Part of the Atlas of Small Regular Polytopes

Polytope of Type {9,6}

Atlas Canonical Name {9,6}*972a

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Overview

Group
SmallGroup(972,101)
Rank
3
Schläfli Type
{9,6}
Vertices, edges, …
81, 243, 54
Order of s0s1s2
6
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*(s1*s2)^2*(s1*s0*s2)^2*s1> of order 3

18 facets

27 vertex figures

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

18 facets

33 vertex figures

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

36 facets

27 vertex figures

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

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

References

None.

to this polytope.

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