Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*324a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(324,37)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
18, 81, 27
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

9-fold

27-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

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> of order 3

15 facets

6 vertex figures

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

9 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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