Polytope of Type {6,3} with group SmallGroup(972,115)

Overview

Group: SmallGroup(972,115)
Rank: 3
Schläfli Type: {6,3}
Vertices, edges, …: 162, 243, 81
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 6
Also known as: {6,3}_(9,0), {6,3}₁₈. if this polytope has another name.

Special Properties

Toroidal
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

{6,3}*324

9-fold

{6,3}*108

27-fold

{6,3}*36

81-fold

{2,3}*12

Covers minimal covers in bold

2-fold

{6,6}*1944c

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

27 facets

27 of {6}*12

54 vertex figures

54 of {3}*6

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

29 facets

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

54 vertex figures

54 of {3}*6

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

9 facets

9 of {6}*12

18 vertex figures

18 of {3}*6

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.