Polytope of Type {6,6} with group SmallGroup(216,162)

Overview

Group: SmallGroup(216,162)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 18, 54, 18
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
Self-Dual
Self-Petrie

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-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)^3> of order 2

12 facets

6 of {3}*6
6 of {6}*12

9 vertex figures

9 of {6}*12

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

9 facets

9 of {6}*12

12 vertex figures

6 of {6}*12
6 of {3}*6

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

12 facets

9 of {2}*4
3 of {6}*12

6 vertex figures

6 of {6}*12

P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3

6 facets

6 of {6}*12

12 vertex figures

3 of {6}*12
9 of {2}*4

Representations

Permutation Representation (GAP)

s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24);;
s1 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24);
s1 := Sym(27)!( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)(20,24)(21,23)(26,27);
s2 := Sym(27)!( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27);
poly := sub<Sym(27)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1 >;

References

None.

to this polytope.