Polytope of Type {6,6,6} with group SmallGroup(1944,2342)

Overview

Group: SmallGroup(1944,2342)
Rank: 4
Schläfli Type: {6,6,6}
Vertices, edges, …: 18, 81, 81, 9
Order of s₀s₁s₂s₃: 6
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Universal
Non-Orientable
Flat

Quotients maximal quotients in bold

3-fold

{6,6,6}*648a

9-fold

{2,6,6}*216

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)^2> of order 3

9 facets

6 of 3-fold non-regular quotient of {6,6}*216d
3 of {6,6}*72a

6 vertex figures

6 of {6,6}*108

Representations

Permutation Representation (GAP)

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,13)(11,15)(12,14)(17,18)(19,25)(20,27)(21,26)(23,24);;
s2 := ( 1,11)( 2,12)( 3,10)( 4,14)( 5,15)( 6,13)( 7,17)( 8,18)( 9,16);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s1*s2*s3*s1*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
s1 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,13)(11,15)(12,14)(17,18)(19,25)(20,27)(21,26)(23,24);
s2 := Sym(27)!( 1,11)( 2,12)( 3,10)( 4,14)( 5,15)( 6,13)( 7,17)( 8,18)( 9,16);
s3 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23);
poly := sub<Sym(27)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s1*s2*s3*s1*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s0*s1 >;

References

None.

to this polytope.