1*s2)^2*s1> of order 6

12 facets

27 vertex figures

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

6 facets

30 vertex figures

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

6 facets

18 vertex figures

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

6 facets

18 vertex figures

Representations

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

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