Part of the Atlas of Small Regular Polytopes

Polytope of Type {9,6}

Atlas Canonical Name {9,6}*972c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(972,103)
Rank
3
Schläfli Type
{9,6}
Vertices, edges, …
81, 243, 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

9-fold

27-fold

81-fold

Covers minimal covers in bold

2-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*s2*s1)^2*s0*(s1*s2)^2*s1> of order 3

18 facets

27 vertex figures

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

18 facets

27 vertex figures

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

18 facets

33 vertex figures

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

6 facets

9 vertex figures

Representations

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

References

None.

to this polytope.

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