Polytope of Type {18,3} with group SmallGroup(972,115)

Overview

Group: SmallGroup(972,115)
Rank: 3
Schläfli Type: {18,3}
Vertices, edges, …: 162, 243, 27
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 18
Also known as: {18,3}₆. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{18,3}*324

9-fold

{6,3}*108

27-fold

{6,3}*36

81-fold

{2,3}*12

Covers minimal covers in bold

2-fold

{18,6}*1944k

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)^6> of order 3

15 facets

9 of {6}*12
6 of {18}*36

54 vertex figures

54 of {3}*6

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

11 facets

9 of {2}*4
2 of {18}*36

18 vertex figures

18 of {3}*6

P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2> of order 9

5 facets

3 of {6}*12
2 of {18}*36

18 vertex figures

18 of {3}*6

Representations

Permutation Representation (GAP)

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