Part of the Atlas of Small Regular Polytopes

Polytope of Type {9,6}

Atlas Canonical Name {9,6}*1296c

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Overview

Group
SmallGroup(1296,1788)
Rank
3
Schläfli Type
{9,6}
Vertices, edges, …
108, 324, 72
Order of s0s1s2
12
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

4-fold

9-fold

12-fold

27-fold

36-fold

54-fold

108-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=<s1*s2*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 2

36 facets

60 vertex figures

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

36 facets

54 vertex figures

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

24 facets

36 vertex figures

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

24 facets

36 vertex figures

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

24 facets

42 vertex figures

P/N, where N=<(s1*s0*s2)^5*s1*s2> of order 4

18 facets

27 vertex figures

P/N, where N=<s0*(s1*s0*s2)^5*s1> of order 4

18 facets

27 vertex figures

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

12 facets

18 vertex figures

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

12 facets

18 vertex figures

P/N, where N=<(s1*s2)^3, s0*(s1*s2)^2*s1*s0*s2> of order 6

12 facets

24 vertex figures

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

12 facets

24 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2, (s0*(s2*s1)^2)^2*s2> of order 12

6 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,13)( 6,15)( 7,14)( 8,16)( 9,21)(10,23)(11,22)(12,24)(25,33)(26,35)(27,34)(28,36)(30,31);;
s2 := ( 1, 2)( 5,10)( 6, 9)( 7,11)( 8,12)(13,14)(17,22)(18,21)(19,23)(20,24)(25,26)(29,34)(30,33)(31,35)(32,36);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);
s1 := Sym(36)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,13)( 6,15)( 7,14)( 8,16)( 9,21)(10,23)(11,22)(12,24)(25,33)(26,35)(27,34)(28,36)(30,31);
s2 := Sym(36)!( 1, 2)( 5,10)( 6, 9)( 7,11)( 8,12)(13,14)(17,22)(18,21)(19,23)(20,24)(25,26)(29,34)(30,33)(31,35)(32,36);
poly := sub<Sym(36)|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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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