Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,6}

Atlas Canonical Name {18,6}*1944r

▶ 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)^9> of order 2

36 facets

81 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

18 facets

54 vertex figures

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

12 facets

27 vertex figures

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

6 facets

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

References

None.

to this polytope.

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