Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*1920b

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Overview

Group
SmallGroup(1920,240996)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
80, 480, 160
Order of s0s1s2
12
Order of s0s1s2s1
5
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

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

80 facets

40 vertex figures

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

80 facets

44 vertex figures

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

84 facets

40 vertex figures

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

56 facets

28 vertex figures

P/N, where N=<(s1*s2)^6, s0*(s1*s2)^2*s1*s0*(s1*s2)^3> of order 4

40 facets

26 vertex figures

P/N, where N=<s0*(s1*s2)^2*s1*s0*(s2*s1)^3, s0*(s2*(s1*s0)^2)^2*(s1*s2)^2> of order 4

40 facets

22 vertex figures

P/N, where N=<s0*(s1*s2)^2*s1*s0*(s2*s1)^3, s0*s1*s2*(s1*s0)^2*s2*s1*s0*(s1*s2)^3> of order 4

40 facets

26 vertex figures

P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 4

44 facets

22 vertex figures

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2, s0*(s1*s2)^2*s1*s0*(s2*s1)^3> of order 4

40 facets

24 vertex figures

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

32 facets

14 vertex figures

P/N, where N=<(s1*s2)^6, (s0*s1)^3*s2*(s1*s0)^2*s1*s2, s0*(s1*s2)^2*s1*s0*(s1*s2)^3> of order 8

20 facets

16 vertex figures

P/N, where N=<(s0*s1)^3, s1*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*(s2*(s1*s0)^2)^2*(s1*s2)^2> of order 8

22 facets

13 vertex figures

P/N, where N=<(s1*s2)^6, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0*s1, s0*(s1*s2)^2*s1*s0*(s1*s2)^3> of order 8

20 facets

14 vertex figures

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

16 facets

10 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 3)( 5, 8)( 6, 7)( 9,10);;
s1 := ( 1, 2)( 4, 6)( 5,10)( 8, 9);;
s2 := ( 2, 5)( 3, 8)( 6,10)( 7, 9);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(10)!( 2, 3)( 5, 8)( 6, 7)( 9,10);
s1 := Sym(10)!( 1, 2)( 4, 6)( 5,10)( 8, 9);
s2 := Sym(10)!( 2, 5)( 3, 8)( 6,10)( 7, 9);
poly := sub<Sym(10)|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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

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