Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,18}

Atlas Canonical Name {6,18}*1944r

▶ Play as a twisty puzzle

Overview

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

81 facets

36 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=<s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 3

54 facets

18 vertex figures

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

54 facets

18 vertex figures

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

54 facets

18 vertex figures

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

27 facets

12 vertex figures

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

18 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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