Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,18}

Atlas Canonical Name {6,18}*648i

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(648,554)
Rank
3
Schläfli Type
{6,18}
Vertices, edges, …
18, 162, 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

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

27 facets

12 vertex figures

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

36 facets

6 vertex figures

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

18 facets

6 vertex figures

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

18 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle